Discontinuity, Nonlinearity, and Complexity

Volume 5 Issue 3 September 2016

ISSN 2164‐6376 (print) ISSN 2164‐6414 (online)

An Interdisciplinary Journal of

Discontinuity, Nonlinearity, and Complexity

Discontinuity, Nonlinearity, and Complexity Editors Valentin Afraimovich San Luis Potosi University, IICO-UASLP, Av.Karakorum 1470 Lomas 4a Seccion, San Luis Potosi, SLP 78210, Mexico Fax: +52 444 825 0198 Email: [email protected]

Lev Ostrovsky Zel Technologies/NOAA ESRL, Boulder CO 80305, USA Fax: +1 303 497 5862 Email: [email protected]

Xavier Leoncini Centre de Physique Théorique, Aix-Marseille Université, CPT Campus de Luminy, Case 907 13288 Marseille Cedex 9, France Fax: +33 4 91 26 95 53 Email: [email protected]

Dimitry Volchenkov The Center of Excellence Cognitive Interaction Technology Universität Bielefeld, Mathematische Physik Universitätsstraße 25 D-33615 Bielefeld, Germany Fax: +49 521 106 6455 Email: [email protected]

Associate Editors Marat Akhmet Department of Mathematics Middle East Technical University 06531 Ankara, Turkey Fax: +90 312 210 2972 Email: [email protected]

Ranis N. Ibragimov Applied Statistics Lab GE Global Research 1 Research Circle, K1-4A64 Niskayuna, NY 12309 Email: [email protected]

Jose Antonio Tenreiro Machado ISEP-Institute of Engineering of Porto Dept. of Electrotechnical Engineering Rua Dr. Antonio Bernardino de Almeida 4200-072 Porto, Portugal Fax: +351 22 8321159 Email: [email protected]

Dumitru Baleanu Department of Mathematics and Computer Sciences Cankaya University, Balgat 06530 Ankara, Turkey Email: [email protected]

Nikolai A. Kudryashov Department of Applied Mathematics Moscow Engineering and Physics Institute (State University), 31 Kashirskoe Shosse 115409 Moscow, Russia Fax: +7 495 324 11 81 Email: [email protected]

Denis Makarov Pacific Oceanological Institute of the Russian Academy of Sciences,43 Baltiiskaya St., 690041 Vladivostok, RUSSIA Tel/fax: 007-4232-312602. Email: [email protected]

Marian Gidea Department of Mathematics Northeastern Illinois University Chicago, IL 60625, USA Fax: +1 773 442 5770 Email: [email protected]

Gennady A. Leonov Department of Mathematics and Mechanics St-Petersburg State University 198504, Russia Email: [email protected]

Josep J. Masdemont Department of Matematica Aplicada I Universitat Politecnica de Catalunya (UPC) Diagonal 647 ETSEIB 08028 Barcelona, Spain Email: [email protected]

Juan Luis García Guirao Department of Applied Mathematics Technical University of Cartagena Hospital de Marina 30203-Cartagena, SPAIN Fax:+34 968 325694 E-mail: [email protected]

E.N. Macau Lab. Associado de Computção e Mateática Aplicada, Instituto Nacional de Pesquisas Espaciais, Av. dos Astronautas, 1758 12227-010 Sáo José dos Campos –SP Brazil Email: [email protected]

Michael A. Zaks Technische Universität Berlin DFG-Forschungszentrum Matheon Mathematics for key technologies Sekretariat MA 3-1, Straße des 17. Juni 136 10623 Berlin Email: [email protected]

Mokhtar Adda-Bedia Laboratoire de Physique Statistique Ecole Normale Supérieure 24 rue Lhomond, 75231 Paris Cedex 05 France Fax: +33 1 44 32 34 33 Email: [email protected]

Ravi P. Agarwal Department of Mathematics Texas A&M University – Kingsville, Kingsville, TX 78363-8202 USA Email: [email protected]

Editorial Board Vadim S. Anishchenko Department of Physics Saratov State University Astrakhanskaya 83, 410026, Saratov, Russia Fax: (845-2)-51-4549 Email: [email protected]

Continued on back materials

An Interdisciplinary Journal of Discontinuity, Nonlinearity, and Complexity Volume 5, Issue 3, September 2016

Editors Valentin Afraimovich Xavier Leoncini Lev Ostrovsky Dimitry Volchenkov

L&H Scientific Publishing, LLC, USA

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Discontinuity, Nonlinearity, and Complexity 5(3) (2016) 199–207

Discontinuity, Nonlinearity, and Complexity https://lhscientificpublishing.com/Journals/DNC-Default.aspx

Modeling of Complex Rheological Fluids with Fractal Structures Rakiz M. Sattarov1†, Ilham R. Sattarzada1 , Sayavur I. Bakhtiyarov2 , and Ranis N. Ibragimov3 1 Azerbaijan

Institute of Oil Industry, Baku, Azerbaijan Houston, TX, USA 3 Department of Mathematics and Physics, University of Wisconsin-Parkside, Kenosha, WI, USA 2 FLOUR,

Submission Info Communicated by A.C.J. Luo Received 21 September 2015 Accepted 23 October 2015 Available online 1 October 2016 Keywords Rheological fluids Fractal Structures Porous media

Abstract An unsteady flow of rheological complex fluid with fractal structure in various pipe geometries was studied when the pressure is a harmonic function of time at a given frequency at the initial cross section of the pipe. An integral type rheological equation by fractional derivative is applied for the first time to model the thixotropic rheological type oils with a high content of wax, resins and asphaltene. The obtained results show that, the relaxation time and the fractal parameters can significantly influence the process of damping pressure along the pipe. It is shown that depending on these parameters the attenuation process may increase and decrease compared to the processes in a viscous damping fluid pressure. The results of simulations are compared to those for resin-asphalt-paraffin oils of two types. ©2016 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction One of the very effective methods of constructing models of flow of real fluids, taking into account the internal microstructures can be considered fractal scaling or ideas which are widely used in complex rheological fluids flow in porous media and tubes [1–4]. The results of the studies of the thixotropic rheological type systems and drilling muds, as well as studies of oils with a high content of wax, resin and asphaltene are provided in numerous of papers [5–7], where the parameters characterizing their thixotropic and relaxation properties are estimated. To describe the above-shown disperse systems the complicated rheological models are used. These models contain a large number of parameters which limits their practical applications. Therefore, the most promising method of the effective description of such complex fluids might be fractal structure analysis technique. In this study we applied an integral type rheological equation with fractional derivative to model the thixotropic rheological fluid flow in pipes. In connection with the foregoing, the following are some of the results on the modeling of complex rheological fluids with fractal structure. A rheological study of stress relaxation in continuous media can be conducted using two methods. Both methods are limited to the measurement of tangential or normal stress over time until an equilibrium is reached: in the first case, for a given constant strain rate and in the second case, for a given permanent deformation. † Corresponding

author. Email address: [email protected],[email protected]

ISSN 2164 − 6376, eISSN 2164 − 6414/$-see front materials © 2016 L&H Scientific Publishing, LLC. All rights reserved. DOI : 10.5890/DNC.2016.09.001

200

Rakiz M. Sattarov, et al. / Discontinuity, Nonlinearity, and Complexity 5(3) (2016) 199–207

Experimental studies with real process fluids used in oil and gas industry suggest that the stress kinetics of these fluids is quite complex. At almost instantaneously applied velocity gradient (shear rate), the flow curve, usually has a falling branch. Figure 1 by the solid lines shows the characteristic flow curves of the kinetics of shear stresses for tar-asphaltparaffin oil (oilfields Varik and Boston, the Republic of Uzbekistan) at immediately applied a constant velocity gradient. Experimental data at different constant strain rates for the tested oils were obtained at the rotational viscometer Reotest-2 [7].

30 25

Ĳ, ɉa

20 15 10 5 0 0

20

40

60

80

100

120

140

160

180

200

220

240

-1

tǜ60 , c Oil field of Varik (experiment);

Oil field of Boston (experiment)

Oil field of Varik (calculated);

Oil field of Boston (calculated)

Fig. 1 Characteristic rheological flow curves of development kinetics of tangential stresses for resin-asphalt-paraffin oils.

2 Mathematical modeling, results and discussion As shown in [7], the flow curves can be described by different spectrum of relaxation times, and with increasing the spectrum an accuracy of the model will be improved. Attempts to consider all of the discrete spectrum of relaxation times lead to more complicated rheological model, and as a result the order of the differential equation increases. In such a situation, it may be more appropriate, the use of integral type models that can take into account the fractal structure of complex rheological fluids. Analysis of the graphs shown in Fig. 1 shows that their behavior is similar to a hyperbolic curve asymptotes in t ≤ 0, which may be described by the following equation of hyperbola B τ = A[1 + ], (1) (t −C)ν where τ -shear stress, t-time, A-parameter characterizing the shear stress at t → ∞ (a steady motion), B-parameter characterizing the relaxation time, C-parameter characterizing the delay time, ν -the fractal parameter.


201

If we assume that as t → ∞ holds steady viscous flow, then (1) can be obtained A = μ γ˙0 (μ -viscosity of the medium, γ˙0 -constant velocity gradient). Equation (1) can be obtained from integral type rheological equation: ˆ −ν d t λν τ = μ [γ˙ + γ˙ t dt ], (2) t − t Γ (1 − ν ) dt C where (t − t )−ν - the core of heredity represented in proportion to a power law with a negative fractional exponent −υ ; λ - the relaxation time; Γ(1 − ν ) - Gamma-function. The proposed equation (2) is analogical to the equations describing the flow of viscoelastic fluids with memory that at long times (t → ∞), as rule, approaches the steady motion of viscous flows [8]. Assuming that γ˙ = γ˙0 = constant (experiments shown in Fig. 1 were carried out to keep the velocity gradients constant) solution (2) has the form:

τ = μ γ˙0 [1 +

1 λν ]. Γ (1 − ν ) (t −C)ν

(3)

Comparison of equations (1) and (3) shows their complete analogy with the following equations: A = μ γ˙0 ,

B=

λν . Γ (1 − ν )

For large time periods, from the equation τ∞ = μ γ˙0 at a given strain rate and the measured shear stress it is not difficult to calculate μ = τγ˙∞0 . For the curves shown in Fig. 1 the viscosity parameters have the following values: Varik μ = 0.11 Pa·s, Boston μ = 0.419 Pa·s, and shear rates were set respectively at 90 s−1 and 12 s−1 . Determination of ν , λ and C was carried out as follows. The parameter C initially is assumed to be zero and experiments are processed in the coordinates of the transformed equation (3): ln(

τ λν − ν lnt. − 1) = ln μ γ˙0 Γ (1 − ν )

(4)

From the slope of the line the value of ν is determined in the coordinates of ln( μτγ˙0 − 1) − lnt, and crossing ν

λ the ordinate axis y at a value of ln(t) = 0 is determined a value ln Γ(1− ν ) by the value of that λ . Then, assuming that the stress curve crosses the ordinate axis y, the value of parameter C is determined from the equation (3) at t = 0 and τ = τ 0 as: 1 1 λν ]ν . (5) C = −[ τ0 Γ (1 − ν ) μ γ˙0 − 1

The crossing of stresses curve the axis of ordinates τ for considered rheological of complex fluids can be explained by the fact that at moment instantly applied shear rate , due to the viscoelastic properties, may be to exist a residual the shear stress. The method of processing the data presented in Fig. 1 according to the equation (4) is shown in Fig. 2. It is not difficult to notice that the both curves are straightened in coordinates ln( μτγ˙0 − 1) − lnt. Determining the slopes and the intercepts with the coordinate axes, the following values of the fractality and relaxation times were obtained: for Varik oilfield ν = 0.605, λ = 27.96 min; for Boston oilfield ν = 0.51, λ = 40.36 min. Since the curves of shear stresses at the initial time (t = 0) have finite values τ0 , then the values of the parameter C were calculated from by the equation (5) for Varik oilfield is 0.892 min, and for Boston oilfield is 0.155 min. The shear stress curves calculated by equation (3) are shown in Fig. 1 and it shows a quite good agreement with the experimental data [7], because their maximum error not exceeding 5%. As shown above, some oils containing tar and asphalt-waxy inclusions, under certain conditions can be satisfactorily described by the integral type rheological model (2), where the function of heredity characterizes

202


2,0 1,5

ln(t / t - 1)

1,0 0,5 0,0 -0,5 -1,0 -1,5 -2,0 0,0

2,0

4,0

6,0

8,0

ln t Oil field of Varik;

Oil field of Boston

Fig. 2 Results of calculations of characteristic rheological curves of kinetics of development of tangential stresses for resin-asphalt-paraffin oils.

at what time the system is subjected to the deformation. It is clear that in the general case, in addition to the function of stress relaxation, it may also a relaxation function (retardation) of velocity gradient can take place. We can assume a analogical behavior of shear rate curve when instantly applied shear stress for the same or the similar rheological of complex fluids. Then, in analogy to equation (2) we can write the following generalizing rheological equation d θα Γ (1 − α ) dt

ˆ e

t

t −t

−α

τ t dt + τ = μ [γ˙ +

d λν Γ (1 − ν ) dt

ˆ

t

t − t

c

−ν γ˙ t dt ],

(6)

where θ -delay time, α - the fractal parameter, e-parameter characterizing the prehistory of shear rate. In particular, if θ = 0, the integral type equation (6) corresponds to the equation (2). If we assume λ = 0, then the equation (6) can be written in the following form: d θα Γ (1 − α ) dt

ˆ

t e

t − t

−α τ t dt + τ = μ γ˙.

(7)

Following [9], we introduced the operation of differentiation of fractional index, which is more commonly known as fractional differentiation. A fractional derivative of order ν of piecewise continuous function f (t) is defined by following expression: Dνc

d d ν f (t) 1 = c ν = dt Γ (1 − ν ) dt

ˆ

t c

−ν f t t − t dt ,

−∞ < ν < 1,

(8)

where c is an arbitrary real number, which can be any value in the interval −∞ < c < +∞. In the special case ν dν when c = 0 in the index in the equation (8) can be omitted and simplified notations Dν0 ≡ Dν and dt0ν ≡ dtd ν will be introduced.


203

Taking into account the introduction of the operator of fractional differentiation (8), a generalized rheological equation (6) can be written as: θ α Dαe τ + τ = μ (γ˙ + λ ν Dνc γ˙) (9) or

ν deα τ ν dc γ˙ ˙ + τ = μ ( γ + λ ). dt α dt ν Consider an unsteady flow of complex rheological dispersive liquid with a fractal structure in a circular cylindrical tube with the radius R along the z-axis under the pressure drop − ∂dzP = f (t) (pressure difference may vary and take a constant value ΔP/L). In this case, the differential equations of motion combined with the rheological relationship (9) can be written as:

θα

ρ

∂ υz ∂ P ∂ τrz τrz =− + + , dt ∂z ∂r r

(10)

ν deα τ ∂ υz ν ∂c ∂ υz + ), (11) + τ = μ ( λ dt α ∂r ∂ tν ∂ r where υz (t, r)-the speed of motion; τrz (t, r)-shear stress; ρ -fluid density. Excluding vz from equations (10) and (11) we obtain:

θα

ρθ α

α +1 ∂ P ν ∂ υz ∂eα +1 υz ∂P 1 ∂ ∂ υz α ∂e ν ∂c ∂ υz = − − + [r( + )]. + ρ θ μ λ ∂ t α +1 ∂t ∂ tα ∂ z ∂ z r ∂r ∂r ∂ tν ∂ r

(12)

Assuming that at the initial time the fluid was at rest and e = 0, c = 0, then (12) can be written in the following form: ∂ υz ∂ α +1 υz ∂ α +1 ∂ P ∂ P 1 ∂ ∂ υz ∂ ν +1 υz = −θ α α − +μ [r( +λν ν )]. (13) ρθ α α +1 + ρ ∂t ∂t ∂t ∂z ∂z r ∂r ∂r ∂t ∂r Taking into consideration the flow symmetry conditions and adhesion fluid on the tube wall, the initial and boundary conditions can be written as:

υz (0, r) = 0,

υz (t, R) = 0,

∂ υz (t, 0) = 0. ∂r

(14)

The solution of the differential equation (13) under the conditions (14) in the Laplace transform plane has the form: √ I1 β R ρ s 2 π F (s) R2 √ ] , (15) [1 − Q (s) = √ ρs β R ρ s I0 β R ρ s where

ˆ Q (s) =

∞ 0

−st

q (t) e

ˆ dt =2π

∞

0

−st

R

e

0

ˆ

u (s, r) =

∞ 0

ˆ F (s) =

β =(

ˆ

υz (t, r) rdrdt =2π

0

R

u (s, r) rdr,

υz (t, r) e−st dt,

∞ 0

ˆ

f (t) e−st dt,

1 1 + θ α sα 1 )2 μ 1 + λ ν sν

.

Direct inverse transformation of the (15) involves considerable difficulties in finding the original √ relationship √ Bessel functions I0 β R ρ s and I1 β R ρ s . Therefore, it might be highly desirable to obtain approximate relationships at small and large time intervals.

204


Based on the known properties of the limiting transitions of the Laplace transformations, the asymptotic behavior at t → 0 and t → ∞ is determined by the behavior of the image at s → ∞ and s → 0, respectively. First, we consider the case when t → 0, which corresponds to the case when s → ∞. Then equation (15) yields: √ π R2 −1 2 μ − α ν ν −α −3 [s − √ θ 2 λ 2 s 2 ]F (s) Q (s) = ρ R ρ for 0 < ν < 1, 0 < α < 1 and Q (s) =

√ π R2 −1 2 μ − 3 [s − √ s 2 ]F (s) ρ R ρ

for ν ≤ 0, α ≤ 0. If we assume a pressure difference is a constant value respectively, have the following forms:

ΔP L ,

then F (s) ≡

(17) ΔP sL

and the originals of (16) and (17),

√ 3+α −ν 2 μ −α ν t 2 ΔP π R2 ] , [t − √ θ 2 λ 2 q (t) ≈ ρ R ρ Γ( 5+2α -ν ) L for 0 < ν < 1, 0 < α < 1 and q (t) ≈

(16)

√ 3 2 μ t 2 ΔP π R2 ] , [t − √ ρ R ρ Γ( 5 ) L 2

(18)

(19)

for ν ≤ 0, α ≤ 0. It is clear from (18) and (19) for small time intervals the flow function at 0 < ν < 1 and 0 < α < 1 depends on the fractal parameters, as well as the relaxation and retardation times, however for ν ≤ 0, α ≤ 0 such dependence is not observed in this particular research. Now consider the case when t → ∞, which corresponds to the case when s → 0. Then from equation (15) we can obtain: π R4 1 + θ α sα F (s) (20) Q (s) = ρ s 2 1+θ α sα . ν ν 4μ 1 + λ s 1 + 4μ R 1+λ ν sν The original of equation (20) can be represented in the form: q (t) ≈

π R4 ∞ ∑ (−1)n [λ ν nDν n + θ α λ ν nDν n+α ] f (t) , 4μ n=0

(21)

for ν < 0 and α < 0. Knowing the specific value of f (t) from (21) we can calculate flow rate q(t). Assuming, that f (t) = ΔP L , (21) can be written as: q (t) ≈

π ΔPR4 ∞ λ ν n −ν n θ α λ ν nt −ν n−α t ]. (−1)n [ + ∑ 4μ L n=0 Γ (1-ν n) Γ (1-ν n-α )

(22)

For large times the original of (20) can be written in the form of the following differential equation: ν α ρ R2 α d α +1 q (t) ρ R2 dq (t) π R4 ν d q (t) α d f (t) θ + q (t) = + λ + [ f (t) + θ ]. 4μ dt α +1 dt ν 4μ dt 4μ dt α

(23)

Prescribing a specific value for f (t), the differential equation (23) can be solved with respect to q(t) and vice versa, knowing q(t), equation (23) can be solved with respect to f (t). Furthermore, in the presence of experimental measurements q(t) and f (t) can be determined based on the model parameters and by setting inverse solutions.


205

Assuming that for a moving fractal parametric fluid flow rate q(0) = q0 , at some initial time there is an instantaneous release of pressure to zero, i.e., f (t) ≡ 0. Then the solution of (23), assuming the conditions ν α 4μ /ρ R2 ≈ λ α −ν /θ α −ν we can obtain the following expression: ν

q (t) ≈ q0 exp(−

λ α −ν α

θ α −ν

t).

(24)

Analysis of equation (24) shows that after resetting the pressure drop liquid fractal structure still continues to move for some time, and the duration of the flow depends on the parameters of relaxation times and fractals. For flows in the circular pipes the integral parameters such as the average velocity over the cross section, density and pressure are utilized. Therefore, it is of practical interest to obtain the averaged differential equations of motion of the rheological fluid with a fractal structure. Hence, we can multiply ´ R the equation (10) to 2π rdr and integrate it along the radius from 0 to R, based on the relationship w = π1R2 0 2πυz (t, r) rdr, we will obtain

ρ

∂w ∂P 2 =− + τR ∂t ∂z R

,

(25)

where τR -wall shear stress in the pipe. If we assume that the pipe wall is performed similar to the quasi-stationary flow of a viscous liquid condition ∂ υz = − R4 w, the equation (11) can be written as ∂t r=R

θα

ν 4μ ∂eα τR ν ∂c w (w + + τ = − λ ). R ∂ tα R ∂ tν

(26)

From the equations (25) and (26), we can obtain

θα

ν 8μ ∂eα +1 w ∂w θ α ∂eα +1 P 1 ∂ P ν ∂c w + 2aw = − − , 2a = + 2a λ + . α +1 ν α ∂t ∂t ∂t ρ ∂t ∂z ρ ∂z ρ R2

(27)

The equation (27) contains two unknown parameters w(t, z) and P(t, z). To close the system of equations we will use the continuity equation for weakly compressible fluid [10]

ρ C02

∂w ∂P =− , ∂z ∂t

(28)

where C0 is the speed of propagation of disturbances. Solving the equations (27) and (28) relative to the P(t, z), we can be obtain

θα

ν +1 P α +2 P 2 ∂eα +2 P ∂ 2P ∂P ν ∂c α 2 ∂e 2∂ P = + 2a λ + + 2a θ C +C . 0 0 ∂ t α +2 ∂ t ν +1 ∂ t2 ∂t ∂ t α ∂ z2 ∂ z2

(29)

The flow start or stop processes, and changing flow regimes of real liquids with fractal structure in different pipelines can be described by a system of differential equations of motion (27) and continuity (28). Assume that at the initial time in the semi-infinite tube filled with rheologically complex liquid with fractal structure, speed and pressure are equal to zero. At some point in time, the pressure in cross section at the z = 0 varies as P(t, 0) = P0 (t), in particular, the pressure may take a constant value equal to P0 . In this case, the unsteady flow above the fluid can be described by a system of differential equations (27) and (28) with the following initial and boundary conditions: w (0, z) = 0, P (t, 0) = P0 (t) ,

P (0, z) = 0; P (t, ∞) = 0.

(30) (31)

206


The equations (27) and (28) expressed in terms of P, at e = 0 and c = 0 has the form similar to equation (29): ∂ α +2 P ∂ ν +1 P ∂ 2 P ∂P ∂ α +2 P ∂ 2P = θ α C02 α 2 +C02 2 . θ α α +2 + 2aλ ν ν +1 + 2 + 2a (32) ∂t ∂t ∂t ∂t ∂t ∂z ∂z In this case, the initial and boundary conditions for P can be written as P (0, z) = 0, P (t, 0) = P0 (t) ,

∂ P(0, z) = 0; ∂t P (t, ∞) = 0.

(33) (34)

Applying the Laplace transformation to equations (32)–(34), we obtain the following value of the velocity in the initial section of the pipe √ s 1 + θ α sα (35) Pˆ0 (s) . wˆ (s) = ρ C02 2a + s + 2aλ ν sν + θ α sα The asymptotic solutions for 0 < α , ν > 1 + α as t → 0 and t → ∞ are obtained from (35) and, accordingly, have the following forms: 1+α −ν θα 1 D 2 P0 (t) (36) w (t) ≈ ρ C02 2aλ ν w (t) ≈

1 √

ρ C02

1

2a

[D 2 −

3 1 1 1 1 D 2 − (λ ν Dν + 2 + θ α Dα + 2 )]P0 (t) . 2 · 2a 2

If we assume that P0 (t) ≡ P0 , then the solutions of (36) and (37), respectively, can be written as θ α − 1+α −ν P0 2 t . w (t) ≈ 2aλ ν ρ C02 Γ 1+α2 −ν w (t) ≈

P0 √ 2

ρ C0

1 1 3 −(α + 21 ) t− 2 t− 2 1 ν t −(ν − 2 ) α t −θ )]}. {[ − 1 − (λ 1 2 Γ −2 Γ 2 −ν Γ 12 − α 2a Γ 12

(37)

(38)

(39)

From the analysis of equations (36)–(39) we can conclude that for small times the flow rate multiplicatively depends on the relaxation time θ , λ , and the parameters of the fractal α , ν , and for long times, this dependence is additive. Now consider the unsteady flow of rheological complex fluid with fractal structure in a semi-infinite pipe when the pressure P is a harmonic function of time at a given frequency at the initial cross section of the pipe. It is assumed that after a sufficiently long time period from the initial moment, there will not be any the effects of the initial conditions on the pressure distribution process over the time. Then the problem reduces to the solution of the differential equation (32) in the following form P (t, z) = P0 exp (iω t + δ z) .

(40)

−θ α ω α +2 iα + 2aλ ν ω ν +1 iν +1 − ω 2 + 2aω i = δ 2C02 (θ α ω α iα + 1) .

(41)

Substituting (40) into (32) we obtain

From equation (41) we can obtain: ω i 1 + θ α ω α iα − 2aλ ν ω ν −1 iν +1 − 2aω −1 i . δ= C0 (1 + θ α ω α iα )

(42)


207

If we assume that θ , λ , ω 2a and small, then 0 < α and ν < 1, it is possible to obtain approximately

δ =−

1 a 2a π π ω π π (1 − θ α ω α cos α + λ ν ω ν cos ν ) − i [1 − (θ α ω α sin α − λ ν ω ν sin ν )] 2 . C0 2 2 C0 ω 2 2

(43)

An analysis of (43) shows that, as the relaxation time θ , λ and the fractal parameters α , ν can significantly influence the process of damping pressure along the pipe. Furthermore, it is obvious that, depending on the parameters α and ν , θ and λ can both increase and decrease the attenuation process compared to processes in a viscous damping fluid pressure. Attenuation coefficient in rheologically complex fluids with a fractal structure is proportional to a π π (1 − θ α ω α cos α + λ ν ω ν cos ν ). C0 2 2 It should be noted that the relaxation times and the fractal parameters also affect the phase shift harmonics pressure boundary value. 3 Conclusions Shown that for modeling of the thixotropic rheological type oils with a high content of wax, resins and asphaltene can use integral type rheological equation or rheological differential equation by fractional derivative. On the basis of these models were obtained differential equations by fractional derivatives for describing the motion of the rheological of complex fluids by fractal structure in pipes. Analysis obtained solutions of equations shows that after resetting the pressure drop liquid fractal structure the fluid still continues to move for some time, and the duration of the flow depends on the parameters of relaxation times and fractals. It is shown that in addition to the function of stress relaxation, one may also consider a relaxation function (retardation) of velocity gradient during the model simulations. Application of the Laplace transformation to the derived equations allowed to obtain the value of the velocity in the initial section of the pipe. The research results showed that in the case if the pressure P is a harmonic function of time at a given frequency at the initial cross section of the pipe, then the relaxation time θ , λ and the fractal parameters α , ν can significantly influence the process of damping pressure along the pipe. Furthermore, it is obvious that, depending on the parameters α and ν , θ and λ can both increase and decrease the attenuation process compared to processes in a viscous damping fluid pressure. It should be noted that the relaxation times and the fractal parameters also affect the phase shift harmonics pressure boundary value. References [1] Gennes, P. de (1979), Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca and London. [2] Mirzadzhanzade, A.Kh., Hasanov, M.M., and Bakhtizin, R.N. (1999), Studies on Modelling of Complicated Systems for Oil Production, Ufa, Gilem, 464. [3] Hasanov, M.M. (1994), Scale invariance of timing hierarchies in the process of relaxations of viscoelastic fluids. APP, 58(1), 110–115. [4] Sattarov, R.M. (1993), Scaling of properties of rheologically complicated liquids used in oil production, Azerbaijan Oil Industry, 9, 13–18. [5] Gubin, V.V. and Pijadin, M.M., and Skripnikov, Y.V. (1973), On thixotropic characteristics of paraffinic oils, Proceedings of VNIISPT Oil, Ufa, XI. [6] Mikhaylov, N.V. (1995), On fluidity and strength of the structured fluids, Colloidal Journal, XVII. [7] Mukuk, K.V. (1980), Element of Hydraulics Relaxing Anomalous Systems, Tashkent, FAN, 116. [8] Lodge, A. (1984), Elastic Fluids, Moscow, Nauka, 443. [9] Babenko, Y.I. (1986), Heat and Mass Transfer, Leningrad, Chemistry, 144. [10] Charniy, I.A. (1975), Unsteady Flow of Real Liquids in Pipes, Moscow, Nedra, 296.



Analytical Prediction of Homoclinic Bifurcations Following a Supercritical Hopf Bifurcation Tanushree Roy1 , Roy Choudhury1†, and Ugur Tanriver2 1 Department 2 Department

of Mathematics, University of Central Florida, Orlando, FL 32816-1364 USA of Mathematics, Texas A&M University, Texarkana, TX 75505 Submission Info

Communicated by Albert C.J. Luo Received 1 October 2015 Accepted 2 November 2015 Available online 1 October 2016 Keywords Supercritical Hopf bifurcations Post-Hopf regimes Collision with neighboring saddle-point Homoclinic orbit formation

Abstract An analytical approach to homoclinic bifurcations at a saddle fixed point is developed in this paper based on high-order, high-accuracy approximations of the stable periodic orbit created at a supercritical Hopf bifurcation of a neighboring fixed point. This orbit then expands as the Hopf bifurcation parameter(s) is(are) varied beyond the bifurcation value, with the analytical criterion proposed for homoclinic bifurcation being the merging of the periodic orbit with the neighboring saddle. Thus, our approach is applicable in any situation where the homoclinic bifurcation at any saddle fixed point of a dynamical system is associated with the birth or death of a periodic orbit. We apply our criterion to two systems here. Using approximations of the stable, post-Hopf periodic orbits to first, second, and third orders in a multiple-scales perturbation expansion, we find that, for both systems, our proposed analytical criterion indeed reproduces the numerically-obtained parameter values at the onset of homoclinic bifurcation very closely. ©2016 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Homoclinic and heteroclinic orbits are important in nonlinear systems for a variety of reasons. For instance, stable homoclinic or heteroclinic orbits in the traveling-wave reduction of any nonlinear partial differential equation (NLPDE) model correspond to pulse or front solutions of the NLPDE, and thus act as carriers or organizing centers for the dynamical information. Alternatively, under certain conditions, their occurrence corresponds to the onset of chaos [1]– [9]. This depends on the eigenvalues of the Jacobian matrix of the flow at the saddlepoint (and also on system symmetries, if any) and may correspond to horseshoes and infinitely many nearby bifurcations following the homoclinic bifurcation. Most often the birth of a homoclinic or heteroclinic orbit is predicted via the use of the Melnikov criterion [1]– [2] based on the vanishing of the distance between the stable and unstable manifolds of the saddle fixed point anchoring the homoclinic or heteroclinic orbit. In this paper, we propose an alternative analytical approach to this global bifurcation problem. This is based on high-order, high-accuracy approximation of the stable periodic orbit created at a supercritical Hopf bifurcation of a fixed point neighboring the saddle fixed point. This periodic orbit then expands (as the Hopf † Corresponding

author. Email address: [email protected]


210

Tanushree Roy, S. Roy Choudhury, Ugur Tanriver / Discontinuity, Nonlinearity, and Complexity 5(3) (2016) 209–222

bifurcation parameter(s) are varied beyond the bifurcation point) till it collides with the neighboring saddle fixed point, thereby giving rise to a homoclinic orbit. Thus, our criterion is applicable in any situation where the homoclinic bifurcation at any saddle fixed point of the dynamical system is associated with the birth or death of a periodic orbit. Of course, even given the high-order analytical approximation of the supercritical-Hopfgenerated stable periodic orbit at a fixed point neighboring the saddle, the analytical predictions are likely to be more accurate if the actual physical distance between the saddle and its adjacent fixed point are not too large. We apply this method to two different systems [7] undergoing homoclinic bifurcations, and find very good agreement with the numerical results. The remainder of this paper is organized as follows. In Section 2, we consider the linear stability analysis of the fixed points of two systems. For both, one of the two fixed points of the system undergoes a Hopf bifurcation at a critical value of the system parameter, while the other fixed point is a saddle for all ranges of the parameter. In Section 3, we employ multiple-scales expansions to construct expressions for the post-Hopf periodic orbits out to the third-order for both systems. As usual, during this analysis, we also determine that these periodic orbits are stable, so the Hopf bifurcations leading to their birth are indeed supercritical for both of our systems. This then puts us in the setting described above, where a stable periodic orbit may expand in size as the system parameters are varied past their Hop bifurcation values. In Section 4, we then enforce the criterion discussed above for the homoclinic bifurcation, i.e., that the extremity of the periodic orbits collide with the adjacent saddle fixed point. Using approximations of the stable post-Hopf periodic orbits to first, second, and third orders in multiple-scales perturbation expansions, we find that, for both systems, our proposed analytical criterion indeed closely reproduces the numerically-obtained parameter values at the onset of homoclinic bifurcation. Section 5 very briefly summarizes the results of the paper, and comments on possible extensions to higher-order systems.

2 Linear stability and Hopf bifurcation analysis 2.1

System I

The first system [7] investigated in this paper (and referred to subsequently as System I) is given by ˙ x¨ = µ x˙ + x − x2 + xx,

(1)

which can also be written as a system of equations as x˙ = y, y˙ = µ y + x − x2 + xy.

(2)

The fixed (alternatively, equilibrium or critical) points of (2) are at (0, 0) and (1, 0). Following standard methods of phase-plane analysis, the Jacobian matrix for the system is given by: 0 1 . (3) J= 1 − 2x + y µ + x Considering the fixed point (0, 0) first, the Jacobian is: 01 J|(0,0) = . 1µ

(4)

As is well-known [7], a negative value of the determinant of the Jacobian matrix is a signature of a saddle fixed point. In this case, the determinant of J is ∆ = −1and so there is a saddle-point at (0, 0).


211

Next, the Jacobian evaluated at (1, 0) yields: J|(1,0) =

0 1 . −1 µ + 1

(5)

The eigenvalues of this matrix satisfy the following characteristic equation

λ 2 − λ (µ + 1) + 1 = 0,

(6)

Hence the eigenvalues are

p (µ + 1)2 − 4 λ1,2 = . (7) 2 We know that the system may undergo a Hopf bifurcation if the eigenvalues become purely imaginary at some critical value of the parameter. Hence, our system might possibly undergo a Hopf bifurcation for µ = −1 where the eigenvalues take the values µ = ±2i. In addition to the criterion of the eigenvalues being purely imaginary, the eigenvalues crossing the imaginary axis must do so with non-zero speed. This behavior of the eigenvalues is ensured by the so-called ‘transversality condition’ (µ + 1) ±

∂ (Re(λ (µ )))|µ =µ0 6= 0. ∂µ In order to compute the partial derivative required in this condition, we differentiate (6) with respect to µ , to obtain 2λ

∂λ ∂λ ∂λ −λ −µ − = 0, ∂µ ∂µ ∂µ

and thus, at λ = ±2i and µ = −1, we have

1 ∂λ = . ∂µ 2

(8)

(9)

Hence,

∂ (10) (Re(λ (µ )))|µ =−1 = 1/2 6= 0, ∂µ and the transversality condition is satisfied. Therefore, a Hopf Bifurcation occurs at the critical value µ = −1. Hence, a periodic solution will then exist near the fixed point (1, 0) in the post-Hopf parameter regime, and both its analytical form as well as its stability will be determined in the next section. 2.2

System II

An analogous analysis is carried out on a second system [7] (refereed to henceforth as System II): x˙ = µ x + y − x2 ,

y˙ = −x + µ y + 2x2 ,

(11)

having fixed points (0, 0) and (

µ 2 + 1 (µ 2 + 1)(1 − 2µ ) , ). µ +2 (µ + 2)2

(12)

The Jacobian of the system is computed as J=

µ − 2x 1 −1 + 4x µ

.

(13)

212


First, the Jacobian is evaluated at the origin, yielding µ 1 J= . −1 µ

(14)

The eigenvalues of this Jacobian matrix satisfy the characteristic equation (λ − µ )2 + 1 = 0,

(15)

λ = µ ± i.

(16)

resulting in eigenvalues

For the possible onset of Hopf Bifurcation, the eigenvalues needs to be purely imaginary. Hence, at µ = 0 we have a possible setting for the origin to undergo Hopf bifurcation since the eigenvalues then have values λ = ±i. As for the previous system, we may verify the occurrence of the Hopf bifurcation at µ = 0 by differentiating (16) with respect to µ and readily verifying the transversality condition

∂ (Re(λ (µ )))|µ =0 = 1 6= 0. ∂µ

(17)

Consequently, the origin undergoes Hopf bifurcation at µ = 0, creating a periodic solution near it. Once again, the shape and stability of this periodic orbit will be considered in the following section. Next, computing the Jacobian at the other fixed point (12) yields   − µ 2 + 2µ − 2 1  µ +2 . (18) J= 2   4µ − µ + 2 µ µ +2 The determinant of this Jacobian is ∆ = −(µ 2 + 1) < 0 which implies the existence of two roots with opposite signs [7]. Hence, this fixed point (12) is a saddle. This creates the possibility that the periodic orbit created at (0, 0) by the Hopf Bifuraction might grow and eventually may collide with this saddle. In the following section, we perform expansions of both of our systems around the Hopf bifurcation points found in this section to obtain approximate solutions for the post-Hopf periodic orbits, as well as to deduce their stability in each case.

3 Multiple-scales expansion for the post-Hopf periodic orbit In this section, the method of multiple scales is used to construct a high-order analytical approximation [10][11] of the periodic orbit formed after the Hopf bifurcations in our two systems (1) and (11) by considering terms upto the third order in the perturbation expansion. As part of this analysis, the post-Hopf periodic orbits are found to be stable, so that the Hopf bifurcations are supercritical for both of our systems. In the next section, these analytical approximations will then be used to find the critical values of the bifurcation parameter where the periodic orbit hits the neighboring saddle-point, thus generating a homoclinic bifurcation of each system. 3.1

Post-Hopf periodic orbit of system I

As usual [10]- [11], the limit cycle surrounding the fixed point (1, 0) of (2) after the Hopf bifurcation may be obtained by expanding the physical variable x about the fixed point as well as the parameter µ (around the bifurcation value µ = −1), using progressively slower time scales. The expansion is of the following form:


213

3

x = x0 + ∑ ε n xn (T0 , T1 , T2 ),

(19)

n=1

where Tn = ε nt, x0 = 1 and ε is a small positive non-dimensional parameter that is used for bookkeeping and will be set to unity in the final analysis. Hence, using the Chain Rule, the time derivative can be expressed as d = D0 + ε D1 + ε 2 D2 + · · · , dt

(20)

where Dn = ∂ /∂ Tn . Using the standard expansion [1], [10], the bifurcation or control parameter is ordered as µ = −1 + ε 2 µa . This is chosen so that the nonlinear terms and the control parameter occur at the same order. Now, using (19) and (20) in (1), we have [D20 + 2ε D0 D1 + ε 2 (D21 + 2D0 D2 ) + · · · ](x0 + ε x1 + ε 2 x2 + · · · )

(21)

= (−1 + ε µa )(D0 + ε D1 + ε D2 + · · · )(x0 + ε x1 + ε x2 + · · · ) + (x0 + ε x1 + ε x2 + ε x3 ) 2

2

2

2

3

−(x20 + 2ε x0 x1 + ε 2 (x21 + 2x0 x1 ) + ε 3 (2x0 x3 + 2x1 x2 ) + · · · ) + (x0 + ε x1 + ε 2 x2 + · · · )

[D0 x0 + ε (D0 x1 + D1 x0 ) + ε 2 (D0 x2 + 2D1 x1 + D2 x0 ) + ε 3 (D0 x3 + D1 x2 + D2 x1 + D3 x0 ) + · · · ].

Separating off the terms order by order, and solving the resulting equations at each order, we obtain: O(1)

D20 x0 + D0 x0 − x0 + x20 − x0 D0 x0 = 0,

(22)

with the zero-order solution x0 = 1, as known from the fixed-point value. O(ε )

D20 x1 + 2D0 D1 x0 = −D0 x1 − D1 x0 + x1 − 2x0 x1 + x0 (D0 x1 + D1 x0 ) + x1 D0 x0 ,

(23)

D20 x1 + x1 = 0,

(24)

x1 = AeiT0 + cc.

(25)

D20 x2 + x2 = −2D0 D1 x1 + D1 x1 − x21 + x1 D0 x1 .

(26)

which yields

with solution of the form

O(ε 2 ) At second order, the terms simplify to

Setting x1 = AeiT0 + cc in this last equation gives, D20 x2 + x2 = (−2i

∂A ∂ A iT0 + )e − (A2 e2iT0 + 2AA∗ ) + i(A2 e2iT0 − (A∗ )2 e−2iT0 ) + cc. ∂ T1 ∂ T1

(27)

214


Suppressing the secular, first-harmonic terms (which occur in the solution x1 of the homogeneous eequation) yields the relation

∂A = 0. ∂ T1

(28)

Next, we assume the second-order particular solution to consist of the standard DC term plus the second harmonics i.e. (0)

(2)

x2 = x2 + x2 e2iT0 + cc.

(29)

Using this in (27) and balancing terms of each type yields both coefficients in the second-order solution x(0) = −2AA∗ , (1 − i) 2 (2) x2 = A . 3

(30) (31)

O(ε 3 ) Using (28-30, the third order terms in (21) simplify to D20 x3 + x3 = (−2i

∂ A iT0 ∂ A iT0 e − e + µa iAeiT0 ) ∂ T2 ∂ T2 (0)

(2)

(32) (2)∗

−2(AeiT0 + A∗ e−iT0 )(x2 + x2 e2iT0 + x2 e−2iT0 ) (0)

(2)

(2)∗

+i(x2 + x2 e2iT0 + x2 e−2iT0 )(AeiT0 − A∗ e−iT0 ) +(AeiT0 + A∗ e−iT0 )(

2i(1 − i)A2 2iT0 ∂ A iT0 e + e ) + cc. 3 ∂ T2

As before, setting the coefficients of the secular, first-harmonic term in the above equation to zero, we get the equation governing evolution of the amplitude of x1 on the slow second-order time scales, i.e., 2i

∂A 11 = iµa A + ( − i)|A|2 A. ∂ T2 3

(33)

Letting A = 1/2aeiβ , (25) yields the first-order solution in the real form x1 = a cos(T0 + β ). Now, separating real and imaginary parts of (33), we have

∂a µa a a3 − = ∂ T2 2 8 2 11a ∂β . =− ∂ T2 24

(34) (35)

In order to find the amplitude of the periodic orbit, we set ∂ a/∂ T2 = 0 (so one may settle onto this fixed point, which corresponds to the periodic orbit of the original system). This results in p (36) a∗ = 4µa , which is real for µa > 0 yielding the side of the bifurcation point µ = 0 where the periodic orbit exists, i.e., the post-Hopf region.


215

The stability of this fixed point a∗ (corresponding to the periodic orbit) is given by the sign of the Jacobian(this is just a number here) at this fixed point, which is given by

∂ µa a a3 ( − ) = −µa < 0, ∂a 2 8

(37)

thus showing that the fixed point (and hence the Hopf-generated periodic orbit)is stable. In order to complete the solution of (32) (the secular, first-harmonic terms have already been suppressed), substituting in the standard form for the third-order particular solution (0)

(3)

(2)

x3 = x3 + x3 e2iT0 + x3 e3iT0 + cc,

(38)

and equating the coefficients of each harmonic, we obtain (0)

x3 = 0, (2)

x3 = 0, (3)

x3 =

(39) (2)

(2A − 3iA)x2 (1 + 5i) 3 =− A . 8 24

This completes our derivation of the stable post-Hopf periodic orbit of our system I or (2) up through O(ε 3 ). The orbit is given by (19), and may be approximated out to third order or O(ε 3 ) using the explicit expressions derived above. In the following section, we shall use this analytical expression for the post-Hopf periodic orbit to various degrees of approximation to predict the onset of the global homoclinic bifurcation, and test the results against actual numerical simulations. 3.2

Post-Hopf periodic orbit of system II

Performing a similar analysis on the second system (11) with fixed point as the origin, both the variables are expanded about the fixed point using progressively slower time scales as x = x0 + ε x1 + ε 2 x2 + ε 3 x3 + · · · ,

y = y0 + ε y1 + ε 2 y2 + ε 3 y3 + · · · ,

(40)

where x0 = 0 and y0 = 0 and the parameter is expanded as µ = ε 2 µa about the Hopf bifurcation value µ = 0 as before. Putting (40) in (11), we obtain the following expansions (D0 + ε D1 + ε 2 D2 + · · · )(ε x1 + ε 2 x2 + ε 3 x3 + · · · )

= ε µa (ε x1 + ε x2 + ε x3 + · · · ) + (ε y1 + ε y2 + ε 2

2

3

2

(D0 + ε D1 + ε 2 D2 + · · · )(ε y1 + ε 2 y2 + ε 3 y3 + · · · )

3

= −(ε x1 + ε x2 + ε x3 + · · · ) + ε µa (ε y1 + ε y2 + ε 2

3

From (41) we have, order by order in ε O(ε ): D0 x1 − y1 = 0 ≡ S11 O(ε 2 ): D0 x2 − y2 = −D1 x1 − x21 ≡ S12

2

2

(41)

y3 + · · · ) − (ε 2 x21 + 2ε 3 x1 x2 + · · · ), (42)

3

y3 + · · · ) + 2(ε 2 x21 + 2ε 3 x1 x2 + · · · ).

216


O(ε 3 ): D0 x3 − y3 = −D1 x2 − D2 x1 + µa x1 − 2x1 x2 ≡ S13 In exactly similar fashion, (42) gives O(ε ) D0 y1 + x1 = 0 ≡ S21 O(ε 2 ): D0 y2 + x2 = −D1 y1 + 2x21 ≡ S22 O(ε 3 ): D0 y3 + x3 = −D1 y2 − D2 y1 + µa y1 + 4x1 x2 ≡ S23 where the source terms on the right hand side have been labeled with the first index representing the equation number, and the second being the order in ε at which each occurs. In general, we note that yi = D0 xi − S1i and D0 yi + xi = S2i at O(ε i ). Eliminating yi , we thus obtain the composite equation at O(ε i ) as D20 xi + xi = S2i + D0 (S1i ).

(43)

We shall now proceed to solve this composite equation order by order in a manner analogous to the previous sub-section. O(ε ) Since, S11 = 0, the first order solution can be taken to be x1 = AeiT0 + cc, y1 = iAeiTo + cc.

(44)

Note that these first-harmonic solutions solve the homogeneous equation (since the composite source S11 = 0 at this leading order). O(ε 2 ) Next, the composite equation at second-order is obtained by substituting these first-order solutions given by (44) into (43) with i = 2, yielding D20 x2 + x2 = S22 + D0 (S21 ), = −2i(D1 A)eiT0 + 2A2 (1 − i)e2iT0 + 4AA∗ + cc.

(45)

In order to obtain a uniform expansion, the secular part of the right-hand side is suppressed by setting the coefficients of the secular first harmonic eiT0 (as noted above, these solve the corresponding homogeneous equation) terms to zero. This yields D1 A = 0.

(46) (0)

Having suppressed the secular first harmonics, the standard second order particular solution x2 = x2 + (2) 2iT0 x2 e + cc is substituted into (45) and, equating coefficients, the second order solution components are found to be (0)

x2 = 4AA∗ , (2)

x2 =

2A2 (i − 1) . 3

(47) (48)


217

Using the above solution and the relation yi = D0 xi − S1i with i = 2 yields y2 = (−

1 + 4i 2 2iT0 )A e + 2AA ∗ +cc. 3

(49)

O(ε 3 ) Employing these second-order solutions as well the first-order solutions in (44), the third order composite equation (given by (43) with i = 3) simplifies to D20 x3 + x3 = −2i(D2 A)eiT0 + 2iµa AeiT0 (0)

(50) (2)∗

(2)

+4[AeiT0 + A∗ e−iT0 ][x2 + x2 e2iT0 + x2 e−2iT0 ] (0)

(2)∗

(2)

−2i[AeiT0 − A∗ e−iT0 ][x2 + x2 e2iT0 + x2 e−2iT0 ] (2)

(2)∗

+2i[AeiT0 + A∗ e−iT0 ][x2 e2iT0 − x2 e−2iT0 ] + cc. Suppressing the secular, first harmonic terms in this equation now yields

44 − 4i)|A|2 A = 0. (51) 3 We solve this third-order secularity condition in a manner similar to that in the previous sub-section for our first system. Setting A = 1/2aeiβ , (44) yields x1 = a cos(T0 + β ) and y1 = −a sin(T0 + β ), Using these in (51) and separating the real and imaginary parts, we have −2i(D2 A) + 2iµa A + (

11 aβ˙ + a3 = 0, 6 (52) 1 a˙ − µa a + a3 = 0. 2 As before, we are interested in the fixed point of the system (52), in order to obtain the radius of the periodic orbit. The fixed point of (52) is p (53) a∗ = 2µa . and it is stable since the Jacobian there is

∂ 1 [µa a − a3 ] = −2µa < 0. ∂a 2

(54) (0)

(2)

(3)

Next, substituting the standard third order particular solution x3 = x3 + x3 e2iT0 + x3 e3iT0 + cc into (50) anad balancing coefficients, we obtain (0)

x3 = 0, (2)

x3 = 0, (3)

A3

(55)

(1 − i). 3 The third-order solution for y3 may now be obtained using the general relation yi = D0 xi − S1i with i = 3. This completes our derivation of the post-Hopf stable periodic orbit in (11) up through O(ε 3 ). The orbit is given by (40), and may be approximated out to third order or O(ε 3 ) using the explicit expressions derived above. In the following section, we shall also use this analytical expression for the post-Hopf periodic orbit to various degrees of approximation to predict the onset of the global homoclinic bifurcation, and test the results against actual numerical simulations. x3 =

218


4 Analytical prediction of homoclinic bifurcations 4.1

Homoclinic bifurcation in system I

The value at which the homoclinic bifurcation occurs for the system (2) is numerically computed in [7] to be µ ≈ −0.8645. In order to predict this analytically, we will gradually improve our prediction by incorporating higher order corrections to the post-Hopf periodic orbits into our analysis. 4.1.1

First order approximation of periodic orbit

As discussed in previous sections, the fixed point (1,0) undergoes supercritical Hopf bifurcation giving rise to a stable periodic orbit. This periodic solution then grows and eventually merges with the saddle, at the origin, located to the left of this fixed point. Hence, we can conclude that that the analytical criterion for this stable orbit to coalesce with the saddle point is achieved at the minimum x value on the periodic orbit. Keeping up to first order terms in the expansion (19) of the periodic orbit we have x = x0 + ε x1 = 1 + ε a∗ cos(To + β )

(56)

and the orbit hits the saddle if xmin = 0. This will occur if cos(T0 + β ) = −1, i.e. if the leftmost point of the orbit 1 1 is the origin, yielding the condition a∗ = . This implies µa = 2 and hence ε 4ε

µcrit = −1 + ε 2 µa = −1 + 1/4 = 0.75. This is our first prediction for the critical value of µ at homoclinic bifurcation from the first order periodic orbit approximation. We can further improve this prediction by incorporating more terms in the expansion for the periodic orbit. 4.1.2

Second order approximation of periodic orbit

Next, we try to improve this approximation for µcritcal by considering the second order approximation (see (19)) for the periodic orbit x = 1 + ε a∗ cos(T0 + β ) + ε 2 [−2AA∗ +

(1 − i) 2 2iT0 (1 + i) ∗2 −2iT0 A e + A e ]. 3 3

(57)

Putting A = 1/2aeiβ as before, we obtain a2∗ a2∗ + ((1 − i)e2i(T0 +β ) + (1 + i)e−2i(T0 +β ) )] 2 12 1 a2 = 1 + ε a∗ cos(t0 + β ) + ε 2[− a2∗ + ∗ (cos 2(T0 + β ) + sin 2(T0 + β ))]. 2 6

x = 1 + ε a∗ cos(t0 + β ) + ε 2[−

(58)

Again, following the reasoning above, we try to find the condition that xmin = 0 numerically. The x on the periodic orbit (58) is numerically plotted versus z = (T0 + β ) for various values of k = ε a∗ , with the objective being to obtain xminpnearly close to zero. For k = ε a∗ = 0.76524 we have xmin very close to zero (see Figure 1). This implies that ε 4µa∗ = 0.76524 and µcrit = −1 + ε 2 µa∗ ≈ −1 + 0.146398 = −0.853602. The above parameter value µcrit = −0.853602 at the homoclinic bifurcation, as predicted at this stage by our analytical criterion, is already very close to the numerically obtained value µ ≈ −0.8645. However, let us see if we can improve this analytical estimate further by keeping third-order terms in the approximation of the periodic orbit.


219

(a) Graph of x vs z = T0 + β for second order approximation (b) Graph of x vs z for third order approximation of the periof the periodic orbit at k = 0.76525. The first minimum has odic orbit at k = 0.77363. The first minimum has just dipped just dipped below zero. below zero.

Fig. 1 Plots showing xmin on the periodic orbit just going negative at the numerically-obtained critical k values.

4.1.3

Third order approximation of periodic orbit

From (19), the third order approximation to the periodic orbit can be readily found to be x = 1 + ε a∗ cos(T0 + β ) 1 a2 +ε 2 [− a2∗ + ∗ (cos 2(T0 + β ) + sin 2(T0 + β ))] 2 6 5 1 +ε 3 a3∗ (− cos 3(T0 + β ) + sin 3(T0 + β )). 96 96

(59)

Enforcing the condition xmin = 0 by numerically plotting this versus z = (T0 + β ) for various values of k = ε a∗ , we obtain the value k = ε a∗ = 0.77363 and thus the third-order prediction for the critical parameter value for homoclinic bifurcation is found to be µcrit ≈ −0.8480949. This is slightly further from the numerically obtained value µ ≈ −0.8645 than the value of -0.853602 predicted by our second-order approximation to the periodic orbit. Hence, our analytical criterion yields values in pretty good agreement with the numerically obtained critical value of parameter at the homoclinic bifurcation, especially given that this is a global phenomenon with the orbit spanning a finite region of the phase-space. In particular, note that the second order correction to the periodic orbit provides the approximate critical value of the parameter µ = −0.853602, which is very close to the numerical value µ = −0.8645. Figure 2 show the phase-plots at these two values of µ showing the approach to the homoclinic bifurcation as the periodic orbit converges towards the neighboring saddle-point. 4.2

Homoclinic Bifurcation in System (11)

For the system in (11), it is well-known [8] that the numerical value at which the homoclinic bifurcation occurs is µcrit ≈ 0.06626. We shall now see how accurately our analytical technique is able to predict this bifurcation point. 4.2.1

First order approximation of the periodic orbit (40)

The saddle is at x = (µ 2 + 1)/(µ + 2) and is to the right of fixed point (0, 0). Thus, we impose our analytical criterion for the homoclinic bifurcation that the stable periodic orbit (40) formed after the supercritical Hopf bifurcation of the fixed point at the origin, will, at its point of maximum rightward swing, collide with this saddle point. Retaining terms in (40) out to O(ε ), we therefore have xmax = (ε a∗ cos(T0 + β ))max = (µ 2 + 1)/(µ + 2).

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Tanushree Roy, S. Roy Choudhury, Ugur Tanriver / Discontinuity, Nonlinearity, and Complexity 5(3) (2016) 209–222 Μ = - 0.8645

y

Μ = - 0.853602

1.0

1.0

0.5

0.5

y

0.0

-0.5

0.0

-0.5

-1.0

-1.0 -2

-1

0

1

2

-2

-1

x

0

1

2

x

(a) Phase plot for system 1 at µ = −0.8645 [7], showing the (b) Phase plot for system 1 at µ = −0.853602, as predicted formation of the homoclinic orbit at the origin. by our second-order approximation of the periodic orbit. Note the periodic orbit almost converging to the neighbouring saddle point at zero.

Fig. 2 Phase plots showing the accuracy of our analytical criterion for homoclinic bifurcation.

This can happen if cos(T0 + β ) = 1 which results in 2ε 2 µa =

µa4 ε 8 + 2µa2 ε 4 + 1 . µa2 ε 4 + 4ε 2 µa + 4

(60)

Retaining only leading order terms in (60), we have

ε µa = 0.125.

(61)

√ To refine this estimate further, let us assume ε µa = v. Then (60) becomes √ √ 2 2v + 2v3 = 1 + v4 .

(62)

√ √ The leading-order solution of (62) yields v = 1/2 2 + w and the first correction gives v = 23/48 2. This results in µ = ε 2 µa ≈ 0.1148 as our first analytical estimate for the parameter value at the homoclinic bifurcation. This is relatively far from the actual value of µ given above for the homoclinic bifurcation. Hence, we proceed to the next higher-order approximation for our periodic orbit. 4.2.2

Second order approximation of periodic orbit

The critical value for the parameter µ can be improved further by introducing the second order correction for the periodic orbit. Consequently, we again enforce the xmax = (µ 2 + 1)/(µ + 2) where the maximum swing or value on the periodic orbit leads to a collision with the neighboring saddle point. Now, using (40) up through second order terms, we have x = [ε a∗ cos(T0 + β ) + ε 2(a2∗ −

a2∗ ε 4µ 2 + 1 (cos 2(T0 + β ) + sin 2(T0 + β )))]max = 2 a . 3 ε µa + 2

(63)


221

(a) Graph of xˆ vs z for second order approximation of the peri- (b) Graph of xˆ vs z for third order approximation of the periodic orbit at k = 0.37388. odic orbit at k = 0.37468.

Fig. 3 Plots showing maximum of xˆ = (xmax − (µ 2 + 1)/(µ + 2)) very close to zero at numerically determined k values for second and third order approximations of the periodic orbit.

We replace µa = 1/2a2∗ from (53), and numerically obtain the value of k = ε a∗ ≈ 0.37388 by plotting (63) versus z = T0 + β and enforcing xmax = µ 2 + 1/µ + 2. This in turn implies µ ≈ 0.0698931 as our improved analytical estimate for the parameter value at homoclinic bifurcation. 4.2.3

Third order approximation of periodic orbit

The third order approximation of the periodic orbit given by (40) retaining terms out to i = 3 simplifies to x = ε a∗ cos(T0 + β ) + ε 2[a2∗ − + sin 2(T0 + β ))] + ε 3

a2∗ (cos 2(T0 + β ) 3

(64)

a3∗ (cos 3(T0 + β ) + sin3(T0 + β )). 12

As before, the condition for the orbit at the origin to merge with the saddle is xmax = (µ 2 + 1)/(µ + 2). After some calculation, this yields k = ε a∗ ≈ 0.37468 (obtained numerically) which results in µ ≈ 0.07019. These results are illustrated graphically in Figure 3. Thus, we see that our analytical criterion employing a high-order approximation to the Hopf-generated periodic orbit and having this merge with the neighboring saddle-point at the homoclinic bifurcation yields very accurate estimates. In particular, the estimate using the second-order approximation to the periodic orbit gives an estimate µ ≈ 0.0698931 which may be considered very close to the actual numerically value µ = 0.06626 [8] (for a global phenomenon like the homoclinic bifurcation, with the orbit spanning a finite region of the phase-space, so that any analytical approximation at all is tricky). The phase plots at both the numerically and analytically predicted µ values are compared in Figure 4 showing the formation of the separatrix as the periodic orbit merges with the neighboring saddle point.

5 Conclusions In this paper, we have developed an analytical criterion for homoclinic bifurcations at a saddle fixed point based on high-order, high-accuracy approximation of the stable periodic orbit created at a supercritical Hopf bifurcation of a neighboring fixed point. This periodic orbit then expands in the post-Hopf bifurcation regime, with the criterion we introduce to predict the onset of homoclinic bifurcation being its collision with the neighboring saddle. Thus, our approach is applicable in any situation where the homoclinic bifurcation at any saddle fixed point of a dynamical system is associated with the birth or death of a periodic orbit. This criterion has been care-

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Tanushree Roy, S. Roy Choudhury, Ugur Tanriver / Discontinuity, Nonlinearity, and Complexity 5(3) (2016) 209–222 Μ = 0.06626

y

Μ = 0.0698931

2

2

1

1

y

0

-1

0

-1

-2

-2 -1.0

-0.5

0.0 x

0.5

1.0

-1.0

-0.5

0.0

0.5

1.0

x

(a) Phase plot for system 2 at µ = 0.06626, as predicted by (b) Phase plot for system 2 at µ = 0.0698931, as predicted by numerical calculation [8], showing the formation of the homo- our second-order approximation to the periodic orbit, showing µ2 + 1 periodic orbits almost converging to the neighbouring saddle clinic orbit at x = µ2 + 1 µ +2 point at x = µ +2

Fig. 4 Phase plots showing the effectiveness of the analytical method to predict homoclinic bifurcation

fully applied to two systems. Using approximations of the stable post-Hopf periodic orbits to first, second, and third orders in multiple-scales perturbation expansions, we find that, for both systems, our proposed analytical criterion indeed reproduces the numerically-obtained parameter values at the onset of homoclinic bifurcation very closely. Clearly, it would be of interest to apply this approach to higher-order systems, including those where the formation of the homoclinic orbit in fact leads to the onset of chaos [1, 2, 6]. This will form the basis of future work in this area.

References [1] Nayfeh, A. and Balachandran, B. (1995), Applied Nonlinear Dynamics, Wiley, New York. [2] Guckenheimer, J. and Holmes, P. (1983), Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York. [3] Afraimovich, V.S., Bykov, V.V., and Shilnikov, L.P. (1977), On the origin and structure of the Lorenz attractor, Sov. Phys. Doklady, 22 253. [4] Sparrow, C.T. (1982), The Lorenz Equations: Bifurcations,Chaos, and Strange Atttractors, Springer-Verlag, New York. [5] Wiggins, S. (1988), Global Bifurcations and Chaos, Springer-Verlag, New York. [6] Shilnikov, L.P. (1965), A case of the existence of a denumerable set of periodic motions, Sov. Math. Doklady, 6, 163. [7] Strogatz, S. (1994), Nonlinear Dynamics and Chaos, Addison-Wesley, Reading (Mass). [8] Dominik Zobel (2013), Nonlinear Dynamics: Some exercises and solutions, Creative Commons License. [9] Glendinning, P. (1994), Stability, Instability and Chaos, Cambridge U. Press, Cambridge. [10] Krise, S. and Choudhury, S. Roy (2003), Bifurcattons and chaos in a predator-prey model with delay and a laser-diode system with self-sustained pulsations, Chaos, Solitons, and Fractals, 16, 59. [11] Cheng, T. and Choudhury, S. Roy (2012), Bifurcations and chaos in a modified driven Chen’s system, Far East Journal of Dynamical Systems, 18 (2012), 1.



Analysis of Stochastic Diffusive Predator Prey Model with Hyperbolic Mortality Rate M. Suvinthra1†, K. Balachandran1 , M. Sambath2 1 Department 2 Department

of Mathematics, Bharathiar University, Coimbatore 641 046, India of Mathematics, Periyar University, Salem 636 011, India Submission Info

Communicated by Albert C.J. Luo Received 31 October 2015 Accepted 26 January 2016 Available online 1 October 2016

Abstract In this work, we establish a Freidlin-Wentzell type large deviation principle for a diffusive predator-prey model with hyperbolic mortality rate perturbed by multiplicative type Gaussian noise. We implement the variational representation developed by Budhiraja and Dupuis to establish the large deviation principle for the solution processes.

Keywords Large deviation principle Laplace principle Stochastic partial differential equations Predator-prey model

©2016 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction The dynamics of interactions between two or more species is of great interest in the mathematical and biological sciences. The first model to describe the density of two populations interacting as a predator-prey system was developed independently by Lotka (1925) and Volterra (1931). Several factors affecting the densities of predator-prey population densities were considered subsequently and several models have been proposed and their qualitative behavior has been analyzed. Chen and Jungel [1] analyzed the existence of non-negative solutions of a parabolic cross-diffusion population model. A predator-prey system with cross-diffusion in heterogeneous habitats was studied by Bendahmane [2]. The existence of solutions for a predator-prey model with mixed boundary conditions was established by Shangerganesh and Balachandran [3]. The study of qualitative behavior such as stability, bifurcation analysis and pattern formations is of great significance to analyze the dynamics of population or cell densities (for instance, see [4, 5]). In [6], Erjaee et al. discussed the stability of equilibrium points and periodic solution for movement of two different species of fish interacting as predator and prey. Also the stability and bifurcations in a diffusive model with predator saturation and competition were analyzed by Sambath et al. [7] and with Smith growth was studied by Sivakumar et al. [8]. Amidst all the factors affecting the biological system, the effect of randomness or probabilistic factors in the predator-prey model allows for better understanding of biodiversity and species interactions and also improves the accuracy of the qualitative properties studied mathematically. Khaminskii et al. [9] studied the qualitative properties of the Lotka-Volterra † Corresponding

author. Email address: [email protected]


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M. Suvinthra, K. Balachandran, M. Sambath / Discontinuity, Nonlinearity, and Complexity 5(3) (2016) 223–237

model with small random perturbations. Li [10] analyzed the impact of noise on pattern formation in a predatorprey model with jump noise. Stochastic factors indeed play a major role in the dynamics of species populations. In nature, there are situations where a drastic change happens affecting a major portion in the environment or a population, but with a very little chance of occurrence. These rare occurrences may have a large impact on the environment and hence cannot be ignored from the study. One is often interested not only in the probability of rare events but also in the characteristic behavior of the system when the rare event occurs. A separate theory termed as “large deviation theory” sprouted from applied probability theory to specifically study the rare events. Large deviation theory is used to study the exponential decline of probability measures of extreme or tail events (see [11–13] for an introductory study). The theory originated in the 1930s when Cramer attempted to tackle the risk of ruin in an insurance company. Subsequent developments were made by the contributions of many mathematicians due to their demand in diverse fields including risk management, information theory, thermodynamics, statistical mechanics and population dynamics. The theory acquired a unified formulation in 1966 after Varadhan framed the large deviation principle (LDP) in terms of the rate function. In the 1970s, Freidlin and Wentzell [14] implemented large deviations to study the small noise asymptotics of dynamical systems perturbed by Markov processes. In the small-noise limit, the solution paths of stochastic differential equations (SDEs) perturbed by a small amount of random effect converge in probability to the trajectory of the corresponding ordinary differential equation (ODE) without randomness. It is large deviations which enable us to quantify the rate at which the solution paths or trajectories converge in the small noise limit. When a critical parameter goes to zero, the probability that the sample path of the diffusion is close to a given rare path decreases exponentially with rate I(·), where the ‘rate function’ I can be expressed in terms of the parameters of the diffusion. The theory developed by Freidlin and Wentzell is based on discretizing the original problem with respect to time, studying the LDP for the discretized problem via contraction principle and finally passing on to the limit. A new technique for establishing the LDP was introduced by Dupuis and Ellis [11] and is widely known in the literature as the weak convergence approach. This approach involves formulation of a controlled system associated with the original process and necessitates the tightness of solutions of the controlled system and in addition the weak convergence of representations of their random perturbations to the deterministic problem. An explicit representation for the rate function is also obtained by implementing this technique. Using the weak convergence approach, Budhiraja and Dupuis [15] developed a variational representation for positive functionals of infinite dimensional Brownian motion using which large deviations can be studied for a variety of stochastic differential equations with small random perturbations. Large deviation methods are used in risk management for the estimation of large portfolio losses in credit risk and portfolio performance in market investment. An LDP for a binomial distribution was studied by Arratia and Gordon [16]. Florens-Landais and Pham [17] established large deviations for an Ornstein-Uhlenbeck model. Champagnat et al. [18] used large deviations argument to analyze dynamical coevolutionary paths. They implemented the theory to study long-time behavior of the diffusion processes when there are multiple attractive evolutionary singularities. Klebaner et al. [19] established a moderate deviation principle for stochastic LotkaVolterra model. The studies on large deviations help to describe the phenomena in the rare occurrence of events as in the case of interacting particle models [20] and protein folding [21]. Zint et al. [22] applied large deviations to predict the way the T-cells recognize foreign antigens and trigger immune response. In population dynamics, large deviation principle is mainly applied to estimate the time of extinction of a species or a disease and thus it helps to take controllability decisions on recovering endangered species and to eradicate the epidemic diseases. Pakdaman et al. [23] applied large deviations to determine the exit points in the diffusion approximations of birth-death processes. Klebaner and Liptser [24] established the large deviations for a stochastic Lotka-Volterra model and applied the results to form a bound for the asymptotics of the time of extinction of prey population. Large deviations for stochastic hybrid systems have been established by Bressloff and Newby [25] using the path-integral representation. The extinction time of an epidemic disease among the affected population was predicted using large deviations and numerical simulations by Kratz et al. [26].


225

In this paper, we intend to establish the large deviation principle for a diffusive predator-prey model with hyperbolic mortality rate perturbed by multiplicative type Gaussian noise using the weak convergence approach via the variational representation introduced by Budhiraja and Dupuis. 2 Abstract formulation We consider the non-dimensionalized predator-prey model with spatial diffusive effect and Holling type-II functional response [27] given by ⎫ suv , x ∈ O,t > 0, ⎪ ⎪ ⎪ β +u ⎪ ⎪ 2 ⎬ uv −γ v + ), x ∈ O,t > 0, vt = η2 Δv + α ( 1 + γv β + u ⎪ ⎪ ⎪ ∂ν u = ∂ν v = 0, x ∈ ∂ O,t > 0, ⎪ ⎪ ⎭ u(0, x) = u0 (x) ≥ 0, v(0, x) = v0 (x) ≥ 0, x ∈ O, ut = η1 Δu + u(1 − u) −

(1)

where O is an open, bounded domain in RN . Here u and v represent the population densities of prey and predator respectively and η1 , η2 are positive constants denoting the diffusion coefficients of the prey-predator populations. Also β is a non-dimensional parameter and corresponds to the ratio of the maximum uptake rate of the prey to its carrying capacity; the parameter s represents the ratio of impact of predation on the relative growth rate of prey to the prey-predator birth rates times the carrying capacity; γ stands for the death rate of the predator. The population densities u and v have been non-dimensionalized respectively via the carrying capacity and the birth rate of predator; and the time t via the birth rate of prey (see [27] for elaborate details). The prescribed Neumann boundary condition means that there is no migration of predators or preys inside or outside the region O. Regarding the initial conditions, the initial values u0 (x) and v0 (x) are assumed to be positive and bounded in the region O. Let T > 0 be a finite time (possibly large) and define J := [0, T ]. Suppose we also take into consideration the effect of randomness for predicting the population densities. Let Ω be the sample space with a filtration F and a probability measure P. Let us assume that the triplet (Ω, F , P) is a probability space with an increasing family {Ft }t∈J of sub-sigma fields of F satisfying the usual assumptions of right continuity and P-completeness. Also assume that the randomness is Gaussian and its coefficients are of multiplicative type, that is, they depend on the predator-prey population densities. Consider the stochastic predator-prey model perturbed by a small parameter ε > 0 as ⎫ √ suv ]dt + εσ1 (t, x, u, v)dW1 (t), x ∈ O,t > 0, ⎪ ⎪ ⎪ β +u ⎪ ⎪ 2 ⎬ √ uv −γ v + ) ]dt + εσ2 (t, x, u, v)dW2 (t), x ∈ O,t > 0, dv = [η2 Δv + α ( 1 + γv β + u ⎪ ⎪ ⎪ ∂ν u = ∂ν v = 0, x ∈ ∂ O,t > 0, ⎪ ⎪ ⎭ x ∈ O, u(0, x) = u0 (x) ≥ 0, v(0, x) = v0 (x) ≥ 0, du = [η1 Δu + u(1 − u) −

(2)

with the population densities u, v and the initial values as random variables. Here σ1 , σ2 represent the multiplicative noise coefficients and W1 (t),W2 (t) are independent Wiener processes taking values in a Hilbert space. We consider the problem (2) for the finite time interval t ∈ J. The function spaces used in this work are as follows: Let L2 (O; R+ ) denote the space of all R+ -valued square integrable functions over the domain O and the norm is defined by ˆ 1 U L2 (O;R+ ) := ( |U (x)|2 dx) 2 . O

226


The space H1 (O; R+ ) is the Sobolev space of all L2 (O; R+ ) valued functions along with their derivatives also belonging to L2 (O; R+ ). Since our domain O is taken to be bounded, the well known Poincare’s inequality can be invoked to obtain an equivalent Sobolev norm: ˆ 1 U H1 (O;R+ ) := ( |∇U (x)|2 dx) 2 . O

For better understanding of the Sobolev spaces and their properties, refer [28]. Likewise L2 (Ω) denotes the space of all random processes with finite expectation. For simplicity of notations, let · be the L2 (O; R+ )−norm We formulate the system (2) into an abstract equation by and ∇ · denote the H1 (O; R+) norm. defining the u0 (x) u . Define the following operators: Let U = ∈ R+ × R+ =: R2+ and correspondingly U0 (x) = v0 (x) v operators −u2 − βsuv ∂ν u − η1 Δ − 1 0 +u , ∂ν U = , F(U ) := , A := −γ v2 uv ∂ν v 0 − η2 Δ α ( 1+ γ v + β +u )

σ (t, x,U ) :=

σ1 (t, x, u, v) 0 σ2 (t, x, u, v) 0

and W (t) :=

W1 (t) . W2 (t)

The system (2) can then be expressed in an abstract form as ⎫ √ dU + AU dt = F(U )dt + εσ (t,U )dW (t), x ∈ O,t ∈ J, ⎬ ∂ν U = 0, x ∈ ∂ O,t ∈ J, ⎭ x ∈ O. U (0, x) = U0 (x),

(3)

Let Q be a symmetric, positive and a trace class operator on L2 (O). Also let W (·) be an L2 -space valued Wiener process with covariance operator Q. Define the space H0 = Q1/2 L2 (O). Then H0 is a Hilbert space with the inner product defined by (U,V )0 = (Q−1/2U, Q−1/2V ), ∀ U,V ∈ L2 (O). Let LQ denote the space of all linear operators S such that SQ1/2 is a Hilbert-Schmidt operator from L2 (O) to L2 (O). Let the norm on the space LQ be S2LQ := Tr(SQS∗ ). We shall also make the following hypotheses on the noise coefficient σ : J × H1 (O) → LQ (H0 ; L2 (O)) : (A1) (Continuity): The function σ ∈ C(J × H1 (O); LQ (H0 ; L2 (O)). (A2) (Lipschitz Continuity): There exists a positive constant, say C1 , such that for all U,V ∈ H1 (O), σ (t,U ) − σ (t,V )2LQ ≤ C1 ∇(U −V )2 , t ∈ (0, T ).

(4)

(A3) (Linear Growth Property): There exists a positive constant C2 such that σ (t,U )2LQ ≤ C2 (1 + ∇U 2 ), t ∈ (0, T ).

(5)

Our aim is to establish the large deviation principle for the solution process {U ε }ε >0 of (3) under the assumption that a positive strong solution to the considered system exists and is pathwise unique. We first intend to prove the following lemmas associated with the linear and nonlinear operators A and F(·).


227

Lemma 1. For U = (u, v) ∈ H1 (O; R2+ ), the following lower estimate holds for the operator A: (AU,U ) ≥ η ∇U 2 − u2 ,

(6)

where η = η1 ∧ η2 (= min{η1 , η2 }). Proof. From the definition of the linear operator A, it follows that (AU,U ) = −η1 (Δu, u) − (u, u) − η2 (Δv, v) = η1 ∇u2 − u2 + η2 ∇v2 ≥ η ∇U 2 − u2 ,

(7)

where η = min{η1 , η2 } and thus (6) holds true for U ∈ H1 (O; R2+ ). The above lemma means that the bilinear form a(U, U˜ ) := (AU, U˜ ) for U, U˜ ∈ H1 (O; R2+ ) associated with the linear operator A is coercive. In the case of the nonlinear operator F(·), we can observe the boundedness and Lipschitz continuity as follows: Lemma 2. For U, U˜ ∈ H1 (O; R2+ ), the following properties are satisfied: 1. Boundedness: Let U = (u, v). Then (F(U ),U ) ≤ α U 2 .

(8)

˜ Then 2. Lipschitz Continuity: Take U˜ = (u, ˜ v) ˜ and let w = (wu , wv ) := U − U. 2 3η ∇w2 + u + u ˜ 2 w2 2(F(U )−F(U˜ ), w) ≤ 4 η 1 2(s2 + α 2 ) + 2{(s2 + α 2 )CO2 + + 2 + γ 2 }w2 + U 2 w2 , η ηβ 2

(9)

where CO = (vol(O))1/4 . Proof. First consider the inner product uv suv −γ v2 , u) + α ( + , v) β +u 1 + γv β + u uv uv γ v2 , u) − α ( , v) + α ( , v). = −(u2 , u) − s( β +u 1 + γv β +u

(F(U ),U ) = (−u2 −

Since U = (u, v) ∈ H1 (O; R2+ ), u takes only positive values from R and so the inner product (u2 , u) is always positive leading to −(u2 , u) ≤ 0. Likewise, the second and third inner products turn out to be zero resulting in uv , v) ≤ α (v, v) ≤ α U 2 , (F(U ),U ) ≤ α ( β +u thus leading to (8). In order to establish the Lipschitz continuity (9), we consider the inner product suv su˜v˜ + u˜2 + , wu ) β +u β + u˜ uv u˜v˜ γ v2 γ v˜2 + + − , wv ) + 2α (− 1 + γ v β + u 1 + γ v˜ β + u˜ su˜v˜ −suv + , wu ) =2(−u2 + u˜2 , wu ) + 2( β + u β + u˜ u˜v˜ −γ v2 uv γ v˜2 + , wv ) + 2α ( − , wv ) + 2α ( 1 + γ v 1 + γ v˜ β + u β + u˜ =IP1 + IP2 + IP3 + IP4 .

˜ w) =2(−u2 − 2(F(U ) − F(U),

(10)

228


First consider the inner product IP1 and use Holder’s and Ladyzhenskaya inequalities along with the Young’s inequality to obtain ˜ − u), ˜ wu ) IP1 = − 2(u2 − u˜2 , wu ) = −2((u + u)(u 2 ˜ ≤ 2u + uu ˜ − u ˜ L4 (O) wu L4 (O) = 2u + uw u L4 (O)

2 2 η ˜ 2 wu 2 ∇wu 2 ≤ ∇wu 2 + wu 2 u + u ≤ √ u + u ˜ 2. 4 η 2

(11)

Now to evaluate the second inner product, first consider |

su˜v˜ −suv(β + u) ˜ + su˜v( ˜ β + u) −suv + | =| | β + u β + u˜ (β + u)(β + u) ˜ ˜ + sβ u˜v˜ + suu˜v˜ −sβ uv − suuv | =| (β + u)(β + u) ˜ suu(v ˜ − v) ˜ ˜ sβ (uv − u˜v) |+| |. ≤| (β + u)(β + u) ˜ (β + u)(β + u) ˜

The second term in the above inequality can be easily bounded as |

suu(v ˜ − v) ˜ u u˜ | ≤ s| || ||v − v| ˜ ≤ s|v − v| ˜ = s|wv |. (β + u)(β + u) ˜ β + u β + u˜

For the first term, consider |

sβ u(v ˜ ˜ − v) ˜ + sβ v(u − u) ˜ sβ (uv − u˜v) |=| | (β + u)(β + u) ˜ (β + u)(β + u) ˜ sβ 2 v(u − u) ˜ − v) ˜ ˜ sβ u(v |+| | ≤| (β + u)(β + u) ˜ β (β + u)(β + u) ˜ s ˜ ≤ s|v − v| ˜ + |v||u − u| β s = s|wv | + |v||wu |, β

as was done to bound the second term. Combining both the bounds, we have |

su˜v˜ s −suv + | ≤ 2s|wv | + |v||wu |. β + u β + u˜ β

Hence the inner product IP2 can be estimated by proceeding similar to that of IP1 and in addition making use of the concept of imbeddings on L p (1 ≤ p ≤ ∞) spaces for bounded domains as su˜v˜ su˜v˜ −suv −suv + , wu ) ≤ 2 + 4/3 wu L4 β + u β + u˜ β + u β + u˜ L s ≤ 22s|wv | + |v||wu |L4/3 wu L4 β s ≤ 2(2swv L4/3 + |v||wu |L4/3 )wu L4 β s ≤ 2(2sCO wv + vwu L4 )wu L4 β 2s = 4sCO wv wu L4 + vwu 2L4 β 2s ≤ 2s2CO2 wv 2 + 2wu 2L4 + vwu 2L4 β

IP2 = 2(


229

2 2s ≤ 2s2CO2 wv 2 + √ wu ∇wu + √ vwu ∇wu 2 β 2 2 2s2 η η ≤ 2s2CO2 wv 2 + wu 2 + ∇wu 2 + v2 wu 2 + ∇wu 2 . 2 η 4 ηβ 4 Thus IP2 ≤ 2s2CO2 wv 2 +

2 2s2 η wu 2 + v2 wu 2 + ∇wu 2 . η ηβ 2 2

(12)

For estimating IP3 , consider the term |

−γ v2 (1 + γ v) γ v˜2 ˜ + γ v˜2 (1 + γ v) γ v2 + γ 2 v2 v˜ − γ v˜2 − γ 2 vv˜2 −γ v2 + | =| | = | | 1 + γ v 1 + γ v˜ (1 + γ v)(1 + γ v) ˜ (1 + γ v)(1 + γ v) ˜ γ (v2 − v˜2 ) γ 2 vv(v ˜ − v) ˜ |+| | ≤| (1 + γ v)(1 + γ v) ˜ (1 + γ v)(1 + γ v) ˜ γ v(v − v) ˜ γ v(v ˜ − v) ˜ γ 2 vv(v ˜ − v) ˜ |+| |+| | ≤| (1 + γ v)(1 + γ v) ˜ (1 + γ v)(1 + γ v) ˜ (1 + γ v)(1 + γ v) ˜ ˜ ≤ |v − v| ˜ + |v − v| ˜ + γ 2 |v − v| ≤ (2 + γ 2 )|wv |,

and at once, IP3 can be bounded as IP3 ≤ 2

−γ v2 γ v˜2 + wv ≤ 2(2 + γ 2 )|wv |wv = 2(2 + γ 2 )wv 2 . 1 + γ v 1 + γ v˜

(13)

An estimate for the fourth inner product IP4 can be obtained by recapitulating the procedure of IP2 as IP4 ≤ 2α 2CO2 wu 2 +

2 2α 2 η wv 2 + u2 wv 2 + ∇wv 2 . 2 η ηβ 2

(14)

Combining all the inequalities (11)-(14), the required inner product in (10) becomes 2 2 η ˜ 2 wu 2 + 2s2CO2 wv 2 + wu 2 2(F(U ) − F(U˜ ), w) ≤ ∇wu 2 + u + u 4 η η 2 2s η + v2 wu 2 + ∇wu 2 + 2(2 + γ 2 )wv 2 + 2α 2CO2 wu 2 ηβ 2 2 2 2 2α η + wv 2 + u2 wv 2 + ∇wv 2 2 η ηβ 2 2 2 η = ∇wu 2 + u + u ˜ 2 wu 2 + 2(s2 + α 2 )CO2 w2 + w2 4 η η 2 2 2(s + α ) η + (u2 + v2 )w2 + ∇w2 + 2(2 + γ 2 )wv 2 2 ηβ 2 3η 2 1 ≤ ∇wu 2 + u + u ˜ 2 wu 2 + 2{(s2 + α 2 )CO2 + + 2 + γ 2 }w2 4 η η 2 2 2(s + α ) + (U 2 )w2 , ηβ 2 which is the required bound (9).

(15)

230


3 Large deviation principle We use the variational representation formulated by Budhiraja and Dupuis for positive functionals of infinite dimensional Brownian motion using which an LDP for the solution processes {U ε }ε >0 of (3) can be established in an elegant way. Let A denote the class of H0 −valued {Ft }−predictable processes φ satisfying ´T 2 0 φ (s)0 ds < ∞ a.s. Define SM := {h ∈ L2 (0, T ; H0 ) :

ˆ

T 0

h(s)20 ds ≤ M}.

Then it could be observed that the set SM endowed with the weak topology is a Polish space (see [29]). Also define the space AM = {φ ∈ A : φ (ω ) ∈ SM , P − a.s.}. Intuitively AM is a class of admissible controls using which we control the solution processes to the desired result. Let H be any Hilbert space, Z be a Polish space and consider a measurable map G ε : C(J; H) → Z . The result developed by Budhiraja and Dupuis originally establishes the Laplace principle for the considered processes {U ε }ε >0 of strong solutions of (3). They made the following assumption on G ε under which Laplace principle holds: Assumption 3.1. There exists a measurable map G 0 : C(J; H) → Z such that the following hold: 1. Consider 0 < M < ∞ and a family {hε } ⊂ÁM such that hε converges in distribution´ (as SM −valued · · random elements) to h. Then G ε (W (·) + √1ε 0 hε (s)ds) converges in distribution to G 0 ( 0 h(s)ds). 2. For every M < ∞, the set ˆ · ΓM := {G ( h(s)ds) : h ∈ SM } 0

(16)

0

is a compact subset of Z . The following is the main theorem developed by Budhiraja and Dupuis [15]. Theorem 3. Let X ε := G ε (W (·)). Suppose that {G ε } satisfies Assumption 3.1. Then the family {X ε }ε >0 satisfies the Laplace principle in Z with rate function I : Z → [0, ∞] defined by 1 { I( f ) := inf ´ · 2 0 {h∈L (0,T ;H0 ): f =G ( 0 h(s)ds)} 2

ˆ 0

T

h(s)20 ds},

(17)

where the infimum over the empty set is taken to be infinity. For our problem, Z is the solution space C(J; L2 (O)) ∩ L2 (0, T ; H1 (O)) and H is the Lebesgue space Since we work on a Polish space, the fact that Laplace principle and LDP are equivalent in a Polish space leads us to the desired result. In [30], Budhiraja and Dupuis have established that solutions of a class of ε stochastic differential equations driven by Hilbert space valued Wiener process ´ · ε can be written as {G (W (·))}. 1 ε ε ε Define G (W (·)) =: U (·), the strong solution of (3). Then G (W (·) + √ε 0 h (s)ds) stands for the solution of the perturbed stochastic differential equation with a control h ∈ AM , 0 < M < ∞: L2 (O).

√ dUhε + AUhε dt = F(Uhε )dt + σ (t,Uhε )hdt + εσ (t,Uhε )dW (t)

(18)


231

with the same initial and boundary conditions as that of (3). By virtue of Girsanov’s theorem, the existence of pathwise unique strong solution to (3) leads to the same existence result for (18), but under a different probability measure (see [29, Lemma 4.1] for a proof of similar kind). Next define the map G 0 : C(J; H0 ) → Z by ´·

Uh , if ζ = 0 h(s) ds for some h ∈ L2 (0, T ; H0 ) (19) G 0 (ζ ) = 0, otherwise, where Uh represents the solution of the deterministic controlled equation dUh + AUh dt = F(Uh )dt + σ (t,Uh )h dt

(20)

with Uh (0) = U0 and ∂ν Uh = 0. To establish the LDP for the solution process {U ε }ε >0 , we use the Assumption 3.1 in the following lemmas. In Assumption 3.1, (ii) is a compactness criterion and it is to be noticed that it has a coincidence with the fact that the level set for a good rate function is compact. Lemma 4 (Compactness). Let M > 0 be a finite number. Define ΓM = {Uh ∈ Z : h ∈ SM },

(21)

where Uh is the unique solution in Z of the equation (20). Then ΓM is compact in Z . Proof. Let {hn } be a sequence in SM and let Uhn denote the solution of the differential equation with control hn as dUhn + AUhn dt = F(Uhn )dt + σ (t,Uhn )hn dt.

(22)

The weak compactness of SM assures that there exists a subsequence of {hn } which converges to a limit h weakly in L2 (0, T : H0 ). Denote this subsequence again by {hn } for notational simplicity. Consider the solution Uh of the equation (20) and take L2 −inner product with Uh to obtain dUh (t)2 + 2(AUh (t),Uh (t))dt = 2(F(Uh (t)),Uh (t))dt + 2(σ (t,Uh (t))h(t),Uh (t))dt. Using (6) and (8), and also the Cauchy-Schwarz inequality, one gets dUh (t)2 + 2η ∇Uh (t)2 ≤ 2Uh (t)2 + 2α Uh (t)2 + 2σ (t,Uh (t))LQ h(t)0 Uh (t). Integrating and applying Young’s inequality along with the hypothesis (5) on σ , we get ˆ t ˆ t 2 2 2 ∇Uh (s) ds ≤U0 + (2 + 2α ) Uh (s)2 ds Uh (t) + 2η 0 0 ˆ t ˆ C2 t 2 h(s)20 Uh (s)2 ds. + η (1 + ∇Uh (s) )ds + η 0 0 Simplifying and applying Gronwall’s inequality yields ˆ T ˆ C2 T 2 2 2 ∇Uh (s) ds ≤ C{U0 + η T } exp(2 + 2α + h(s)20 ds), sup Uh (t) + η η 0 t∈J 0 ´T where C is an arbitrary positive constant. Since h ∈ SM , we have 0 h(s)20 ds ≤ M and so the above bound becomes ˆ T C2 T M 2 ∇Uh (s)2 ds ≤ C{U0 2 + η T } exp(2 + 2α + ) (23) sup Uh (t) + η η t∈J 0

232


and hence the bound is uniform in n. Next let wn = Uhn −Uh . Then wn would be the solution of the equation dwn + Awn dt = [F(Uhn ) − F(Uh )]dt + [σ (t,Uhn )hn − σ (t,Uh )h]dt. Taking inner product of the above equation with wn and then integrating lead to ˆ

2

wn (t) + 2

0

t

ˆ (Awn (s), wn (s))ds =2

t

(F(Uhn (s)) − F(Uh (s)), wn (s))ds ˆ t + 2 (σ (s,Uhn (s))hn (s) − σ (s,Uh (s))h(s), wn (s))ds. 0

0

Making use of the estimates (6) and (9) on the linear and nonlinear operators A and F(·), we have 2

ˆ

t

∇wn (s)2 ds wn (t) + 2η 0 ˆ ˆ t 3η t 2 wn (s) ds + ∇wn (s)2 ds ≤2 4 0 0 ˆ t ˆ 1 2 t 2 2 2 2 2 2 Uhn (s) +Uh (s) wn (s) ds + 2{(s + α )CO + + 2 + γ } wn (s)2 ds + η 0 η 0 ˆ t ˆ 2(s2 + α 2 ) t 2 2 Uhn (s) wn (s) ds + 2 (σ (s,Uhn (s))hn (s) − σ (s,Uh (s))h(s), wn (s))ds. + ηβ 2 0 0

(24)

Consider the integrand in the last term of the above estimate and apply Cauchy-Schwarz inequality, the Lipschitz continuity assumption (4) on σ and also the Young’s inequality to get subsequently 2(σ (s,Uhn (s))hn (s) − σ (s,Uh (s))h(s), wn (s)) = 2((σ (s,Uhn (s)) − σ (s,Uh (s)))hn (s), wn (s)) + 2(σ (s,Uh (s))(hn (s) − h(s)), wn (s)) ≤ 2σ (s,Uhn (s)) − σ (s,Uh (s))LQ hn (s)0 wn (s) + 2σ (s,Uh (s))(hn (s) − h(s))wn (s) 4C1 η hn (s)20 wn (s)2 + σ (s,Uh (s))(hn (s) − h(s))2 + wn (s)2 . ≤ ∇wn (s)2 + 4 η Therefore (24) becomes ˆ t ∇wn (s)2 ds wn (t)2 + η ˆ ˆ t 0 4 t 2 wn (s) ds + (Uhn (s)2 + Uh (s)2 )wn (s)2 ds ≤2 η 0 0 ˆ t ˆ 1 2(s2 + α 2 ) t 2 2 2 2 2 wn (s) ds + Uhn (s)2 wn (s)2 ds + 2{(s + α )CO + + 2 + γ } 2 η ηβ 0 ˆ0 t ˆ t ˆ 4C1 t 2 2 2 hn (s)0 wn (s) ds + σ (s,Uh (s))(hn (s) − h(s)) ds + wn (s)2 ds. + η 0 0 0 On simplifying, we get 2

wn (t) + η

ˆ

t 0

2

ˆ

t

ˆ

t

wn (s) ds + K2 (Uhn (s)2 + Uh (s)2 )wn (s)2 ds 0 0 ˆ t ˆ 4C1 t 2 2 hn (s)0 wn (s) ds + σ (s,Uh (s))(hn (s) − h(s))2 ds, + η 0 0

∇wn (s) ds ≤ K1

2

(25)


where K1 = 2( 72 + (s2 + α 2 )CO2 + obtain wn (t)2 + η

ˆ

t 0

1 η

2

233

2

+α ) + γ 2 ); K2 = 2( η2 + (s ηβ 2 ). Finally applying Gronwall’s inequality, we

∇wn (s)2 ds ≤ C

ˆ

T

σ (s,Uh (s))(hn (s) − h(s))2 0 ˆ t ˆ 4C1 t 2 2 × exp(K1 T + K2 (Uhn (s) + Uh (s) )ds + hn (s)20 ds) η 0 0

(26)

for some positive constant C. Since σ (·, ·)Q1/2 is a Hilbert-Schmidt operator on L2 (O), it is compact and so the weak convergence of hn → h in SM implies the convergence of σ (·, ·)hn → σ (·, ·)h in L2 (O). This convergence along with the uniform boundedness in (23) results in 2

sup wn (t) + η t∈J

ˆ

t 0

∇wn (s)2 ds → 0 as n → ∞.

(27)

Hence wn → 0 in Z leading to the compactness of ΓM in Z . of (18) with control hε in´ place of h. Then the solution Uhεε admits the Let Uhεε denote the strong solution ´ · · representation Uhεε = G ε (W (·) + √1ε 0 hε (s)ds) . Consider G 0 ( 0 h(s)ds) as defined in (19). With these, the Assumption 3.1 - (i) of weak convergence of the representations G ε (·) and G 0 (·) is proved in the forthcoming lemma. Lemma 5. Let´{hε : ε > 0} ⊂ AM converge in distribution´ to h as SM −valued random element. Then · · G ε (W (·) + √1ε 0 hε (s)ds) converges in distribution to G 0 ( 0 h(s)ds) in Z as ε → 0. Proof. Since SM is a Polish space, using the Skorokhod representation theorem, we could construct processes ˜ W˜ ε ) such that the joint distribution of (h˜ ε , W˜ ε ) is the same as that of (hε ,W ε ) and the distribution of h˜ (h˜ ε , h, coincides with that of h and h˜ ε → h˜ a.s. in the topology (refer [29]). Let wε = Uhεε −Uh . Then wε represents the solution of the stochastic controlled differential equation √ dwε + Awε dt = (F(Uhεε ) − F(Uh ))dt + (σ (t,Uhεε )hε − σ (t,Uh )h)dt + εσ (t,Uhεε )dW (t). Applying Itô formula [31] to the process wε 2 , we get ε

ˆ

t

w (t) + 2 (Awε (s), wε (s))ds ˆ t 0 =2 (F(Uhεε (s)) − F(Uh (s)), wε (s))ds 0 ˆ t + 2 (σ (s,Uhεε (s))hε (s) − σ (s,Uh (s))h(s), wε (s))ds ˆ0 t ˆ t √ ε ∗ ε Tr(σ (s,Uhε (s))Qσ (s,Uhε (s)))ds + 2 ε (wε (s), σ (s,Uhεε (s))dW (s)). +ε 2

0

0

Using the coercivity (6) on A, the Lipschitz continuity (9) on F(·) and also the Cauchy-Schwarz inequality, one

234


gets ε

ˆ

t

∇wε (s)2 ds w (t) + 2η 0 ˆ ˆ t 3η t wε (s)2 ds + ∇wε (s)2 ds ≤2 4 0 0 ˆ t ˆ 1 2 t ε 2 ε 2 2 2 2 2 Uhε (s) +Uh (s) w (s) ds + 2((s + α )CO + + 2 + γ ) wε (s)2 ds + η 0 η 0 ˆ ˆ t 2(s2 + α 2 ) t ε 2 ε 2 Uhε (s) w (s) ds + 2 σ (s,Uhεε (s)) − σ (s,Uh (s))LQ hε (s)0 wε (s)ds + ηβ 2 0 0 ˆ t ˆ t ε ε Tr(σ (s,Uhεε (s))Qσ ∗ (s,Uhεε (s)))ds + 2 σ (s,Uh (s))(h (s) − h(s))w (s)ds + ε 0 0 ˆ t √ ε ε + 2 ε (w (s), σ (s,Uhε (s))dW (s)). 2

0

On using the Lipschitz continuity (A2) and linear growth assumption (A3) on σ (·, ·), applying Young’s inequality and simplifying, we get ε

ˆ

t

∇wε (s)2 ds w (t) + η 0 ˆ t ˆ t ε 2 w (s) ds + K2 (Uhεε (s)2 + Uh (s)2 )wε (s)2 ds ≤K1 0 0 ˆ t ˆ 4C1 t ε 2 ε 2 h (s)0 w (s) ds + σ (s,Uh (s))(hε (s) − h(s))2 ds + η 0 0 ˆ t ˆ t √ ε 2 + ε C2 (1 + ∇Uhε (s) )ds + 2 ε | (wε (s), σ (s,Uhεε (s))dW (s))|, 2

0

(28)

0

2

2

+α ) where K1 = 2( 72 + (s2 + α 2 )CO2 + η1 + γ 2 ); K2 = 2( η2 + (s ηβ 2 ). Define the stopping time

ˆ

τN,ε := inf{t :

t 0

[∇Uhεε (s)2 + ∇Uh (s)2 ]ds > N or sup Uh (s)2 > N or sup Uhεε (s)2 > N}. 0≤s≤t

0≤s≤t

Taking supremum in (28) over the interval [0, T ∧ τN,ε ] yields sup

0≤t≤T ∧τN,ε

ˆ

wε (t)2 + η ˆ

T ∧τN,ε

0 T ∧τN,ε

∇wε (s)2 ds ˆ

T ∧τN,ε

(Uhεε (s)2 + Uh (s)2 )wε (s)2 ds ˆ T ∧τN,ε ˆ 4C1 T ∧τN,ε ε 2 ε 2 h (s)0 w (s) ds + σ (s,Uh (s))(hε (s) − h(s))2 ds + η 0 0 ˆ t √ + ε C2 (T + N) + 2 ε { sup | (wε (s), σ (s,Uhεε (s))dW (s))|}.

≤K1

0

ε

2

w (s) ds + K2

0

0≤t≤T ∧τN,ε

(29)

0

Using the Burkholder-Davis-Gundy inequality to bound the expectation of the last stochastic integral term on


the right hand side, we get √ 2 εE

ˆ |

sup

0≤t≤T ∧τN,ε

ˆ √ ≤ 4 ε E[

√ ≤ 4 ε E[ √ ≤2 ε

t 0

(wε (s), σ (s,Uhεε (s))dW (s))|

T ∧τN,ε 0

1/2 wε (s)2 σ (s,Uhεε (s))2LQ ds ] ε

sup

0≤t≤T ∧τN,ε

E(

235

sup

2

w (t)

0≤t≤T ∧τN,ε

ˆ 0

T ∧τN,ε

1/2 σ (s,Uhεε (s))2LQ ds

]

wε (t)2 ) +C2 (T + N)

< ∞.

(30)

Using Gronwall’s inequality in (29) and the definition of τN,ε yields

ˆ T ∧τN,ε ˆ T ∧τN,ε ∇wε (s)2 ds ≤ C σ (s,Uh (s))(hε (s) − h(s))2 ds + ε C2 (T + N) sup wε (t)2 + η 0≤t≤T ∧τN,ε

0

0

ˆ

√ +2 ε [

sup

0≤t≤T ∧τN,ε

|

t 0

(w

ε

(s), σ (s,Uhεε (s))dW (s))|]

4C1 exp K1 T + 2K2 N + M η

(31)

for arbitrary positive constant C. Let N be fixed. Then it could be observed that C . N Thus we obtain T ∧ τN,ε → T as N → ∞. Now recalling that hε → h a.s. in the weak topology of SM and that the operator σ (·, ·) is compact in L2 (O), it is clear that ˆ T ∇wε (s)2 ds] → 0 as ε → 0. E[ sup wε (t)2 + η lim inf P{τN,ε = T } ≥ 1 − ε →0

0≤t≤T

0

Using Markov’s inequality for any positive constant δ > 0,

ˆ T ˆ T 1 ε 2 ε 2 ε 2 ∇w (s) ds ≥ δ ≤ E[ sup w (t) + η ∇wε (s)2 ds] → 0 as ε → 0. P sup w (t) +η δ 0≤t≤T 0≤t≤T 0 0 Thus sup

0≤t≤T

Uhεε (t) −Uh (t)2 +η

ˆ 0

T

∇(Uhεε (s) −Uh (s))2 ds → 0

(32)

in probability as ε → 0 and hence the weak convergence of Uhεε → Uh in Z follows and the proof is complete. Thus we have arrived at the large deviation principle for the solution processes of (3) and the main result we have established is the following: Theorem 6. Let {U ε (·)} be the pathwise unique strong solution of the stochastic system (3). Then with the assumptions (A1)-(A3) on σ , the family {U ε } satisfies the large deviation principle in Z := C([0, T ]; L2 (O)) ∩ L2 (0, T ; H1 (O)) with a good rate function

ˆ T

1 2 h(s)0 ds , (33) I( f ) = inf ´ {h∈L2 (0,T :H0 ): f =G 0 ( 0· h(s) ds)} 2 0 ´· where the infimum over an empty set is taken as ∞ and G 0 ( 0 h(s) ds) denotes the solution Uh of the system dUh + AUh dt = F(Uh )dt + σ (t,U ) h dt with Uh (0) = U0 and h ∈ AM .

(34)

236


Acknowledgements The first author would like to thank the Department of Science and Technology (DST), Government of India, New Delhi for their financial support under the INSPIRE Fellowship Scheme. The work of the second author is supported by Defence Research and Development Organization (DRDO), Government of India. The work of the third author is supported by University Grants Commission (UGC), New Delhi, Government of India under the Special Assistance Programme (SAP - 1). References [1] Chen, L. and Jungel, A. (2006), Analysis of a parabolic cross-diffusion population model without self-diffusion, Journal of Differential Equations, 224, 39–59. [2] Bendahmane, M. (2010), Weak and classical solutions to predator-prey system with cross-diffusion, Nonlinear Analysis, 73, 2489–2503. [3] Shangerganesh, L. and Balachandran, K. (2011), Existence and uniqueness of solutions of predator-prey type model with mixed boundary conditions, Acta Applicandae Mathematicae, 116, 71–86. [4] Leonetti, M., Boedec, G. and Jaeger, M. (2013), Breathing instability in biological cells, patterns of membrane proteins, Discontinuity, Nonlinearity, and Complexity, 2, 75–84. [5] Sambath, M. and Balachandran, K. (2013), Spatiotemporal dynamics of a predator-prey model incorporating a prey refuge, Journal of Applied Analysis and Computation, 3, 71–80. [6] Erjaee, G.H., Ostadzad, M.H., Okuguchi K. and Rahimi, E. (2013), Fractional differential equations system for commercial fishing under predator-prey interaction, Journal of Applied Nonlinear Dynamics, 2, 409–417. [7] Sambath, M., Gnanavel, S. and Balachandran, K. (2013), Stability and Hopf bifurcation of a diffusive predator-prey model with predator saturation and competition, Applicable Analysis, 92, 2439-2456. [8] Sivakumar, M., Sambath, M. and Balachandran, K. (2015), Stability and Hopf bifurcation analysis of a diffusive predator-prey model with Smith growth, International Journal of Biomathematics, 8, 1550013. [9] Khaminskii, R.Z., Klebaner, F.C. and Liptser, R. (2003), Some results on the Lotka-Volterra model and its small random perturbations, Acta Applicandae Mathematicae, 78, 201–206. [10] Li, A-W. (2011), Impact of noise on pattern formation in a predator-prey model, Nonlinear Dynamics, 66, 689–694. [11] Dupuis, P. and Ellis, R.S. (1997), A Weak Convergence Approach to the Theory of Large Deviations, WileyInterscience: New York. [12] Dembo, A. and Zeitouni, O. (2007), Large Deviations Techniques and Applications, Springer, New York. [13] Varadhan, S.R.S. (2008), Large deviations, The Annals of Probability, 36, 397–419. [14] Freidlin, M.I. and Wentzell, A.D. (1970), On small random perturbations of dynamical systems, Russian Mathematical Surveys, 25, 1–55. [15] Budhiraja, A. and Dupuis, P. (2000), A variational representation for positive functionals of infinite dimensional Brownian motion, Probability and Mathematical Statistics, 20, 39–61. [16] Arratia, R. and Gordon, L. (1989), Tutorial on large deviations for the binomial distribution, Bulletin of Mathematical Biology, 51, 125–131. [17] Florens-Landais, D. and Pham, C.H. (1999), Large deviations in estimation of an Ornstein-Uhlenbeck model, Journal of Applied Probability, 36, 60-77. [18] Champagnat, N., Ferriere, R. and Meleard, S. (2006), Unifying evolutionary dynamics: From individual stochastic processes to macroscopic models, Theoretical Population Biology, 69, 297–321. [19] Klebaner, F.C., Lim, A. and Liptser, R. (2007), FCLT and MDP for stochastic Lotka-Volterra model, Acta Applicandae Mathematicae, 97, 53–68. [20] Bertini, L., De Sole, A., Gabrielli, D., Jona-Lasinio, G. and Landim, C. (2007), Large deviations of the empirical current in interacting particle systems, Theory of Probability and its Applications, 51, 2–27. [21] Weber, J.K., Jack, R.L., Schwantes, C.R. and Pande, V.S. (2014), Dynamical phase transitions reveal amyloid-like states on protein folding landscapes, Biophysical Journal, 107, 974–982. [22] Zint, N., Baake, E. and den Hollander, F. (2008), How T-cells use large deviations to recognize foreign antigens, Journal of Mathematical Biology, 57, 841–861. [23] Pakdaman, K., Thieuller, M. and Wainrib, G. (2010), Diffusion approximation of birth-death processes: Comparison in terms of large deviations and exit points, Statistics and Probability Letters, 80, 1121–1127. [24] Klebaner, F.C. and Liptser, R. (2001), Asymptotic analysis and extinction in a stochastic Lotka-Volterra model, The Annals of Applied Probability, 11, 1263–1291. [25] Bressloff, P.C. and Newby, J.M. (2014), Path integrals and large deviations in stochastic hybrid systems, Physical Review E, 89, 042701.


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Nonlinear Parametrizations of Outgoing Longwave Radiation in Zero-Dimensional Energy Balance Models Dmitry V. Kovalevsky† Nansen International Environmental and Remote Sensing Centre, 14th Line 7, office 49, Vasilievsky Island, 199034 St. Petersburg, Russia Saint Petersburg State University, Universitetskaya Emb. 7-9, 199034 St. Petersburg, Russia Submission Info Communicated by Dimitry Volchenkov Received 15 December 2015 Accepted 26 January 2016 Available online 1 October 2016 Keywords Energy balance model Outgoing longwave radiation Nonlinearity Exact analytical solution

Abstract A one-layer and two-layer zero-dimensional (0D) energy balance models (EBMs) of the global climate system with different approximations for parametrization of outgoing longwave radiation (OLR) are considered. Three alternative approximations for parametrizing the OLR are explored in detail: (i) the (conventional) linear approximation, (ii) the quadratic approximation, and (iii) the ‘exact’ (power 4) model. In case of one-layer 0D EBM, exact analytical solutions are derived in closed form for all three alternative approximations for parametrizing the OLR. In the numerical examples provided, the deviations of the linear approximation from the ‘exact’ model are visible, while the quadratic approximation is virtually indistinguishable from the ‘exact’ model. ©2016 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Starting from seminal papers by Budyko [1] and Sellers [2], energy balance models (EBMs) have been playing a prominent role in climate science. Although EBMs cannot produce as detailed climate information as climate models of further generations (e.g. as the atmosphere–ocean coupled general circulation models (AOGCMs)), they are still successfully used in basic climate research until now [3–5]. Compared to AOGCMs, EBMs are much simpler from mathematical standpoint (to the extent that the simplest EBMs sometimes allow derivation of analytical solutions in closed form) and much less demanding from computational resources standpoint. It is well known that the climate system is strongly nonlinear. These pronounced nonlinearities are accounted for in existing models of the climate system (and of its various components) at different levels of complexity [6– 11]. However, many existing EBMs [1, 12, 13] adopt the linear approximation for outgoing longwave radiation (OLR). At the same time, modelling results reported in [14–18] indicate that accounting for OLR in nonlinear approximatios is important in many cases, particularly for properly describing the meridional heat transfer. The aim of the present paper is to contribute to the discussion on the importance of accounting for nonlinearities in parametrizations of OLR in simple zero-dimensional (0D) EBMs. The rest of the paper is organized † Corresponding

author. Email address: [email protected], d v [email protected] ISSN 2164 − 6376, eISSN 2164 − 6414/$-see front materials © 2016 L&H Scientific Publishing, LLC. All rights reserved. DOI : 10.5890/DNC.2016.09.004

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Dmitry V. Kovalevsky/ Discontinuity, Nonlinearity, and Complexity 5(3) (2016) 239–249

as follows. In Sec. 2 a simple (probably, the simplest possible) one-layer 0D EBM is considered in detail in three alternative approximations for parametrizing the OLR: (i) the (conventional) linear approximation, (ii) the quadratic approximation, and (iii) the ‘exact’ (power 4) model. In Sec. 3 a similar analysis is made for nonlinear extensions of a slightly more complex two-layer 0D EBM (the linear version of the model is considered in [5]). Sec. 4 concludes. 2 A one-layer EBM with nonlinear OLR 2.1

Alternative parametrizations of OLR in EBMs

The simplest 0D EBM can be written in a general form CT˙ = F − I

(1)

where T is the global mean temperature, C is the effective heat capacity (per unit area), F is the sum of incoming shortwave radiation and the radiative forcing (e.g. caused by greenhouse effect), and I is the OLR. Following [14, 15], we adopt a parametrization for I of the form I = δT4

(2)

where δ is an empirical constant (in numerical estimates provided in Sec. 2.4 below it will be set equal to the Stefan–Boltzmann constant: δ = σ = 5.67 · 10−8 W/m2 K4 ). By substituting Eq. (2) into Eq. (1), we get a nonlinear ordinary differential equation (ODE) CT˙ = F − δ T 4 .

(3)

If F is constant (F = F0 ), then the stationary solution T0 of Eq. (3) obeys an equation F0 = δ T04 .

(4)

Below we assume that at t < 0 the system was in its stationary state: F(t) = F0 , T (t) = T0 , and consider the transient response of simple EBMs to a stepwise increase of F: t < 0, F0 , (5) F(t) = F0 + ΔF, t ≥ 0, where ΔF = const. By combining Eqs. (3)-(5), we come to the Cauchy problem

δ T 4 + ΔF δ 4 − T , T˙ = 0 C C T (0) = T0 .

(6) (7)

We now introduce the deviation Θ(t) of temperature from its unperturbed value T0

Clearly,

Θ(t) = T (t) − T0 .

(8)

T 4 = (T0 + Θ)4 = T04 + 4T03 Θ + 6T02 Θ2 + 4T0 Θ3 + Θ4 .

(9)

Below we consider three alternative approximations of Eq. (9) (and, consequently, of the Cauchy problem (6)–(7)).


241

a) Linear approximation We retain in the r.h.s. of Eq. (9) only the first and the second term: T 4 T04 + 4T03 Θ.

(10)

Then, by substituting Eq. (10) into Eq. (6), we get: 3 ˙ = ΔF − 4δ T0 Θ, Θ C C Θ(0) = 0.

(11) (12)

b) Quadratic approximation Alternatively, we can retain in the r.h.s. of Eq. (9) the first three terms: T 4 T04 + 4T03 Θ + 6T02 Θ2 .

(13)

Then, by substituting Eq. (13) into Eq. (6), we get: 3 2 ˙ = ΔF − 4δ T0 Θ − 6δ T0 Θ2 , Θ C C C Θ(0) = 0.

(14) (15)

c) The ‘exact’ model (power 4) Finally, we can retain all terms in the r.h.s. of Eq. (9). In such a case, it is more convenient to return from temperature deviation Θ(t) to temperature T (t) itself and to consider the ‘exact’ model (6)-(7):

δ T 4 + ΔF δ 4 − T , T˙ = 0 C C T (0) = T0 .

(16) (17)

As shown below, for all three proposed approximations the exact analytical solutions can be derived in closed form. 2.2

Exact analytical solutions of alternative approximations

2.2.1

Linear approximation

Consider first the linear EBM (Eqs. (11)-(12)). For brevity, we introduce the linear relaxation time

τ=

C 4δ T03

(18)

ξ=

ΔF . 4δ T04

(19)

and an auxiliary non-dimensional parameter

Then the Cauchy problem (11)–(12) can be rewritten in the form ˙ = 1 (Θ(1) − Θ), Θ τ ∞ Θ(0) = 0, where

Θ(1) ∞ = ξ T0

(20) (21) (22)

is a stationary (asymptotic) value of temperature increase in linear case. Obviously, the exact analytical solution of the linear Cauchy problem (20)-(21) is Θ(t) = Θ(1) ∞ [1 − exp(−t/τ )] .

(23)

242


2.2.2

Quadratic approximation

Consider now the quadratic EBM (Eqs. (14)-(15)). Making use of Eqs. (18)-(19), we can rewrite it in the form ˙ = 1 (ξ T0 − Θ − 3 Θ2 ), Θ τ 2T0 Θ(0) = 0.

(24) (25)

To find the stationary (asymptotic) value of temperature increase in the quadratic approximation, we should take the positive root of a quadratic equation 3 2 Θ + Θ − ξ T0 = 0. 2T0 This yields

(26)

1 + 6ξ − 1 T0 . (27) 3 By comparing Eqs. (22) and (27) we observe that the stationary (asymptotic) temperature increase is less in quadratic approximation than in linear approximation: Θ(2) ∞

=

(1) Θ(2) ∞ < Θ∞ .

(28)

The nonlinear ODE (24) has exact analytical solution. Indeed, by introducing the auxiliary variable Z(t) =

T0 + Θ(t) 3

(29)

we can rewrite the Cauchy problem (24)-(25) in the form 3 2 1 + 6ξ Z = T0 , Z˙ + 2τ T0 6τ T0 Z(0) = . 3

(30) (31)

Eq. (30) is a particular case of the special Riccati equation [19]. For the initial condition (31), its solution takes the form √ 1+6ξ 1 + 6ξ + (1 + 6ξ ) tanh( 2τ t) T0 √ . (32) Z(t) = 1+6ξ 3 1 + 6ξ + tanh( 2τ t) Coming back from the auxiliary variable Z(t) to temperature deviation Θ(t), we get the solution in the form √ 1+6ξ 2ξ tanh( 2τ t) √ T0 (33) Θ(t) = 1+6ξ 1 + 6ξ + tanh( 2τ t) that can be simplified if we divide both the numerator and the denominator of Eq. (33) by tanh(( 1 + 6ξ /2τ )t). Given that, by definition, 1/ tanh x = coth x, we find Θ(t) =

1+

2ξ T0 1 + 6ξ coth(

√

1+6ξ 2τ t)

.

(34)

As coth(at) → 1 when t → +∞ for any positive a, we easily obtain from Eq. (34) the asymptotic limit at t = +∞ found previously (Eq. (27)).


2.2.3

243

The ‘exact’ model (power 4)

It can be easily shown that the ‘exact’ model (Eqs. (16)-(17))

δ T 4 + ΔF δ 4 − T , T˙ = 0 C C T (0) = T0 . has the stationary (asymptotic) value of the temperature T∞ = 4 1 + 4ξ T0

(35) (36)

(37)

where ξ is defined by Eq. (19). So the Cauchy problem (35)-(36) can be rewritten in the form

δ T˙ = (T∞4 − T 4 ), C T (0) = T0 .

(38) (39)

Equation (38) is a separable ODE, hence dT T∞4 − T 4 and

ˆ

=

δ dt, C

(40)

d T˜ δ = t. T∞4 − T˜ 4 C

(41)

a + x 1 dx + 1 tan−1 x +C = 3 ln 4 4 a −x 4a a − x 2a3 a

(42)

T

T0

Using the standard integral [20] ˆ

we obtain from Eq. (41) an implicit solution t = t(T ) of the form T∞ + T0 T∞ + T C −1 T −1 T0 + 2 tan − ln ln . − 2 tan t(T ) = 4δ T∞3 T∞ − T T∞ T∞ − T0 T∞

(43)

As seen from Eq. (43), T∞ is indeed reached asymptotically at t = +∞. Finally, the stationary (asymptotic) temperature increase for the ‘exact’ model can be easily obtained from Eq. (37): 4 (44) Θ(4) ∞ = T∞ − T0 = ( 1 + 4ξ − 1)T0 . 2.3

First-order approximations of exact analytical solutions

The exact analytical solutions for the linear, quadratic and ‘exact’ EBM derived above (Eqs. (23), (34), and (43), respectively) might look like three completely different formulae having nothing in common in their structure. To overcome this false impression, in the present section we show that in the limit of small ξ (that is, of small extra forcing ΔF, see Eq. (19)) these three formulae actually become identical. 2.3.1

Linear approximation

As follows from Eqs. (22)–(23), in the linear approximation the exact solution is Θ(t) = ξ T0 [1 − exp(−t/τ )] .

(45)

244

2.3.2


Quadratic approximation

To compare Eq. (34) with Eq. (45), we first replace

1 + 6ξ in Eq. (34) by unity,

2ξ T0 , 1 + coth(t/2τ )

(46)

exp(x) + exp(−x) 1 + exp(−2x) ≡ 1 + 2 exp(−2x), exp(x) − exp(−x) 1 − exp(−2x)

(47)

ξ T0 , 1 + exp(−t/τ )

(48)

Θ(t) and notice that for large x coth x ≡ hence for large t Θ(t) or, with the same accuracy,

Θ(t) ξ T0 [1 − exp(−t/τ )] ,

(49)

which coincides with the exact solution (45) in the linear case. 2.3.3

The ‘exact’ model (power 4)

To compare Eq. (43) with Eq. (45), we first rewrite Eq. (45) in the equivalent implicit form: t(Θ) = −τ ln(1 −

Θ ). ξ T0

(50)

We then expand Eq. (37) to the first order in ξ : T∞ (1 + ξ )T0 .

(51)

Then by replacing in Eq. (43) T by an equivalent expression T0 + Θ, we get with the same accuracy the exact solution of the linear model in the form (50). Consequently, in the first order in ξ (or, equivalently, in ΔF) all three solutions derived in Sec. 2.2 indeed coincide. 2.4

Numerical example

Below we provide a numerical example to illustrate the accuracy of alternative approximations derived above. Consider first the stationary (asymptotic) limits in different approximations. Following [21], we adopt the value T0 = 288 K for global mean temperature. Then, as mentioned in Sec. 2.1, we imply that the empirical constant δ in Eq. (2) is equal to the Stefan–Boltzmann constant (δ = σ = 5.67 · 10−8 W/m2 K4 ). We assume the ‘four-degree’ scenario, namely, that the stationary (asymptotic) temperature increase in the (4) (4) ‘exact’ model (Eqs. (35)-(36)) is precisely 4 K. This means that we should equate Θ∞ in Eq. (44) to 4 K. If Θ∞ and T0 are given, then the corresponding value of ξ can be easily derived from Eq. (44): (4)

Θ∞ 4 1 ξ = [(1 + ) − 1]. 4 T0 (4)

(52)

For the values T0 = 288 K and Θ∞ = 4 K this yields ξ = 0.01418. It follows from Eq. (19) that the corresponding extra forcing (53) ΔF = 4δ T04 · ξ is equal in the case under study to ΔF = 22.1264 W/m2 .


245

Table 1 Stationary (asymptotic) temperature increases in different approximations (one-layer model) Approximation

Asymptotic temperature increase

Linear

Θ∞ = 4.08404 K

+2.101%

= 4.00067 K

+0.017%

= 4.00000 K

0.000%

Quadratic ‘Exact’

(1)

(2) Θ∞ (4) Θ∞

Relative error

Stationary (asymptotic) temperature increases in linear and quadratic approximations are given by Eqs. (22) and (27), respectively. Making calculations and putting the results altogether, we finally get the values provided in Table 1. The transient regime for the first 100 years after the stepwise increase of forcing in the one-layer model is shown in different approximations on Fig. 1. The blue line corresponds to linear approximation (Θ(1) (t)); the red line corresponds to the ‘exact’ model (Θ(4) (t)); the black line (indistinguishable on Fig. 1 from the red line) corresponds to quadratic approximation (Θ(2) (t)). The value C = 100 W · year/m2 K has been adopted for the effective heat capacity (per unit area) for transient calculations presented on Fig. 1; all other parameters are the same as described above. We see that the linear approximation leads to a certain overestimation of the temperature increase as compared with the ‘exact’ EBM that is visible on Fig. 1, while the quadratic approximation yields the results virtually indistinguishable from that of the ‘exact’ EBM.

Fig. 1 The transient regime for the first 100 years after the stepwise increase of forcing in different approximations of the one-layer EBM: blue line – linear approximation (Θ(1) (t)); red line – the ‘exact’ model (Θ(4) (t)); black line (indistinguishable from the red line) – quadratic approximation (Θ(2) (t)).

3 A two-layer EBM with nonlinear OLR As the next example, we consider a two-layer linear 0D EBM as described in [5] and explore its nonlinear extensions.

246


The linear version of the two-layer model as described and calibrated using the atmosphere–ocean coupled general circulation model (AOGCM) step-forcing experiment in [5] isa ˙ = ΔF − λ Θ − γ (Θ − Θ∗), CΘ ˙ ∗ = γ (Θ − Θ∗ ), C0 Θ

(54) (55)

∗

Θ(0) = 0,

Θ (0) = 0.

(56)

The model consists of two layers. The upper layer corresponds to the atmosphere, the land surface, and the upper ocean, while the lower layer represents the deep ocean. In Eqs. (54)-(56) Θ is the global mean surface air temperature perturbation from the control climate, Θ∗ is the characteristic temperature perturbation of the deep ocean, ΔF is the radiative forcing amplitude parameter, λ is the (linear) radiative forcing feedback, C and C0 are the effective heat capacities (per unit area) of the upper ocean and the deep ocean respectively, and γ is the heat exchange coefficient. We adopt from [5] the following values of model parameters: ΔF = 3.9 W/m2 , λ = 1.3 W/m2 K, C = 8.0 W · year/m2 K, C0 = 100.0 W · year/m2 K, γ = 0.7 W/m2 K. Obviously, Eq. (54) implies accounting for OLR in linear approximation only. By comparing Eq. (54) with Eq. (11) we find λ = 4δ ∗ T03 (57) where δ ∗ is a new value of an empirical constant in a nonlinear parametrisation of I given by Eq. (2). Taking, as before, T0 equal to 288 K, we find from Eq. (57) δ ∗ = 1.36 · 10−8 W/m2 K4 that is substantially smaller than the value of the Stefan–Boltzmann constant (δ = σ = 5.67 · 10−8 W/m2 K4 ) adopted for δ in Sec. 2.4. In full analogy with Sec. 2, the quadratic and ‘exact’ (power 4) extensions of the linear two-layer EBM (54)–(56) will be: Quadratic approximation: 3λ 2 Θ − γ (Θ − Θ∗ ), 2T0 ˙ ∗ = γ (Θ − Θ∗ ), C0 Θ

˙ = ΔF − λ Θ − CΘ Θ(0) = 0,

(58) (59)

∗

Θ (0) = 0.

(60)

The ’exact’ model (power 4):

λ (T 4 − T04 ) − γ (T − T ∗ ), 4T03 C0 T˙ ∗ = γ (T − T ∗ ), T ∗ (0) = T0 . T (0) = T0 , ˙ = ΔF − CΘ

(61) (62) (63)

Note that in Eqs. (61)–(63) we have returned from temperature increases Θ(t), Θ∗ (t) to temperatures themselves (T (t), T ∗ (t)). The linear two-layer EBM (54)–(56) can be solved analytically, and its analytical solution is obtained and explored in depth in [5]. Unfortunately, in the two-layer case neither the quadratic model (Eqs. (58)-(60)) nor the ‘exact’ model (Eqs. (61)-(63)) allows deriving transient solutions analytically. However, the stationary (asymptotic) values of temperature increases (Θ∞ , Θ∗∞ ) at t → +∞ can be found analytically exactly as in Sec.2.2. Indeed, if, following Eq. (57), we rewrite Eq. (19) in the form

ξ=

ΔF , λ T0

(64) (1)

∗(1)

then we will easily find that the stationary (asymptotic) temperature increase in linear (Θ∞ ≡ Θ∞ ), quadratic (2) ∗(1) (4) ∗(4) (Θ∞ ≡ Θ∞ ), and the ‘exact’ model (Θ∞ ≡ Θ∞ ) are provided by the same formulae as before (Eqs. (22), a For

the sake of consistency with Sec. 2, the notation used in the equations of Sec. 3 differs from the original notation used in [5].


247

(27), and (44), respectively). Note that the asymptotic temperature increases are identical for the upper and the lower layers. Making calculations, we get in the case of the two-layer model the asymptotic results as provided in Table2: Table 2 Stationary (asymptotic) temperature increases in different approximations (two-layer model) Approximation

Asymptotic temperature increase (upper=lower layer)

Linear

(1) Θ∞ (2) Θ∞ (4) Θ∞

Quadratic ‘Exact’

∗(1) ≡ Θ∞ ∗(2) ≡ Θ∞ ∗(4) ≡ Θ∞

Relative error

= 3.00000 K

+1.550%

= 2.95451 K

+0.011%

= 2.95418 K

0.000%

The transient regime for the first 1000 years after the stepwise increase of forcing in different approximations of the two-layer model is shown on Fig. 2 (panel a – the upper layer temperature increase, panel b – the lower layer temperature increase). As on Fig. 1, the blue line corresponds to linear approximation (Θ(1) (t), Θ∗ (1) (t)); the red line corresponds to the ‘exact’ model (Θ(4) (t), Θ∗ (4) (t)); the black line (indistinguishable on Fig. 2 from the red line) corresponds to quadratic approximation (Θ(2) (t), Θ∗ (2) (t)). As before, we see that the linear approximation leads to a certain overestimation of the temperature increase as compared with the ‘exact’ EBM that is visible on Fig. 2, while the quadratic approximation yields the results virtually indistinguishable from that of the ‘exact’ EBM. 4 Conclusions In the present paper, the dynamics of simple 0D EBMs are studied analytically and numerically both within and beyond the conventional linear approximation for parametrizing the OLR. As two alternative candidates to substitute the linear approximation, both the quadratic approximation and the ‘exact’ (power 4) model (enrooted in the ‘first-principle’ Stefan–Boltzmann law) are considered. From a theoretical standpoint, a quite surprising finding of the paper is that both of two alternative approximations, despite their pronounced structural nonlinearities, allow for deriving the exact analytical transient solutions in closed form for the one-layer EBM. Unfortunately, the transient dynamics of nonlinear extensions of the two-layer EBM can be studied only numerically, though the stationary (asymptotic) limits of temperature increases in stepwise forcing experiments still can be found analytically. When it comes to numerics, the deviation of the linear approximation from the ‘exact’ model is visible (though not much pronounced) on the resultant graphs, while the quadratic approximation yields the results virtually indistinguishable from the ‘exact’ model. It should be mentioned that previous studies [14–18] indicate that accounting for effects of nonlinearity of OLR is much more important in spatially resolved models. So in the future we are planning to come from the aspatial aggregate 0D EBMs considered in the present paper first to ‘discrete’ several-box, several-layer models, and then at least to one-dimensional (1D) continuous spatial EBMs. Taking the previous research [14–18] as the starting point, we are intending to explore alternative approximations for nonlinear parametrization of OLR in more complex and more detailed models. Acknowledgements The author is indebted to Prof. Genrikh V. Alekseev for helpful comments. The reported study was supported by the Russian Foundation for Basic Research, research project No. 15-05-03512-a.

248


Fig. 2 The transient regime for the first 1000 years after the stepwise increase of forcing in different approximations of the two-layer model. Panel a: the upper layer temperature increase. Panel b: the lower layer temperature increase. Blue line – linear approximation (Panel a – Θ(1) (t), Panel b – Θ∗(1) (t)); red line – the ‘exact’ model (Panel a – Θ(4) (t), Panel b – Θ∗(4) (t)); black line (indistinguishable from the red line) – quadratic approximation (Panel a – Θ(2) (t), Panel b – Θ∗(2) (t)).

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Exact Analytical Solutions of Selected Behaviourist Economic Growth Models with Exogenous Climate Damages Dmitry V. Kovalevsky† Nansen International Environmental and Remote Sensing Centre, 14th Line 7, office 49, Vasilievsky Island, 199034 St. Petersburg, Russia Saint Petersburg State University, Universitetskaya Emb. 7-9, 199034 St. Petersburg, Russia Submission Info Communicated by Xavier Leoncini Received 17 December 2015 Accepted 2 February 2016 Available online 1 October 2016 Keywords Economic growth Climate change Climate damage Analytical solution

Abstract Capital dynamics are calculated for (i) the AK model with output reduced by climate damages, (ii) the AK model with climate-dependent depreciation rate, and (iii) the Solow–Swan model with output given either by the Cobb–Douglas production function or by the constant elasticity of substitution (CES) production function and reduced by climate damages. The climate projections used as model inputs are exogenous. Simple analytical parametrisations for temperature dynamics are assumed (either linear or exponential temperature growth). The quadratic and the Nordhaus climate damage functions are considered. Exact analytical solutions for capital dynamics are derived in closed form (with the exception of the Solow–Swan model with CES production function). Numerical examples are provided for illustrative purposes. As the unabated climate change with unlimited temperature growth is assumed, the long-run model dynamics are dramatic: the capital converges to zero at infinite time, and the economy collapses. ©2016 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Integrated Assessment models (IAMs) are the main tool for assessing the efficiency and potential economic impact of climate mitigation policies [1–18]. Typically an IAM consists of two main modules: the economic module that describes the world economy, and the climate module that describes the global climate system. These two modules are coupled. The global economy influences the climate system through anthropogenic greenhouse gases (GHG) emissions. There is also a feedback from the climate module usually parametrised through climate damage function(s) reducing the effective output(s) of the global (regional) economy(-ies). It is not surprising that the most detailed IAMs with high degree of regional and sectoral disaggregation can be solved only numerically and often require substantial computational resources to perform model runs [10,12,13,18]. But even ‘toy’ models of this kind with just few equations normally have to be solved numerically as well. To a great extent, analytical non-tractability is caused by non-linearities inherent to most economic † Corresponding

author. Email address: [email protected], d v [email protected] ISSN 2164 − 6376, eISSN 2164 − 6414/$-see front materials © 2016 L&H Scientific Publishing, LLC. All rights reserved. DOI : 10.5890/DNC.2016.09.005

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Dmitry V. Kovalevsky / Discontinuity, Nonlinearity, and Complexity 5(3) (2016) 251–261

modules, most climate modules, and most specifications of climate damage functions, leaving little room for paper-and-pencil work. The situation changes when we try incorporating climate damages in a model describing the economic growth of a single country and not of the whole world in the aggregate. It is known that global climate change is driven by aggregate global GHG emissions. Therefore, even for countries that are large emitters the projections of climate variables that enter as inputs into the regional climate damage functions should be treated as exogenous. So, one option to build an analytically tractable model is to choose a simple economic growth model and a simple analytical approximation for the projected dynamics of relevant climate variables. Then in certain cases the exact analytical formulae for projections of national macroeconomic indicators corrected for climate damages can be derived. In the present paper we employ this program for some ‘behaviourist’ economic growth models (a term coined by Robert M. Solow for those neoclassical growth models that avoid the intertemporal utility maximization paradigm [19]), namely, for two alternative modifications of the AK model and for the Solow–Swan model. The rest of the paper is organized as follows. In Secs. 2, 3, and 4 we derive exact analytical solutions for capital dynamics in case of (i) the AK model with output reduced by climate damages (Sec. 2.1), (ii) the AK model with climate-dependent depreciation (Sec. 3.1), and (iii) the Solow–Swan model with output given by the Cobb–Douglas production function and reduced by climate damages (Sec. 4.1). Analytical solutions derived are supplemented by numerical examples for illustrative purposes (Secs. 2.2, 3.2, and 4.2, respectively). Additionally, in Sec. 4.2 the case of the constant elasticity of substitution (CES) production function in the Solow–Swan model is considered numerically. Sec. 5 concludes. 2 The AK model with climate damages 2.1

Analytical solutions

The standard AK model [20–22] has the forma K˙ = (sA − δ )K

(1)

where K is capital, s is the savings rate, A is the technology parameter, and δ is the depreciation rate. The solution of Eq. (1) is straightforward: (2) K = K0 exp(r0 t) where K0 is the initial capital stock and r0 = sA − δ

(3)

is the growth rate. The modified AK model with output reduced by climate damages takes the form K˙ = [(1 − d(T ))sA − δ ]K

(4)

where d(T ) is temperature-dependent climate damage function.b If the precise form of the climate damage function is specified and the exogenous temperature projections T (t) are given, then variables in Eq. (4) separate and the solution takes the form ˆ K(t) = K0 exp[r0t − sA a Regarding the history of the AK model, the authors

t

0

d(T (t))dt]

(5)

of the world-famous textbook on economic growth [20] mention: “We think that the first economist to use a production function of the AK type was von Neumann (1937)” [21]. They also acknowledge the contribution of Knight [22] in developing models of this kind. b By temperature we mean the global mean surface air temperature as is usually assumed in IAMs.


253

where r0 is the same baseline growth rate as in Eq. (2). In what follows we will explore two alternative analytical approximations for temperature growth: (i) Linear temperature growth TL (t) = T0 + Γt

(6)

where T0 is the initial value of temperature and Γ is the constant temperature change rate; and (ii) Exponential temperature growth TE (t) = T0 exp(ξ t)

(7)

where T0 is again the initial value of temperature and ξ is the constant temperature growth rate. For climate damages we also use two alternative specifications: (i) The quadratic climate damage function broadly used in theoretical literature [23, 24]: dQ (T ) = β (T − T¯)2 ,

(8)

and (ii) The Nordhaus climate damage function developed by William D. Nordhaus for his seminal IAM DICE [9] and later used by many other authors: dN (T ) = 1 −

1 2 1 + αN (T − T¯)

.

(9)

In Eqs. (8)–(9) αN and β are constant parameters and T¯ is the pre-industrial temperature level. By substituting Eqs. (6)–(9) into Eq. (5) and performing the integration we easily get four alternative analytical projections for capital: 1. Linear temperature growth, the quadratic climate damage function (TL (t), dQ (T )): Γ K(t) = K0 exp[(r0 − sAβ Θ2 )t − sAβ Γ(Θt 2 + t 3 )]; 3

(10)

2. Linear temperature growth, the Nordhaus climate damage function (TL (t), dN (T )): √ sA −1 √ tan ( αN (Θ + Γt)) − tan−1 ( αN Θ) − δ t]; K(t) = K0 exp[ √ αN Γ

(11)

3. Exponential temperature growth, the quadratic climate damage function (TE (t), dQ (T )): sAβ T0 T0 ( (exp(2ξ t) − 1) − 2T¯(exp(ξ t) − 1))]; K(t) = K0 exp[(r0 − sAβ T¯2 )t − ξ 2

(12)

4. Exponential temperature growth, the Nordhaus climate damage function (TE (t), dN (T )): K(t) = K0 exp[(

2 1 1 + αN (T0 − T¯) sA sA { ln − δ )t + + 1 + αN T¯2 ξ (1 + αN T¯2 ) 2 1 + αN (T0 exp(ξ t) − T¯)2 √ √ √ αN T¯ tan−1 ( αN (T0 exp(ξ t) − T¯)) − tan−1 ( αN (T0 − T¯)) . (13)

In Eqs. (10)–(11) the notation is used Θ = T0 − T¯, i.e. Θ is the initial temperature increase above the pre-industrial level.

(14)

254

2.2


Numerical examples

To provide some numerical examples based on analytical solutions obtained in Sec. 2.1 illustrating the magnitude of potential impacts of climate change on long-term economic growth, we first quantify the temperature scenarios. The start year of all projections provided in the present paper is 2015, while the end year is 2200. Following the recent press release of the British Met Office,c we set the initial value of the temperature T0 = 1.0◦ C in Eqs. (6)-(7) that means a one-degree temperature increase above the pre-industrial level in 2015. Then, for linear temperature growth scenario (Eq. (6)) we adopt Γ = 0.03◦ C/year that corresponds to threedegree temperature increase per century. For exponential temperature growth scenario we assume (somewhat arbitrarily) ξ = (1/75) = 0.013 year−1 , an exponent about half of the value recently derived from historical data series in [25]. This would correspond to T = 3.1◦ C in 2100 and to very high T = 11.8◦ C in 2200. These quantitative linear and exponential temperature growth scenarios are visualised on Fig. 1.

Fig. 1 Linear (blue line) and exponential (red line) temperature growth scenarios

Regarding the economic parameters of the model (cf. Eq. (1)), we take the conventional value δ = 0.05 year−1 for capital depreciation rate. We also assume the baseline growth rate r0 = sA − δ (Eq. (3), neglecting the climate impacts) equal to 2% per annum; this yields sA = 0.07 year−1 . For the parameter β in the quadratic climate damage function (Eq. (8)) we take the value β = 0.00144 (◦ C)−2 as was adopted in an earlier version of the DICE model [24]. For the parameter αN in the Nordhaus climate damage function (Eq. (9)) the value αN = (1/20.46)2 (◦ C)−2 is used. This value for αN was adopted in the later version of the DICE model [9], as well as in many other Integrated Assessment models. As all ordinary differential equations (ODEs) in Sec. 2.1 are essentially linear, we can measure the capital K in arbitrary (e.g. dimensionless) units, so for convenience we take K0 = 1.0 (dimensionless) as the initial climate in context as the world approaches 1◦ C above pre-industrial for the first time. The British Met Office. URL: http://www.metoffice.gov.uk/research/news/2015/global-average-temperature-2015 (accessed 15 December 2015). c Global


255

condition for capital in year 2015. The projections of capital dynamics by the end of the 22nd century are shown on Figs. 2a and 2b for linear and exponential temperature growth scenarios respectively. As seen from the graphs, through the 21st century the climate damages (for given specifications of climate damage functions) are quite modest. However, closer to the end of the 22nd century the slowdown of economics growth due to the adverse impacts of climate change becomes pronounced. The background behind the very dramatic economic scenarios indicated by black lines on Figs. 2a and 2b will be explained in Sec. 3.2 below. Finally, it should be mentioned that in our previous work [26] we explored the AK model as in Eq. (4) with the weakly nonlinear Nordhaus climate damage function (Eq. (9)) and the strongly nonlinear Weitzman climate damage function [27] within a different model setup. Instead of stylized analytical approximations like in Eqs. (6) or (7), ‘realistic’ exogenous temperature projections provided by state-of-the-art global climate models were used. That modelling exercise of course required numerical simulations. 3 The AK model with climate-dependent depreciation 3.1


In a detailed analysis of a multi-region Solow–Swan model Sorger [28] pointed out that, generally speaking, different values of depreciation rates should be assigned to different model regions. Interestingly, this heterogeneity of regional depreciation rates was supported in Sorger’s work by a climate-related argument. Particularly, it was mentioned that the diverse regional climate conditions affecting the physical capital should be taken into account in multi-region economic growth models. This fruitful idea of climate-dependent constant depreciation rates was later transformed to a concept of time-varying climate-dependent endogenous depreciation rates used in several theoretical IAMs [29, 30]. In the present section we consider the AK model with the depreciation rate linear in temperature: K˙ = [sA − δ (1 + ε (T − T0 ))]K

(15)

where ε is the constant sensitivity of depreciation to temperature increases. As with Eq. (4), variables in Eq. (15) easily separate provided that temperature projections are known, and, analogously to Eq. (5), we get ˆ t (16) K(t) = K0 exp[r0 t − δ ε ( T (t)dt − T0t)] 0

where r0 is again given by Eq. (3). We use the same temperature projections as in Sec. 2 (Eqs. (6)–(7)) and easily get from Eq. (16): 1. Linear temperature growth (TL (t)): K(t) = K0 exp[r0 t −

δ εΓ 2 t ]; 2

(17)

2. Exponential temperature growth (TE (t)): K(t) = K0 exp[(r0 + δ ε T0 )t − 3.2

δ ε T0 (exp(ξ t) − 1)]. ξ

(18)

Numerical examples

Following our previous work [31], for a numerical example we choose the value ε = 0.2 (◦ C)−1 for temperature sensitivity of the depreciation rate in Eq. (15). This means that the depreciation rate would double if the temperature increases by 5◦ C above the initial level. All other values of model parameters are as in Sec. 2.2.

256


Fig. 2 Capital dynamics in case of linear (panel a) and exponential (panel b) temperature growth scenarios: blue line – baseline growth (no climate damages); red line – the quadratic climate damage function (Eq. (8)); olive line – the Nordhaus climate damage function (Eq. (9)); black line – the case of temperature-dependent depreciation rate (Sec. 3, dynamic equation (15))

The projections of capital dynamics provided by the temperature-dependent depreciation model (Eq. (15)) are shown by black lines on Figs. 2a and 2b for linear and exponential temperature growth scenarios respectively. As clearly seen from the graphs, the temperature-dependent depreciation model yields much more dramatic, not to say catastrophic, economic dynamics projections for the 22nd century.


257

4 The Solow–Swan model with climate damages 4.1


The capital per unit of effective labour k in the standard Solow–Swan model [20, 32, 33] obeys the dynamic equation (19) k˙ = s f (k) − (n + δ + γ )k where s is the savings rate, f (k) is output per unit of effective labour, n is the labour force growth rate, δ is the depreciation rate, and γ is the labour productivity growth rate (s, n, δ and γ are assumed to be constant). In this section, we assume the Cobb–Douglas form of the production function that implies f (k) = Akα ,

A = const,

0 < α < 1.

(20)

By substituting Eq. (20) into Eq. (19) we get k˙ = sAkα − (n + δ + γ )k.

(21)

Climate damage function can be introduced into Eq. (21) in the same way as it was done with the AK model in Sec. 2.1: (22) k˙ = (1 − d(T ))sAkα − (n + δ + γ )k. It is known that in a particular case of Cobb–Douglas production function the standard Solow–Swan model can be solved analytically by reducing it to a linear ordinary differential equation [34].d We apply the same method to Eq. (22). Namely, we introduce an auxiliary variable z = k1−α

(23)

and, after substituting it into the nonlinear equation (22) and with some rearrangements, get a linear equation z˙ + (1 − α )(n + δ + γ )z = (1 − α )sA(1 − d(T )).

(24)

b = (1 − α )(n + δ + γ ),

(25)

z˙ + bz = (1 − α )sA(1 − d(T )).

(26)

Denote for brevity then Eq. (24) can be rewritten in the form

Provided that the temperature projections T (t) are known, the solution of Eq. (26) takes the form ˆ t z(t) = exp(−bt)[z0 + (1 − α )sA (1 − d(T )) exp(bt)dt]

(27)

0

where z0 is the initial value of z:

z0 = k01−α .

(28)

After the integral is taken in Eq. (27), k(t) can then be found from Eq. (23). Unlike in Sec. 2, in the present section we will explore only the quadratic climate damage function dQ (t) (Eq. (8)) since for the Nordhaus climate damage function the required integrals cannot be taken in closed form. Again, we use either linear or exponential temperature projections as inputs to Eq. (27). d Exact

analytical solutions of certain modifications of the standard Solow–Swan model (provided that the Cobb-Douglas form of the production function is assumed) are reported in the literature. E.g. the solution of the model with a generalized logistic labour force growth law replacing the standard exponential growth law has a closed-form expression via Hypergeometric functions [35].

258


1. Linear temperature growth, the quadratic climate damage function (TL (t), dQ (T )): k(t) = ({k01−α −

(1 − α )sA (1 − β M)} exp(−bt)+ b

1 Γ (1 − α )sA (1 − β [M + 2Γ(Θ − )t + Γ2t 2 ])) 1−α , (29) b b

where an auxiliary constant is introduced for brevity Γ Γ M = Θ2 − 2Θ + 2( )2 . b b

(30)

2. Exponential temperature growth, the quadratic climate damage function (TE (t), dQ (T )): T2 2T0 T¯ T¯2 1 + )]} exp(−bt)+ k(t) = ({k01−α − (1 − α )sA[ − β ( 0 − b 2ξ + b ξ + b b 1 T2 1 2T0 T¯ T¯2 exp(ξ t) + )]) 1−α . (31) (1 − α )sA[ − β ( 0 exp(2ξ t) − b 2ξ + b ξ +b b 4.2

Numerical examples

In numerical examples provided in this section we assume that in Eq. (19) γ = 0, i.e. that there is no labour productivity growth. For the sake of consistency with the results discussed in Sec. 2-3, in this section we will again provide the projections of capital K(t) itself, and not of capital per unit of labour k(t) for which the analytical solutions (29) and (31) were derived in Sec. 4.1. To do so, we first note that, by definition, k=

K , L

(32)

and that the exponential growth of labour is assumed in the standard Solow–Swan model: L(t) = L0 exp(nt).

(33)

Therefore to obtain K(t) one should multiply analytical solutions (29) and (31) by the r.h.s. of Eq. (33). In full analogy with what has been done in Sec. 2-3, we again measure K(t) and L(t) in dimensionless units, and take K0 = 1.0 and L0 = 1.0 as initial conditions in 2015. Then, we take the value of the exponent n in Eq. (33) equal to 0.01 year−1 (which corresponds to labour force growth rate of 1% per annum). For the parameter α appearing in the Cobb–Douglas production function (Eq. (20)) we choose the value α = 0.3 referred to in [36] as a good approximation for modelling the output of the US economy. All other values of model parameters are as in Sec. 2.2 and 3.2. The projections of the capital dynamics in the Solow–Swan model with the Cobb–Douglas production function are shown on Fig. 3a for the baseline case (no climate damages) and for both alternative temperature scenarios. As mentioned in Sec. 4.1, the Solow–Swan model is generally a nonlinear model; however, in the particular case of the Cobb–Douglas production function we were able to reduce it to a linear ODE. We would like to conclude this section with a ‘truly nonlinear’ version of the Solow–Swan model where the ‘linearising trick’ (Eq. (23)) no longer holds. To do this, we replace in the Solow–Swan model the Cobb–Douglas production function, that is written in the extensive form as Y = FCD (K, L) = AK α L1−α ,

(34)


259

by the constant elasticity of substitution (CES) production function [20] of the form Y = FCES (K, L) = A[α K ψ + (1 − α )Lψ ]1/ψ

(35)

that approaches the Cobb–Douglas production function in the limit ψ → 0. We choose the value ψ = 8.0 for our numerical example, all other values of model parameters are as before. Unfortunately we are no longer able to derive an analytical solution in the CES case, so the numerical simulations are now necessary. The simulation results in the CES case are provided on Fig. 3b (in situations parallel with the Cobb–Douglas case considered above and presented on Fig. 3a). 5 Conclusions In the present paper we incorporated simple analytical approximations of exogenous temperature projections into simple behaviourist economic growth models with simple parametrisations of climate damages. This allowed us to derive analytical projections for capital dynamics in closed form in many cases. As the unabated climate change with unlimited temperature growth was assumed, it is not surprising that in all particular cases considered above the capital decays to zero in the long run, and the economy collapses (Eqs. (10)–(13), (17)–(18), (29), (31)). It should be mentioned that in the case of the quadratic climate damage function (Eq. (8)) the situation is more dramatic. Unlike the Nordhaus climate damage function (Eq. (9)) that converges to unity in the long run but actually never reaches it at any finite time, the quadratic climate damage function reaches unity at finite time. This implies that at that moment the effective output of the economy (corrected for climate damages) is strictly equal to zero. We believe that the approach developed in the present paper and exact analytical solutions derived in several particular cases might be of use for fast ‘zero-approximation’ estimations of projected regional economic development under conditions of adverse anthropogenic climate change. Acknowledgements The reported study was supported by the Russian Foundation for Basic Research, research project No. 13-0600368-a. References [1] Capellán-Pérez, I., González-Eguino, M., Arto, I., Ansuategi, A., Dhavala, K., Patel, P. and Markandya, A. (2014), New climate scenario framework implementation in the GCAM integrated assessment model. BC3 Working Paper Series 2014-04, Basque Centre for Climate Change (BC3), Bilbao, Spain. [2] Edenhofer, O., Lessmann, K., Kemfert, C., Grubb, M. and Köhler, J. (2006), Induced technological change: Exploring its implications for the economics of atmospheric stabilization. Synthesis Report for the Innovation Modeling Comparison Project, The Energy Journal, Special Issue: Endogenous Technological Change, 57-107. [3] Hasselmann, K. (2013), Detecting and responding to climate change, Tellus B, 65, 20088. [4] Hasselmann, K., Cremades, R., Filatova, T., Hewitt, R., Jaeger, C., Kovalevsky, D., Voinov, A. and Winder, N. (2015), Free-riders to forerunners, Nature Geoscience, 8, 895-898. [5] Hasselmann, K. and Kovalevsky, D.V. (2013), Simulating animal spirits in actor-based environmental models, Environmental Modelling & Software, 44, 10-24. [6] Kovalevsky, D.V., Kuzmina, S.I. and Bobylev, L.P. (2015), Impact of nonlinearity of climate damage functions on longterm macroeconomic projections under conditions of global warming, Discontinuity, Nonlinearity, and Complexity, 4, 25-33. [7] Moss, S., Pahl-Wostl, C. and Downing, T. (2001), Agent-based integrated assessment modelling: the example of climate change, Integrated Assessment, 2, 17-30. [8] Nordhaus, W.D. (1993), Rolling the ‘DICE’: An optimal transition path for controlling greenhouse gases, Resource and Energy Economics, 15, 27-50.

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Fig. 3 Capital dynamics in the Solow–Swan model with the Cobb–Douglas production function (panel a) and the constant elasticity of substitution (CES) production function (panel b): black line – baseline growth (no climate damages); blue line – the linear temperature growth scenario; red line – the exponential temperature growth scenario [9] Nordhaus, W.D. (2008), A Question of Balance, Yale University Press: New Haven & London. [10] Nordhaus, W.D. and Yang, Z. (1996), RICE: A regional dynamic general equilibrium model of alternative climatechange strategies, The American Economic Review, 86, 741-765. [11] Rovenskaya, E. (2010), Optimal economic growth under stochastic environmental impact: Sensitivity analysis. In: Dynamic Systems, Economic Growth, and the Environment, Vol. 12 of the series Dynamic Modeling and Econometrics in Economics and Finance, J.C. Cuaresma, T. Palokangas, A. Tarasyev (eds.), Springer, 79-107. [12] Stanton, E.A., Ackerman, F., and Kartha, S. (2009), Inside the integrated assessment models: four issues in climate


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On Symmetric Strictly non-Volterra Quadratic Stochastic Operators U.U. Jamilov † Institute of Mathematics, National University of Uzbekistan, 29, Do’rmon Yo’li str., 100125, Tashkent, Uzbekistan Submission Info Communicated by Dimitry Volchenkov Received 11 July 2015 Accepted 11 February 2016 Available online 1 October 2016 Keywords

Abstract For a symmetric strictly non-Volterra quadratic stochastic operator on the three-dimensional simplex it is proved that this operator has a unique fixed point. A sufficient condition of attractiveness for the unique fixed point is found. For such operators we describe the set of ω − limit points. We proved that some classes of such operators have infinitely many periodic points. Also it is shown that there are trajectories which are asymptotically cyclic with period two.

Quadratic stochastic operator Volterra and non-Volterra operators Simplex Trajectory


1 Introduction Let Sm−1 = {x = (x1 , x2 , . . . , xm ) ∈ Rm : for any i, xi ≥ 0, and

m

∑ xi = 1}

(1)

i=1

be the (m − 1)- dimensional simplex. As shown by Jenks [16], the following homogeneous differential system on (1) dxi = ∑ aijk x j xk , (2) dt j,k with (i) aijk = aik j for all i, j, k ∈ {1, . . . , m}; (ii) ∑ aijk = 0 for all j, k ∈ {1, . . . , m}; (iii) aijk ≥ 0 for all j = i, i

k = i ∈ {1, . . . , m}, governs mathematical models for large interacting populations of m constituents. Here the numbers xi represent a fraction of constituents of type i, i = 1, . . . , m and satisfy the conservation law ∑ xi = 1, i

e.g. the original Lotka-Volterra predator-prey equations. As shown in [6] the discrete time system corresponding to (2) is defined by the mapping V : Sm−1 → Sm−1 with (V x)k =

m

∑

pi j,k xi x j

i, j=1 † Corresponding

author. Email address: [email protected] ISSN 2164 − 6376, eISSN 2164 − 6414/$-see front materials © 2016 L&H Scientific Publishing, LLC. All rights reserved. DOI : 10.5890/DNC.2016.09.006

(3)

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U.U. Jamilov / Discontinuity, Nonlinearity, and Complexity 5(3) (2016) 263–283

where pi j,k = aijk + (δik + δ jk )/2, and δi j is Kronecker delta. It is evident that i)pi j,k ≥ 0,

ii)pi j,k = p ji,k

m

for all i, j, k; iii) ∑ pi j,k = 1.

(4)

k=1

A mapping (3) with conditions (4), is called a quadratic stochastic operator (QSO), which was first introduced by Bernstein [2]. Such operators frequently arise in many models of mathematical genetics, namely, theory of heredity(e.g. [18]). Let us give some details of appearance of a QSO: Consider a biological population, that is a community of organisms closed with respect to reproduction. Assume that each individual in this population belongs to precisely one species (genotype) E = {1, . . . , m}. The scale of the species is such that the species of the parents i and j unambiguously determines the probability pi j,k of every species k for the first generation of direct descendants of the i and j. This probability is called the heredity coefficient. It is then obvious that pi j,k ≥ 0 for all i, j, k and ∑m k=1 pi j,k = 1 for any pair i, j. The state of the population can be described by the m-tuple (x1 , x2 , . . . , xm ) of species probabilities, that is xk is the fraction of the species k in the total population. In the case of panmixia (random interbreeding) the parent pairs i and j arise for a fixed state x = (x1 , x2 , . . . , xm ) with probability xi x j . Hence the total probability of the species k in the first generation of direct descendants is defined by m

∑

pi j,k xi x j ,

(k = 1, . . . , m).

i, j=1

Note that the linear operators are a special case of quadratic operators. If Π = (ri j ) be a stochastic matrix of size m × m, i.e., ri j ≥ 0 and ∑ j ri j = 1 then with pi j,k = (rik + r jk )/2 we get m

(V x)k = ∑ rik xi . i=1

For example, if pi j,k = (δik + δ jk )/2, then the corresponding quadratic stochastic operator is an identity transformation. Note that the operator (3) is a non-linear (quadratic) operator, and its dimension increases with m. Higher dimensional dynamical systems are important but there are relatively few dynamical phenomena that are currently understood ([4], [5], [22]). The special case when the relation (iii) in (2) holds with equality then the equations for a Volterra’s treatise on the biological struggle for life [28] are distinguished by the form dxi = xi ∑ ai j x j , i = 1, . . . , m. dt j

(5)

Here the ai j are biological constants satisfying ai j = −a ji , i.e. the m × m matrix A = (ai j ) is skew-symmetric. The system of equation (5) is called Volterra dynamical system [7]. As shown [1], [12] the discrete time system corresponding to a Volterra dynamical system (5) is defined by the Volterra operator V : Sm−1 → Sm−1 with m

(V x)k = xk (1 + ∑ aki xi ),

(6)

i=1

where A = (ai j )m 1 is a skew-symmetric matrix with |ai j | ≤ 1. Here i, j ∈ {1, 2, . . . , m}. The dynamical systems (5), (6) were studied by E. Akin and V. Losert [1] in terms zero-sum game dynamics and they used to explain mammalian ovulation control, in particular regulation for a prescribed number of mature eggs in biology. Also, there shown the radical difference between continuous (5) and discrete (6) models in the skew-symmetric case. Note also that Nagylaki [20], [21] has introduced (6) with antisymmetric ai j as a model for a gene conversion.


265

In nonlinear discrete models (3) for a given x(0) ∈ Sm−1 the trajectory x(n) , n = 0, 1, 2, . . . of x(0) under the action of the QSO (3) is defined by x(n+1) = V (x(n) ), where n = 0, 1, 2, . . . . One of the main problems in mathematical biology consists in the study of the asymptotical behavior of the trajectories. Denote by ω (x0 ) the / set of ω − limiting points of trajectory x(n) . Since Sm−1 is a compact and {x(n) } ⊂ Sm−1 it follows that ω (x0 ) = 0. 0 0 It is clear that if ω (x ) consists of a single point, then the trajectory converges and ω (x ) is a fixed point of (3). It should be noted that the limit behavior of the trajectories of QSO on one-dimensional space was fully studied by Yu.I. Lyubich [19]. However, the problem is still open even in two-dimensional simplex. In works [12–14] this problem was particularly solved for a class of Volterra QSO. A class of Volterra QSO is defined by (3),(4) and the additional assumption / {i, j}. pi j,k = 0 if k ∈

(7)

Obviously, the condition (7) means that each individual can inherit only the species of the parents. In [7, 8, 10, 11, 27, 29] the ergodicity problems of the Volterra operators considered. In [9, 25] a Volterra operator of a bisexual population was investigated. However, in the non-Volterra case (i.e. where condition (7) is violated), many questions remain open and there seems to be no general theory available. To the best of our knowledge, there are few papers devoted to such operators. In [3, 16, 17, 23, 24] the dynamics of some classes of non-Volterra quadratic operators were studied. See [15] for a recent review of QSOs. It seems natural to consider a nonlinear model non-Volterra case and explore the dynamical system resulting from it. In [24] it was introduced the conception of strictly non-Volterra QSOs and proved that an arbitrary strictly non-Volterra quadratic operator on the two-dimensional simplex has a unique fixed point, which is established as being non-attracting. It is shown that some strictly non-Volterra operators, as distinct from the Volterra operators, have periodic trajectories. The present paper is continuation of [24]. We consider a class of strictly non-Volterra operators defined on the three-dimensional simplex. The paper is organized as follows. In § 2 we give the definition of a symmetric strictly non-Volterra operator on the three-dimensional simplex. Fixed points of such operators are examined in § 3, where we show that an arbitrary symmetric strictly non-Volterra QSO has a unique fixed point. In § 4 we show that this point cannot be a repelling point. In § 5 we describe the ω -limit set of a symmetric strictly non-Volterra QSO on S3 . 2 Definitions Definition 1 ( [24]). A quadratic stochastic operator (3),(4) is called strictly non-Volterra if pi j,k = 0, for k ∈ {i, j}, i, j, k = 1, . . . , m.

(8)

Remark 1. A strictly non-Volterra operator exists only if m ≥ 3. In [24] the case m = 3 was studied. In present paper we will study the case m = 4 and show that dynamics of strictly non-Volterra QSO on S3 differ from dynamics of such type QSO defined on S2 . An arbitrary strictly non-Volterra QSO defined on the three-dimensional simplex has the form ⎧ x1 = ax22 + bx23 + cx24 + 2dx2 x3 + 2ex2 x4 + 2gx3 x4 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ x = a x2 + b x2 + c x2 + 2d x x + 2e x x + 2g x x ⎪ 1 1 1 3 1 4 1 1 3 1 1 4 1 3 4 ⎪ ⎨ 2 V: x = a2 x21 + b2 x22 + c2 x24 + 2d2 x1 x2 + 2e2 x1 x4 + 2g2 x2 x4 ⎪ ⎪ ⎪ 3 ⎪ ⎪ ⎪ ⎪ ⎪ x4 = a3 x21 + b3 x22 + c3 x23 + 2d3 x1 x2 + 2e3 x1 x3 + 2g3 x2 x3 , ⎪ ⎩

(9)

266


where a, ai , b, bi , c, ci , d, di , e, ei , g, gi ≥ 0, i = 1, 2, 3 and a1 + a2 + a3 = 1, a + b2 + b3 = 1, b + b1 + c3 = 1, c + c1 + c2 = 1, d + g3 = 1, d1 + e3 = 1, d2 + d3 = 1, e + g2 = 1, g + g1 = 1, e1 + e2 = 1.

(10)

In this paper we consider the operators which have a symmetry in the coefficients. Namely, we shall consider the class of strictly non-Volterra QSOs which satisfy the following conditions: the probabilities won’t change by permutation 1234 π1 = , 2143 i.e. for the coefficients pi j,k the following relations hold p11,2 = p22,1 p33,4 = p44,3 p13,4 = p14,3 , p11,3 = p11,4 p44,1 = p44,2 p14,2 = p24,1 , p22,3 = p22,4 p12,3 = p12,4 p23,4 = p24,3 , p33,1 = p33,2 p13,2 = p23,1 p34,1 = p34,2 . Then we have

a = a1 , b = b1 , c = c1 , d = d1 , e = e1 , g = g1 , a2 = a3 , b2 = b3 , c2 = c3 , d2 = d3 , e2 = e3 , g2 = g3 .

(11)

Therefore for the coefficients (10) of the QSO (9) from (4),(8),(10) and (11) we get a = a1 , b = b1 = c = c1 , d = d1 = e = e1 , g = g1 = 12 , a2 = a3 = b2 = b3 , c2 = c3 , d2 = d3 = 12 , e2 = e3 = g2 = g3 .

(12)

Then QSO (9) has the following form

V:

⎧ x1 = ax22 + bx23 + bx24 + 2dx2 x3 + 2dx2 x4 + x3 x4 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ x = ax2 + bx2 + bx2 + 2dx1 x3 + 2dx1 x4 + x3 x4 2

⎪ ⎪ ⎪ x3

1

=

3

4

a2 x21 + a2 x22 + c2 x24 + x1 x2 + 2e2 x1 x4 + 2e2 x2 x4

(13)

⎪ ⎪ ⎪ ⎩ x = a x2 + a x2 + c x2 + x x + 2e x x + 2e x x . 2 1 2 2 2 3 1 2 2 1 3 2 2 3 4 where a, a2 , b, c2 , d, e2 ≥ 0 and a + 2a2 = 1, 2b + c2 = 1, d + e2 = 1. Definition 2. The strictly non-Volterra QSO (13) defined on the three-dimensional simplex is called symmetric strictly non-Volterra quadratic stochastic stochastic operator (SSnVQSO). One can check that QSOs generated by permutations

π2 =

1234 3412

, π3 =

1234

4321

are coincide with QSO (13) up to a renaming of the coordinates and parameters. Also obviously the trajectories of them coincide up to a renaming of the parameters and coordinates of the limit points.


267

3 Fixed point of operator Definition 3. A point x ∈ Sm−1 is called a fixed point of a QSO V if V (x) = x and the set of all fixed points denoted by Fix(V ). Theorem 1. Any SSnVQSO (i.e. (13)) has a unique fixed point x∗ = (x∗1 , x∗2 , x∗3 , x∗4 ) ∈ S3 , where the coordinates of x∗ equal to following: i) if a + 2b + 1 − 4d = 0 then x∗1 = x∗2 =

2b + 1 2−a , x∗3 = x∗4 = ; 8(1 + b − d) 8(1 + b − d)

ii) if a + 2b + 1 − 4d = 0 then x∗1

=

x∗3

x∗2

2(b + 1 − d) − (2b + 1)(2 − a) + (1 − 2d)2 , = 2(a + 2b + 1 − 4d)

= x∗4

a − 1 − 2d + (2b + 1)(2 − a) + (1 − 2d)2 . = 2(a + 2b + 1 − 4d)

Proof. The equation V (x) = x has the following form ⎧ x1 = ax22 + bx23 + bx24 + 2dx2 x3 + 2dx2 x4 + x3 x4 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ x2 = ax21 + bx23 + bx24 + 2dx1 x3 + 2dx1 x4 + x3 x4 ⎪ ⎪ x3 = a2 x21 + a2 x22 + c2 x24 + x1 x2 + 2e2 x1 x4 + 2e2 x2 x4 ⎪ ⎪ ⎪ ⎪ ⎩ x4 = a2 x21 + a2 x22 + c2 x23 + x1 x2 + 2e2 x1 x3 + 2e2 x2 x3 .

(14)

From (14) we get

x1 − x2 = a(x22 − x21 ) + 2dx3 (x2 − x1 ) + 2dx4 (x2 − x1 ), x3 − x4 = c2 (x24 − x23 ) + 2e2 x1 (x4 − x3 ) + 2g2 x2 (x4 − x3 ).

(15)

From the first equation of (15) we get x1 = x2 and a(x1 + x2 ) + 2d(x3 + x4 ) = −1. Since a(x1 + x2 ) + 2d(x3 + x4 ) ≥ 0 the equation a(x1 + x2 ) + 2d(x3 + x4 ) = −1 hasn’t solutions in the three-dimensional simplex. Similarly from second equation of (15) we obtain x3 = x4 . Since x ∈ S3 we have x1 + x3 = 12 and from (14) we get 2b + 1 = 0. (16) (a + 2b + 1 − 4d)x21 + 2(d − b − 1)x1 + 4 Let a + 2b + 1 − 4d = 0 then the equation (16) has a unique solution x∗1 =

2b + 1 . 8(1 + b − d)

Obviously that x∗1 > 0. Indeed 1+b−d = 0 ⇔ b = 0, d = 1. Since a+2b+1−4d = 0 we obtain 1+b−d = 0. Using a + 2b + 1 − 4d = 0 and a − 2 < 0 we get x∗1 ≤ 12 . Using x1 + x3 = 12 and again a + 2b + 1 − 4d = 0 we get x∗3 =

2−a . 8(1 + b − d)

268


Since −4 − 4b < 0 it is easy to verify that 0 ≤ x∗3 ≤ 12 . This completes proof of i). Let a + 2b + 1 − 4d < 0. From (16) and D ≡ D(a, b, d) = (2b + 1)(2 − a) + (1 − 2d)2 ≥ 1

(17)

we get the following two solutions 2(b + 1 − d) + (2b + 1)(2 − a) + (1 − 2d)2 , = 2(a + 2b + 1 − 4d) 2(b + 1 − d) − (2b + 1)(2 − a) + (1 − 2d)2 (2) . x1 = 2(a + 2b + 1 − 4d)

(1) x1

(18)

(1)

Since a + 2b + 1 − 4d < 0 by (18) it is easy to see that x1 < 0. The checking of the second solution shows (2) 0 ≤ x1 ≤ 12 and a − 1 − 2d + (2b + 1)(2 − a) + (1 − 2d)2 (2) x3 = 2(a + 2b + 1 − 4d) also belong to [0, 12 ]. (1)

Let a + 2b + 1 − 4d > 0. From (16) we get also two solutions in form (18), but using x1 > 0 we obtain 1 a − 1 − 2d − (2b + 1)(2 − a) + (1 − 2d)2 (1) (1) < 0. x3 = − x1 = 2 2(a + 2b + 1 − 4d) (2)

Similarly one can prove that 0 ≤ x1 ≤

1 2

(2)

and 0 ≤ x3 ≤ 12 .

Remark 2. A Volterra QSO for m = 4 has at least 4 fixed points [12]. The following example is due to M. Scheutzow and it shows that in generally in a high dimensional simplex the strictly non-Volterra operator might have more than one fixed point. Example. ⎧ ⎪ x = 1 x2 + 1 x2 + 2x2 x3 + 23 x4 x5 , ⎪ ⎪ 1 2 2 2 3 ⎪ ⎪ 1 2 1 2 2 ⎪ ⎪ ⎪ x2 = 2 x1 + 2 x3 + 2x1 x3 + 3 x4 x5 , ⎨ (19) V : x3 = 12 x21 + 12 x22 + 2x1 x2 + 23 x4 x5 , ⎪ ⎪ ⎪ 2 ⎪ x4 = x5 + 2x1 x5 + 2x2 x5 + 2x3 x5 , ⎪ ⎪ ⎪ ⎪ ⎩ x = x2 + 2x x + 2x x + 2x x . 1 4 2 4 3 4 4 5 It is clear that points of the form (1/3, 1/3, 1/3, 0, 0) and (1/9, 1/9, 1/9, 1/3, 1/3) are fixed points for the operator (19). 4 The type of the fixed point Definition 4 ( [4]). A fixed point x∗ of the operator V is called hyperbolic if its Jacobian J at x∗ has not eigenvalues on the unit circle. Definition 5 ( [4]). A hyperbolic fixed point x∗ is called:


269

i) attracting if all the eigenvalues of the Jacobian J(x∗ ) are less than 1 in absolute value; ii) repelling if all the eigenvalues of the Jacobian J(x∗ ) are greater than 1 in absolute value; iii) a saddle otherwise. To find the type of a fixed point using x4 = 1 − x1 − x2 − x3 we rewrite QSO (13) as follows:

V:

⎧ x1 = bx21 + (a + b − 2d)x22 + (2b − 1)x23 + 2(b − d)x1 x2 + (2b − 1)x1 x3 + ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (2b − 1)x2 x3 − 2bx1 + 2(d − b)x2 + (1 − 2b)x3 + b; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ x = (a + b − 2d)x2 + bx2 + (2b − 1)x2 + 2(b − d)x1 x2 + (2b − 1)x1 x3 + ⎨ 2 1 2 3 ⎪ (2b − 1)x2 x3 + 2(d − b)x1 − 2bx2 + (1 − 2b)x3 + b; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ x = (a2 + c2 − 2e2 )x21 + (a2 + c2 − 2e2 )x22 + c2 x23 + (2c2 + 1 − 4e2 )x1 x2 + ⎪ ⎪ ⎪ 3 ⎪ ⎪ ⎩ 2(c2 − e2 )x1 x3 + 2(c2 − e2 )x2 x3 + 2(e2 − c2 )x1 + 2(e2 − c2 )x2 − 2c2 x3 + c2 ,

where (x1 , x2 , x3 ) ∈ {(x, y, z) : x, y, z ≥ 0, 0 ≤ x + y + z ≤ 1} and x1 , x2 , x3 − are the first three coordinates of a point lying in the three-dimensional simplex. The Jacobian at the fixed point has the representation ⎞ a11 (x∗ ) a12 (x∗ ) a13 (x∗ ) J(x∗ ) = ⎝ a21 (x∗ ) a22 (x∗ ) a23 (x∗ ) ⎠ , a31 (x∗ ) a32 (x∗ ) a33 (x∗ ) ⎛

where a11 (x∗ ) = a22 (x∗ ) = (1 + 2b − 2d)x∗1 −

(20)

2b + 1 , 2

a12 (x∗ ) = a21 (x∗ ) = (2a + 2b − 6d + 1)x∗1 + 2d −

2b + 1 , 2

a31 (x∗ ) = a32 (x∗ ) = (2a2 + 2c2 + 1 − 6e2 )x∗1 + e2 − c2 , a13 (x∗ ) = a23 (x∗ ) = 0, a33 (x∗ ) = 2(c2 − 2e2 )x∗1 − c2 . The Jacobian (20) has the following three eigenvalues

λ1 = a33 (x∗ ), λ2 = a11 (x∗ ) − a12 (x∗ ), λ3 = a11 (x∗ ) + a12 (x∗ ). or

λ1 = 2x∗1 (c2 − 2e2 ) − c2 , λ2 = 2x∗1 (2d − a) − 2d, λ3 = 2x∗1 (a + 1 + 2b − 4d) + 2d − 2b − 1. After simple calculus one has: |λ1 | = |2x∗1 (c2 − 2e2 ) − c2 | = |c2 (2x∗1 − 1) − 4e2 x∗1 | = | − 2c2 x∗3 − 4e2 x∗1 | = 2c2 x∗3 + 4e2 x∗1 = 2x∗1 (2e2 − c2 ) + c2 , |λ2 | = |2x∗1 (2d − a) − 2d| = |2d(2x∗1 − 1) − 2ax∗1 | = | − 4dx∗3 − 2ax∗1 | = 2ax∗1 + 4dx∗3 = 2x∗1 (a − 2d) + 2d. Further if a + 2b − 1 − 4d = 0 then |λ3 | = |2d − 2b − 1| = |2d − (2b √ + 1)| = |a − 2d|, and if a + 2b + 1 − 4d = 0 then we have |λ3 | = |2x∗1 (a + 1 + 2b − 4d) + 2d − 2b − 1| = |1 − D|. Theorem 2. Let d = 1/2 and if a + 2b = 1, a = c2 < 1 or a + 2b = 1, a < 1, c2 < 1, D < 4 then the unique fixed point of SSnVQSO is attracting.

270


Proof. Using 0 ≤ 2x∗1 ≤ 1, e2 = 1 − d and conditions of the Theorem it is easy to verify that: |λ1 | = 2c2 x∗3 + 4e2 x∗1 = 2c2 x∗3 + 2x∗1 < 2x∗3 + 2x∗1 = 1 and |λ2 | = 2ax∗1 +√4dx∗3 = 2ax√∗1 + 2x∗3 < 2x∗3 + 2x∗1 = 1 |λ3 | = |a − 1| = 1 − a < 1 if a + 2b = 1, a = c2 < 1 and |λ3 | = |1 − D| = 1 − D < 1 if or a + 2b = 1, a < 1, c2 < 1, D < 4. Remark 3. The Theorem 2 shows substantially difference of the case m = 4 from the case m = 3. For m = 3 a strictly non-Volterra QSO has a unique fixed point [24] and the type of the hyperbolic fixed point might be repelling or saddle, i.e. the attracting type is impossible. The SSnVQSO (for m = 4) also has a unique fixed point but the type of the hyperbolic fixed point might be attracting. 5 The ω -limit set The problem of describing the ω − limit set of a trajectory is of great importance in the theory of dynamical systems. In this section we will solve this problem for the trajectory of SSnVQSO (13) for several special cases. We denote by δi = (δi1 , δi2 , δi3 , δi4 ), i = 1, 2, 3, 4− the vertexes of the simplex S3 , where δi j is Kronecker’s symbol and the sets M12 = {x ∈ S3 : x1 = x2 }, M34 = {x ∈ S3 : x3 = x4 } and Si3j = {x ∈ S3 : xi ≥ x j }, i, j = 1, 2, 3, 4. It is easy to see that the sets M12 , M34 are invariant with respect to the operator (13). CASE 1. Let x(0) ∈ M = M12 ∩ M34 . The restriction of (13) on the set M has the form ⎧ ⎨ x1 = ax21 + (2b + 1)x23 + 4dx1 x3 ⎩ x3 = (2a2 + 1)x21 + c2 x23 + 4e2 x1 x3 .

(21)

Using 2x1 + 2x3 = 1 and x = 2x1 , x = 2x1 from (21) we obtain f (x) ≡ x =

a + 2b + 1 − 4d 2 2b + 1 x + (2d − 2b − 1)x + . 2 2

(22)

Proposition 3. Let a + 2b + 1 − 4d = 0. i) If |2d − 2b − 1| < 1 then all trajectories of (22) converge to the fixed point x∗ = 2x∗1 ; ii) If |2d − 2b − 1| ≥ 1 then any point of M is periodic point with period 2. Proof. i) Let a + 2b + 1 − 4d = 0 and |2d − 2b − 1| < 1. It is clear that 2b + 1 n−1 2b + 1 k . (2d − 2b − 1) = lim f n (x) = lim (2d − 2b − 1)n x + ∑ n→∞ n→∞ 2 k=0 4(1 + b − d) ii) Let a + 2b + 1 − 4d = 0 and |2d − 2b − 1| ≥ 1. From (12) and (10) follows d ∈ [0, 1], b ∈ [0, 12 ] and −2 ≤ 2d − 2b − 1 ≤ 1. We consider all possible cases. We assume that 2d − 2b − 1 < −1, i.e. d < b then from a + 2b + 1 − 4d = 0 we get b > d > 0.5 which is contradiction to b ∈ [0, 12 ]. We assume that 2d − 2b − 1 = 1 i.e. d − b = 1 which is equivalent to d = 1, b = 0. From a + 2b + 1 − 4d = 0 we get a = 3 which is contradiction to a ∈ [0, 1]. We assume 2d − 2b − 1 = −1 i.e. d = b then from a + 2b + 1 − 4d = 0 we get b ≥ 0.5. Since b ∈ [0, 12 ] we have b = d = 0.5 and a = 0. For a = 0 and b = d = 0.5 the function (22) has the form f (x) = 1 − x and any point from segment [0, 1] is a solution of equation f 2 (x) = x. Further we need some notations and results from the well known dynamical system generated by quadratic function Fμ (x) = μ x(1 − x).


271

Definition 6 ( [4]). Let h : A → A and g : B → B be two maps. h and g are said to be topologically conjugate if there exists a homeomorphism ϕ : A → B such that ϕ ◦ h = g ◦ ϕ . The homeomorphism ϕ is called a topological conjugacy. Theorem 4 ( [4]). Let 1 < μ < 3. 1. Fμ (x) has an attracting fixed point pμ =

μ −1 μ

and repelling fixed point 0.

2. If 0 < x < 1 then lim Fμn (x) = pμ . n→∞

1. For μ = 3 the fixed point pμ =

Theorem 5 ( [26]).

μ −1 μ

lim F n (x) n→∞ μ 2. For 3 < μ ≤

√ 6 the fixed point pμ =

{Fμ−n (pμ )}∞ n=0

is non hyperbolic but for 0 < x < 1

= pμ .

μ −1 μ

is repelling and for any initial point x(0) ∈ [0, 1] \ {0, 1} \ the trajectory tends to periodic trajectory {λˆ , λ˜ }, where 2 2 ˆλ = 1 + μ − μ − 2μ − 3 , λ˜ = 1 + μ + μ − 2μ − 3 . 2μ 2μ

Proposition 6. Let (a + 2b + 1 − 4d) = 0, i) if D ≤ 4 (see (17)) then the trajectory of (22) tends to fixed point x∗ = 2x∗1 ; ii) if 4 < D ≤ 5 then the trajectory of (22) tends to a periodic trajectory with period 2. Proof. At the line of real number R one can examine that there is a topological conjugacy ϕ such that ϕ ◦ f = Qc ◦ ϕ , where a + 2b + 1 − 4d x + d − b − 0.5, Qc (x) = x2 + c, 2 (2 − a)(2b + 1) 1 − D = . c = d(1 − d) − 4 4 As c ≤ 0 also one can check that there is a topological conjugacy ψ such that ψ ◦ Qc = Fμ ◦ ψ , where

ϕ (x) =

ψ (x) = − μx + 0.5, Fμ (x) = μ x(1 − x), √ √ μ = 1 + 1 − 4c = 1 + D. Obviously that 1 ≤ D < 4 is equivalent to 2 ≤ μ < 3 and from Theorem 4 we obtain that the function (22) on R has a repelling fixed point λ ∗ and attracting fixed point λ ∗∗ and the trajectory tends to λ ∗∗ , where

λ ∗ = ϕ −1 Qc ϕψ −1 Fμ ψ (0), λ ∗∗ = x∗ = ϕ −1 Qc ϕψ −1 Fμ ψ (pμ ). But the function (22) defined on the segment [0, 12 ]. From proof of the Theorem 1 directly follows that if (a + 2b + 1 − 4d) > 0 then λ ∗ < 0 and if (a + 2b + 1 − 4d) ≥ 0 then λ ∗ > 1. Therefore the function (22) has a unique attracting fixed point x∗ and all trajectories tends to x∗ . For D = 4 from item 1 of Theorem 5 follows that √ the fixed point x∗ still attractive. The case 4 < D ≤ 5 is equivalent to 3 < μ ≤ 1 + 5. From item 2 of Theorem 5 we have the unique fixed point is repelling and all trajectories of (22) starting an any x(0) ∈ [0, 1] \ {0, 1} \ { f −n (x∗ )}∞ n=0 tends to periodic trajectory {α , β }, where

α = ϕ −1 Qc ϕψ −1 Fμ ψ (λˆ ), β = ϕ −1 Qc ϕψ −1 Fμ ψ (λ˜ ). Moreover the subsequence { f 2n (x(0) )}∞ n=0 converges to one point of the periodic trajectory and the subse− converges to another [26]. quence { f 2n+1 (x(0) )}∞ n=0

272


From Proposition 3 and Proposition 6 we obtain respectively for the trajectories of SSnVQSO starting at any initial point from the set M the following Theorem: Theorem 7. i) If a = 0, b = d =

1 2

then all points of M are periodic points of the operator (13) with period 2;

ii) if a + 2b + 1 − 4d = 0 and 4 < D ≤ 5 then the trajectory of (13) tends to the periodic trajectory {x, ˆ x}, ˜ where α α 1−α 1−α β β 1−β 1−β , , , , x˜ = , , , ; xˆ = 2 2 2 2 2 2 2 2 iii) In other cases the trajectory of (13) tends to the unique fixed point x∗ . Remark 4. The SSnVQSO might has infinitely many periodic points with period 2. CASE 2. Let x(0) ∈ M = M34 \ M. Then the restriction of QSO (13) on the set M34 has the following form ⎧ x1 = ax22 + (2b + 1)x23 + 4dx2 x3 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ x2 = ax21 + (2b + 1)x23 + 4dx1 x3 , (23) V: = a x2 + a x2 + c x2 + x x + 2e x x + 2e x x , ⎪ ⎪ x 2 1 2 2 2 3 1 2 2 1 3 2 2 3 ⎪ 3 ⎪ ⎪ ⎪ ⎩ x4 = a2 x21 + a2 x22 + c2 x23 + x1 x2 + 2e2 x1 x3 + 2e2 x2 x3 . The following theorem describes ω − limiting points for the SSnVQSO (23). Theorem 8.

a) If max{a, 2d} < 1 then ω (x(0) ) ⊂ M;

b) If 2d < a = 1 and x(0) ∈ M \{δ1 , δ2 } then ω (x(0) ) ⊂ M. Moreover V 2 (δ2 ) = V (δ1 ) = δ2 . Also if a < 2d = 1 then ω (x(0) ) ⊂ M; c) If a = 2d = 1 then the trajectory of corresponding SSnVQSO tends to some periodic trajectory with period 2; d) If 2d > a = 1 then ω (x(0) ) = {δ1 , δ2 }; e) If a = d = 1 − α , c2 < 4/11, 5/11 < α < 1/2 then ω (x(0) ) ⊂ M; f) If 0 < a∗ < a < 5/8, 2d = 2 − a, 0 < c2 < 1/2, a∗ = 2 − c2 − (2 − c2 )2 − 2 then ω (x(0) ) = {δ1 , δ2 }. Proof. a) Let max{a, 2d} < 1. We consider a function θ (x) = |x1 − x2 |. We claim that lim θ (x(n) ) = 0.

n→∞

Indeed denoting ξ = max{a, 2d} and using x1 + x2 + 2x3 = 1 from (23) we get

θ (V (x)) = aθ (x)(x1 + x2 ) + 4dx3 θ (x) < ξ θ (x)(x1 + x2 + 2x3 ) = ξ θ (x) and consequently

θ (x(n+1) ) < ξ θ (x(n) ) < ξ n+1 θ (x(0) ), n = 0, 1, 2, . . .

whence follows (24). So ω (x(0) ) ⊂ M. b) Let 2d < a = 1 and x(0) ∈ M \ {δ1 , δ2 } then according operator (23) has the form

(24)


V:

273

⎧ x1 = x22 + (2b + 1)x23 + 4dx2 x3 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ x2 = x21 + (2b + 1)x23 + 4dx1 x3 ,

(25)

⎪ ⎪ x3 = c2 x23 + x1 x2 + 2e2 x1 x3 + 2e2 x2 x3 , ⎪ ⎪ ⎪ ⎪ ⎩ x4 = c2 x23 + x1 x2 + 2e2 x1 x3 + 2e2 x2 x3 .

We consider the function θ (x) = |x1 − x2 |. Using 2d < a = 1 and x1 + x2 + 2x3 = 1 from (25) we have

θ (V (x)) = θ (x)(x1 + x2 + 4dx3 ) ≤ θ (x) < 1, and the following (n)

(n)

(n)

θ (V (x(n) )) = θ (x(n) )(x1 + x2 + 4dx3 ) ≤ θ (x(n) ) n = 0, 1, 2, . . . , i.e. θ (x(n) ) is bounded from below and a monotone decreasing sequence. Therefore there exists following limit lim θ (x(n) ) = τ .

n→∞

Obviously 0 ≤ τ < 1. We claim that τ = 0. We assume that τ > 0, then (n)

θ (x(n) )(1 + 2(2d − 1)x3 ) θ (x(n+1) ) (n) = lim = 1 + 2(2d − 1) lim x3 . (n) n→∞ θ (x(n) ) n→∞ n→∞ θ (x )

1 = lim

(n)

3 ) ⊂ S3 and V (S3 ) ⊂ S3 . So we consider a operator Therefore we have lim x3 = 0. From (25) one gets V (S12 21 21 12 n→∞

W (y) = V (V (y)) and it is easy to verify that: i) Fix(W ) = {δ1 , δ2 , x∗ }, where δ1 , δ2 − are vertexes of simplex and x∗ is the fixed point of QSO V ; 3 ) ⊂ S3 and W (S3 ) ⊂ S3 . ii) W (S12 12 21 21 3 be an initial point then for the trajectory y(n+1) = W (y(n) ), n = 0, 1, . . . from W (S3 ) ⊂ S3 Let y(0) ∈ S12 12 12 (n) (n) (n) (n) and {y(n) } ⊂ {x(n) } we obtain lim θ (y(n) ) = lim (y1 − y2 ) = τ and from lim y3 = lim x3 = 0 follows (n)

(n)

n→∞

n→∞

n→∞

n→∞

lim (y + y2 ) = 1. n→∞ 1 Accordingly exists the following limits 1 1+τ (n) (n) (n) (n) , = lim (y1 + y2 ) + (y1 − y2 ) = 2 n→∞ 2 1 1−τ (n) (n) (n) (n) (n) , lim y2 = lim (y1 + y2 ) − (y1 − y2 ) = n→∞ 2 n→∞ 2 1+τ 1−τ (n) and it should be the fixed point of operator W . For i.e., the trajectory {y } tends to λ = 2 , 2 , 0, 0 (n) lim y n→∞ 1

2d < a = 1 we shall show that λ = x∗ . Indeed in this case a + 2b + 1 − 4d = 2 + 2b − 4d = 2(1 + b − 2d) > 0 then from Theorem 1 in the case λ = x∗ and a + 2b + 1 − 4d = 0 we have −2d + (2b + 1) + (1 − 2d)2 ∗ ∗ =0 x3 = x4 = 2(2b + 2 − 4d) and we get 2(b + 1 − 2d) = 0 which is contradiction to a + 2b + 1 − 4d > 0.

274

U.U. Jamilov / Discontinuity, Nonlinearity, and Complexity 5(3) (2016) 263–283 3 ⇒ λ = δ remains only λ = δ i.e. we get τ = 1 which contradicts to 0 ≤ τ < 1. Since δ2 ∈ / S12 2 1

Similarly for y(0)

3 ∈ S21

we get that the trajectory

y(n+1)

= W (y(n) ),

1−τ 1+τ 2 , 2 , 0, 0

n = 0, 1, . . . tends to λ =

and λ = δ1 . Again we will get τ = 1 which contradicts to 0 ≤ τ < 1. So we have lim θ (y(n) ) = lim θ (x(2n) ) = n→∞

τ = 0 and since 0 ≤ θ (x(2n+1) ) ≤ θ (x(2n) ) we obtain lim θ (x(2n+1) ) = τ = 0.

n→∞

n→∞

Consequently τ = 0 and for an initial point from M34 \ {M ∪ {δ1 , δ2 }} we obtain ω (x(0) ) ⊂ M and it is easy to see that V 2 (δ2 ) = V (δ1 ) = δ2 . In the case a < 1, 2d = 1 and similarly one can prove that for any initial point from M34 \ M follows ω (x(0) ) ⊂ M. c) Let a = 2d = 1 and x(0) ∈ M . Then corresponding SSnVQSO (13) has the form ⎧ x = x22 + (2b + 1)x23 + 2x2 x3 , ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎨ x2 = x21 + (2b + 1)x23 + 2x1 x3 , V: ⎪ ⎪ x3 = c2 x23 + x1 x2 + x1 x3 + x2 x3 , ⎪ ⎪ ⎪ ⎪ ⎩ x4 = c2 x23 + x1 x2 + x1 x3 + x2 x3 .

(26)

We consider the function θ (x) = x1 − x2 . From (26) we get

θ (V (x)) = −θ (x) and for any initial point we have

θ (x(n) ) = (−1)n θ (x(0) ), n = 0, 1, 2, . . . For the trajectory of (26) we get (2n+1)

x(2n+1) = (x2

(2n)

x(2n) = (x2

(0)

(0)

(2n+1)

− x1 + x2 , x2 (0)

(0)

(2n)

(0)

(0)

(2n+1)

, x1 + x3 − x2 (0)

(0)

(2n)

(0)

(0)

(2n+1)

, x1 + x3 − x2

(0)

(0)

(2n)

+ x1 − x2 , x2 , x2 + x3 − x2 , x2 + x3 − x2 )

i) We suppose b = 0 then corresponding SSnVQSO (26) has the form ⎧ x1 = x22 + 2x2 x3 + x23 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ x2 = x21 + 2x1 x3 + x23 , V: ⎪ ⎪ x3 = x23 + x1 x2 + x1 x3 + x2 x3 , ⎪ ⎪ ⎪ ⎪ ⎩ x4 = x23 + x1 x2 + x1 x3 + x2 x3 .

)

(27) (28)

(29)

4

Using xi = 1 − ∑ x j we rewrite (29) as follow j=1 j=i

V:

⎧ x1 = x2 − x1 x2 + x23 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ x2 = x1 − x1 x2 + x23 , ⎪ ⎪ x3 = x3 + x1 x2 − x23 , ⎪ ⎪ ⎪ ⎪ ⎩ x4 = x3 + x1 x2 − x23 .

(30)


275

We consider a function σ (x) = x1 x2 − x23 and denote σ ≡ σ (x(0) ) then from (30) we get

σ (x ) = x1 x2 − (x3 )2 = (x2 − σ (x))(x1 − σ (x)) − (x3 + σ (x))2 = = x2 x1 − (x1 + x2 )σ (x) + (σ (x))2 − x23 − 2x3 σ (x) − (σ (x))2 = 0. ˆ x}, ˜ where Consequently the image of an initial point from set S3 \ M belong to the periodic trajectory {x, (0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

xˆ = (x2 − σ , x1 − σ , x3 + σ , x3 + σ ), x˜ = (x1 − σ , x2 − σ , x3 + σ , x3 + σ ). We claim that the SSNVQSO (30) is ergodic. Indeed we denote by yn =

x(0) + x(1) + · · · + x(n−1) n

and using it we get y1 = x(0) , y2 = y2n = and

x(0) + xˆ x(0) + xˆ + x˜ , y3 = ,..., 2 3

x(0) + nxˆ + (n − 1)x˜ x(0) + nxˆ + nx˜ , y2n+1 = ,... 2n 2n + 1

xˆ + x˜ . 2 ii) We suppose b > 0 and rewrite the operator (26) in the following form ⎧ x1 = x2 + σ (x), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ x2 = x1 + σ (x), V: ⎪ ⎪ x3 = x3 − σ (x), ⎪ ⎪ ⎪ ⎪ ⎩ x4 = x3 − σ (x), lim y2n = lim y2n+1 =

n→∞

n→∞

where σ (x) = (2b + 1)x23 − x1 x2 . Using (31) one has

σ (x ) = (2b + 1)(x3 )2 − x1 x2 = (2b + 1)(x3 − σ (x))2 − (x1 + σ (x))(x2 + σ (x)) = 2b(x3 − σ (x))2 − 2bx23 = 2b · σ (x)(σ (x) − 2x3 ). So we get

σ (x) − 2x3 = (2b + 1)x23 − 2x3 − x1 x2 ≤ 2x23 − 2x3 − x1 x2 ≤ 0

and using it we obtain sgn(σ (x )) = −sgn(σ (x)) and σ (x(n+1) ) (n) (n) (n) 2 (n) (n) (n) σ (x(n) ) = 2b|σ (x ) − 2x3 | ≤ 2b|2x3 − 2(x3 ) + x1 x2 | ≤ (n) 2 (n) (1 − 2x (n) (n) (n) 2 2 3 ) 2b2x3 − 2(x3 ) + = 2b|x3 − (x3 ) + 1/4| ≤ b ≤ 1/2. 4 Therefore

lim |σ (x(n) )| = 0

n→∞

(31)

276


and from the second relation of (31) sgn(x2 − x2 ) = sgn(σ (x) + σ (x )) = sgn(σ (x)).

(32)

It is clear that if σ (x(0) ) = 0 then (0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

ω (x(0) ) = {(x1 , x2 , x3 , x3 ); (x2 , x1 , x3 , x3 )}. (2n)

If σ (x(0) ) > 0(σ (x(0) ) < 0) then due to equation (32) we have x2 is a monotone increasing (decreasing). (2n) Consequently there exists lim x2 = x∗2 and from (28) follows existence of the following limits n→∞

(2n) lim x n→∞ 1

(2n)

= x∗1 , lim x3 n→∞

= x∗3 .

Using them we obtain (2n+1) lim x n→∞ 2

(2n)

= lim x1 n→∞

+ lim σ (x(2n) ) = x∗1 n→∞

and similarly from (27) follows existence of other limits (2n+1) lim x n→∞ 1

Finally we obtain

(2n+1)

= x∗2 , lim x3 n→∞

= x∗3 .

ω (x(0) ) = {(x∗1 , x∗2 , x∗3 , x∗3 ); (x∗2 , x∗1 , x∗3 , x∗3 )}.

d) Let a = 1, 2d > 1 and x(0) ∈ M . The corresponding operator has the form ⎧ x = x22 + (2b + 1)x23 + 4dx2 x3 , ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎨ x2 = x21 + (2b + 1)x23 + 4dx1 x3 , V: ⎪ ⎪ x3 = c2 x23 + x1 x2 + 2e2 x1 x3 + 2e2 x2 x3 , ⎪ ⎪ ⎪ ⎪ ⎩ x4 = c2 x23 + x1 x2 + 2e2 x1 x3 + 2e2 x2 x3 .

(33)

We consider θ (x) = |x1 − x2 |. From (33) we get

θ (V (x)) = θ (x)(x1 + x2 + 4dx3 ) ≥ θ (x) > 0 and

(n)

(n)

(n)

θ (x(n+1) ) = θ (x(n) )(x1 + x2 + 4dx3 ) ≥ θ (x(n) ), n = 0, 1, 2, . . .

Obviously that the sequence θ (x(n) ) is a monotone increasing and bounded from above. Therefore there is exists the following limit lim θ (x(n) ) = τ . n→∞

It is clear 0 < τ ≤ 1. We claim τ = 1. We suppose τ < 1 then (n)

1 − θ (x(n) )(1 + 2(2d − 1)x3 ) 1 − θ (x(n+1) ) = lim = 1 = lim n→∞ 1 − θ (x(n) ) n→∞ 1 − θ (x(n) )

θ (x(n) ) (n) τ (n) x3 = 1 − 2(2d − 1) lim x (n) n→∞ 1 − θ (x ) 1 − τ n→∞ 3

= 1 − 2(2d − 1) lim


277

(n)

So we have lim x3 = 0 and using x1 + x2 = 1 − 2x3 from (33) we get n→∞

(n) (n) lim x x n→∞ 1 2

= lim

n→∞

(n+1) (n) (n) (n) − c2 (x3 )2 − 2e2 (1 − 2x3 )x3 x3

=0

(34)

We consider the operator W (y) = V (V (y)) and as was showed in the item b), there exist limits (n)

lim y1 =

n→∞

1+τ 1−τ (n) , lim y2 = . n→∞ 2 2

Therefore we obtain

1 − τ2 >0 n→∞ 2 It is contradiction to (34) because by assumption we have τ < 1. So τ = 1 then from max θ (x) = 1 ⇔ x = δ1 or x = δ2 , (n) (n)

0 = lim y1 y2 =

x∈S3

follows ω (x(0) ) = {δ1 , δ2 } . e) Let a = d = 1 − α , c2 < 4/11, 5/11 < α < 1/2 and x(0) ∈ M then the according operator (23) has form ⎧ x1 = (1 − α )x22 + (2b + 1)x23 + 4(1 − α )x2 x3 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ x2 = (1 − α )x21 + (2b + 1)x23 + 4(1 − α )x1 x3 , (35) V: = α x2 + α x2 + c x2 + x x + 2α x x + 2α x x , ⎪ ⎪ x 2 1 2 1 3 2 3 ⎪ 3 2 1 2 2 3 ⎪ ⎪ ⎪ ⎩ α 2 α 2 x4 = 2 x1 + 2 x2 + c2 x23 + x1 x2 + 2α x1 x3 + 2α x2 x3 . We consider the function θ (x) = |x1 − x2 |. Using a = 1 − α , 2d = 2 − 2α and x1 + x2 + 2x3 = 1 from (35) we have θ (V (x)) = θ (x)|a(x1 + x2 ) + 4dx3 | = θ (x)|a + 2(2d − a)x3 | =

θ (x)|1 − α + 2(1 − α )x3 | = θ (x)(1 − α )(1 + 2x3 ) ≤ θ (x)(1 + x3 − α + (1 − 2α )x3 ) From (35) we have x3 =

α 2 α 2 x + x + c2 x23 + x1 x2 + 2α x1 x3 + 2α x2 x3 = 2 1 2 2

α (x1 + x2 )2 + c2 x23 + (1 − α )x1 x2 + 2α (x1 + x2 )x3 = 2 α (1 − 2x3 )2 + c2 x23 + (1 − α )x1 x2 + 2α (1 − 2x3 )x3 = 2 1−α α α (1 − 2x3 )2 + ≤ (c2 − 2α )x23 + (1 − α )x1 x2 + ≤ (c2 − 2α )x23 + 2 4 2 1+α (c2 + 1 − 3α )x23 − (1 − α )x3 + 4 Using the last inequality we obtain x3 − α + (1 − 2α )x3 ≤ (c2 + 1 − 3α )x23 − (1 − α )x3 +

1+α − α + (1 − 2α )x3 ≤ 4

278


(c2 + 1 − 3α )x23 − (1 − α )x3 +

5 − 11α . 4

Since c2 < 4/11, 5/11 < α < 1/2 it is easy to verify that −2 < (c2 + 1 − 3α )x23 − (1 − α )x3 +

5 − 11α 0, then (n)

θ (x(n) )(1 − α )(1 + 2x3 ) θ (x(n+1) ) (n) = lim = 1 + 2 lim x3 . 1 = lim (n) n→∞ θ (x(n) ) n→∞ n→∞ θ (x ) Therefore we have

(n) lim x n→∞ 3

(n)

(n)

= 0, lim (x1 + x2 ) = 1. n→∞

Consequently using formulas 1 (n) 1 (n) (n) (n) (n) (n) (n) (n) (n) (n) x1 = ((x1 + x2 ) + (x1 − x2 )), x2 = ((x1 + x2 ) + (x1 − x2 )), 2 2 we get existence of the following limits (2n) lim x n→∞ 1

=

1+τ 1−τ (2n) (0) (0) , lim x2 = , if x1 ≥ x2 n→∞ 2 2

1−τ 1+τ (2n) (0) (0) , lim x2 = , if x1 ≤ x2 n→∞ 2 2 So from the third equation of (35) one gets (2n) lim x n→∞ 1

(2n+1)

x3

=

=

α (2n) 2 α (2n) 2 (2n) (2n) (2n) (2n) (2n) (2n) (2n) (x ) + (x2 ) + c2 (x3 )2 + x1 x2 + 2α x1 x3 + 2α x2 x3 , 2 1 2

and

(2n+1)

0 = lim (x3 n→∞

(2n)

(2n)

− c2 (x3 )2 − 2α (x1

(2n)

(2n)

+ x2 )x3 ) =

α (2n) α (2n) (2n) (2n) lim ( (x1 )2 + (x2 )2 + x1 x2 ) > n→∞ 2 2 α (2n) α (2n) α α (2n) (2n) (2n) (2n) lim ( (x1 )2 + (x2 )2 + α x1 x2 ) = lim (x1 + x2 )2 = > 0 n→∞ 2 2 2 n→∞ 2 Since the last relation is contradiction so we have lim θ (x(2n) ) = 0 and using n→∞

0 ≤ lim θ (x(2n+1) ) ≤ lim θ (x(2n) ) n→∞

n→∞

we get τ = 0. Consequently for an initial starting point x(0) ∈ M the set of limiting points is a subset of M, i.e. ω (x(0) ) ⊂ M.


279

f) Let x(0) ∈ M and 0 < a∗ < a < 5/8, 2d = 2 − a, 0 < c2 < 1/2, a∗ = 2 − c2 − (2 − c2 )2 − 2 then the according operator (23) has the form ⎧ x1 = ax22 + (2b + 1)x23 + 4dx2 x3 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ x2 = ax21 + (2b + 1)x23 + 4dx1 x3 , (36) V: ⎪ ⎪ x3 = 12 (1 − a)x21 + 12 (1 − a)x22 + c2 x23 + x1 x2 + ax1 x3 + ax2 x3 , ⎪ ⎪ ⎪ ⎪ ⎩ x4 = 12 (1 − a)x21 + 12 (1 − a)x22 + c2 x23 + x1 x2 + ax1 x3 + ax2 x3 , We consider the function θ (x) = |x1 − x2 |. Using 0 < a < 1, 2d = 2 − a, a2 = (1 − a)/2, 2e2 = a and x1 + x2 + 2x3 = 1 from (36) we have

θ (V (x)) = θ (x)(a(x1 + x2 ) + 4dx3 ) = θ (x)(a(x1 + x2 ) + (2 − a)2x3 ) = θ (x)(a + 4(1 − a)x3 ) We claim that x3 ≥ 1/4 for all x ∈ intS3 = {x ∈ S3 : x1 x2 x3 x4 > 0}. Indeed, if x ∈ intS3 then from (35) we have 1−a 2 1−a 2 x + x + c2 x23 + x1 x2 + ax1 x3 + ax2 x3 2 1 2 2 1−a (x1 + x2 )2 + c2 x23 + a(x1 + x2 )x3 + ax1 x2 = 2 1−a (1 − 2x3 )2 + c2 x23 + a(1 − 2x3 )x3 + ax1 x2 = 2 1−a + ax1 x2 . = (c2 + 2 − 4a)x23 + (3a − 2)x3 + 2

x3 =

Using last equation we get 1 1 − 2a = (c2 + 2 − 4a)x23 + (3a − 2)x3 + + ax1 x2 . 4 4 Since c2 ≤ 1/2 it is easy to check that x3 −

a2 − 2(2 − c2 )a + 2 ≤ 0, ∀a ∈ (a∗ , 5/8),

where a∗ = 2 − c2 − Therefore

(2 − c2 )2 − 2 ∈ [0, 1]

(3a − 2)2 − (1 − 2a)(c2 + 2 − 4a) − 4a(c2 + 2 − 4a)x1 x2 = a2 − 2(2 − c2 )a + 2 − 4a(c2 + 2 − 4a)x1 x2 < 0. Since c2 + 2 − 4a > 0 for a ∈ (a∗ , 5/8) we obtain x3 ≥ 1/4. Consequently θ (V (x)) ≥ θ (x) and

θ (V (x(n) )) ≥ θ (x(n) ) n = 1, 2, . . . , i.e. the θ (x(n) ) is bounded from above and monotone increasing sequence. Therefore there exists the following limit lim θ (x(n) ) = τ . n→∞

Similarly as the case d) one can proof that τ = 1. Consequently for any initial point x(0) ∈ M the set of limiting points is equal ω (x(0) ) = {δ1 , δ2 }.

280


Numerical calculations suggest the following conjecture Conjecture 9.

i) If a ≈ 0, 2d ≈ 1 and x(0) ∈ M then ω (x(0) ) ⊂ M;

ˆ x} ˜ ⊂ M34 . ii) If a ≈ 1, 2d ≈ 2 and x(0) ∈ M then ω (x(0) ) = {x, The case x(0) ∈ M12 \ M can be handled in the similar way, i.e. in this case corresponding SSnVQSO (13) has the form ⎧ x1 = ax21 + bx23 + bx24 + 2dx1 x3 + 2dx1 x4 + x3 x4 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ x2 = ax21 + bx23 + bx24 + 2dx1 x3 + 2dx1 x4 + x3 x4 (37) V: ⎪ ⎪ x3 = c2 x24 + (2a2 + 1)x21 + 4e2 x1 x4 ⎪ ⎪ ⎪ ⎪ ⎩ x4 = c2 x23 + (2a2 + 1)x21 + 4e2 x1 x3 . and for the operator (37) the following assertions hold Theorem 10.

a) If max{c2 , 2e2 } < 1 then ω (x(0) ) ⊂ M;

b) If 2e2 < c2 = 1 and x(0) ∈ M \ {δ3 , δ4 } then ω (x(0) ) ⊂ M. Moreover V 2 (δ4 ) = V (δ3 ) = δ4 . Also if c2 < 2e2 = 1 then ω (x(0) ) ⊂ M; c) If c2 = 2e2 = 1 then the trajectory of corresponding SSnVQSO tends to some periodic trajectory with period 2; d) If 2e2 > c2 = 1 then ω (x(0) ) = {δ3 , δ4 }; e) If c2 = e2 = 1 − β , a < 4/11, 5/11 < β < 1/2 then ω (x(0) ) ⊂ M; f) If 0 < c∗2 < c2 < 5/8, 2e2 = 2 − c2 a, 0 < a < 1/2, c∗2 = 2 − a − (2 − a)2 − 2 then ω (x(0) ) = {δ3 , δ4 }. ˜ CASE 3. Let M˜ = {M12 ∪ M34 } and an arbitrary x(0) ∈ S3 \ M. Theorem 11. b) ω (x(0) ) ⊂

a) If a < 1, 2d = 1, c2 < 1 then ω (x(0) ) ⊂ M ; M12 , a < 1, 2d = c2 = 1 or a < 1, 2d < 1, c2 < 1 M34 , a = 2d = 1, c2 < 1 or a < 1, 2d > 1, c2 < 1

;

c) If a = 2d = c2 = 1 then x = V (x(0) ) belong to periodic trajectory by period 2; d) 1) If a = 1, 2d < 1, c2 < 1, x(0) ∈ S3 \ (M˜ ∪ {δ1 , δ2 }) then ω (x(0) ) ⊂ M12 , moreover V 2 (δ1 ) = V (δ2 ) = δ1 ; 2) If a < 1, 2d > 1, c2 = 1, x(0) ∈ S3 \ (M˜ ∪ {δ3 , δ4 }) then ω (x(0) ) ⊂ M34 , moreover V 2 (δ3 ) = V (δ4 ) = δ3 ; {δ1 , δ2 }, a = 1, 2d > 1, c2 < 1, ; e) ω (x(0) ) ⊂ {δ3 , δ4 }, a < 1, 2d < 1, c2 = 1, f) 1) If a = 1, 2d < 1, c2 = 1, x(0) ∈ S3 \ (M˜ ∪ {δ1 , δ2 }) then ω (x(0) ) = {δ3 , δ4 }, moreover V 2 (δ2 ) = V (δ1 ) = δ2 ; 2) If a = 1, 2d > 1, c2 = 1, x(0) ∈ S3 \ (M˜ ∪ {δ3 , δ4 }) then ω (x(0) ) = {δ1 , δ2 }, moreover V 2 (δ3 ) = V (δ4 ) = δ3 ;


281

Proof. a) Let a < 1, 2d = 1, c2 < 1. We consider the functions ϕ (x) = |x1 − x2 | and ψ (x) = |x3 − x4 |. Repeating the proof as section b) Case 2 we obtain lim ϕ (x(n) ) = lim ψ (x(n) ) = 0.

n→∞

n→∞

Therefore ω (x(0) ) ⊂ M. b) Let a = 1, 2d = 1, c2 < 1. We consider the function ψ (x) = |x3 − x4 |. Similarly as in the section b) Case 2 we obtain lim ψ (x(n) ) = 0. n→∞

Therefore for any point x(0) ∈ S3 \ M˜ we get ω (x(0) ) ⊂ M34 . Similarly when a < 1, 2d = 1, c2 = 1 using the function ϕ (x) = |x1 − x2 | one can get ω (x(0) ) ⊂ M12 . Let a < 1, 2d < 1, c2 < 1. Using the function ϕ (x) = |x1 − x2 | and results of section a) Case 2 we obtain lim ϕ (x(n) ) = 0,

n→∞

and so ω (x(0) ) ⊂ M12 . Similarly when a < 1, 2d > 1, c2 < 1 using function ψ (x) = |x3 − x4 | we get ω (x(0) ) ⊂ M34 . c) Let a = 2d = c2 = 1, Then corresponding SSnVQSO (13) has the form ⎧ x1 = x22 + x2 x3 + x2 x4 + x3 x4 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ x2 = x21 + x1 x3 + x1 x4 + x3 x4 , V: 2 ⎪ ⎪ ⎪ x3 = x4 + x1 x2 + x1 x4 + x2 x4 , ⎪ ⎪ ⎪ ⎩ x4 = x23 + x1 x2 + x1 x3 + x2 x3 .

(38)

4

Using xi = 1 − ∑ x j we rewrite (38) as j=1 j=i

V:

get

⎧ x1 = x2 − x1 x2 + x3 x4 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ x2 = x1 − x1 x2 + x3 x4 ,

(39)

⎪ ⎪ x3 = x4 + x1 x2 − x3 x4 , ⎪ ⎪ ⎪ ⎪ ⎩ x4 = x3 + x1 x2 − x3 x4 .

We consider a function σ (x) = x1 x2 − x3 x4 and denote σ ≡ σ (x(0) ) then from (39) as in item c) Case 2 we

σ (x ) = x1 x2 − x3 x4 = 0. ˆ x}, ˜ where Consequently the image of an initial point from the set S3 \ M˜ belong to the periodic trajectory {x, (0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

xˆ = (x2 + σ , x1 + σ , x4 − σ , x3 − σ ), x˜ = (x1 + σ , x2 + σ , x3 − σ , x4 − σ ).

282


So it is easy to see that the SSNVQSO (39) is ergodic, too. d) Let a = 1, 2d < 1, c2 < 1. We consider the function ϕ (x) = |x1 − x2 |. From results shown in the section b) case 2 we get lim ϕ (x(n) ) = 0. n→∞

Consequently for any initial point x(0) ∈ S3 \ (M˜ ∪ {δ1 , δ2 }) we get ω (x(0) ) ⊂ M12 it is easy to see that 2 ) = V (δ1 ) = δ2 .

V 2 (δ

Similarly when a < 1, 2d > 1, c2 = 1 using the function ψ (x) = |x3 − x4 | one can prove that for any initial point x(0) ∈ S3 \ (M˜ ∪ {δ3 , δ4 }) it follows ω (x(0) ) ⊂ M34 and V 2 (δ4 ) = V (δ3 ) = δ4 . e) Let a = 1, 2d > 1, c2 < 1. Using the functions ϕ (x) = |x1 − x2 |, ψ (x) = |x3 − x4 | and the results of sections a), d) Case 2 we obtain lim ψ (x(n) ) = 0, n→∞

and so for an initial point x(0) ∈ S3 \ M˜ the trajectory of QSO (13) tends to a periodic trajectory {δ1 , δ2 }. Similarly in the case a < 1, 2d < 1, c2 = 1 one can show that ω (x(0) ) = {δ3 , δ4 }. f) Let a = 1, 2d < 1, c2 = 1. We consider the functions ϕ (x) = |x1 − x2 | and ψ (x) = |x3 − x4 |. Using the results of sections b), d) case 2 we get lim ϕ (x(n) ) = 0 and for an initial point x(0) ∈ S3 \ (M˜ ∪ {δ1 , δ2 }) it follows n→∞

that ω (x(0) ) = {δ3 , δ4 } and it is easy to see that V 2 (δ2 ) = V (δ1 ) = δ2 . Similarly when a = 1, 2d > 1, c2 = 1 one can show that for arbitrary initial x(0) ∈ S3 \ (M˜ ∪ {δ3 , δ4 }) one has ω (x(0) ) = {δ1 , δ2 } and V 2 (δ4 ) = V (δ3 ) = δ4 . Remark 5. The Volterra QSO has no periodic trajectories [12]- [14] and the above results show that SSnVQSO has periodic trajectories and claim c) of Theorem 11 shows that periodic trajectories might be infinitely many. 6 Conclusion We have considered the SSnVQSOs defined on three dimensional simplex and studied their trajectory behaviors. The condition of Definition 1 has the following biological interpretation: ‘offspring’ k doesn’t repeat properties of its ‘parents’ i and j. From a biological point of view it is natural to expect that ‘offsprings’ of these populations will not have a stable structure, as ‘offsprings’ lose all the properties of their ‘parents’. In [24] a strict mathematical proof of this phenomenon in the case m = 3 is given. In the case SSnVQSOs it is proven that any SSnVQSO has a unique fixed point and it can be an attracting point, that is the above mentioned populations might have a stable future. Moreover we showed that SSnVQSO (unlike the Volterra QSO) has the periodic trajectories and such trajectories might be infinitely many. Acknowledgements The Author grateful to professors U. A. Rozikov, J. Blath and M. Scheutzow for helpful discussions. He thanks IMU Berlin Einstein Foundation Program (EFP), Berlin Mathematical School(BMS) for scholarship and for support of his visit to Technical University(TU) Berlin and TU Berlin for kind hospitality.


283

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On the Existence of Stationary Solutions for Some Systems of Integro-Differential Equations with Anomalous Diffusion Vitali Vougalter1†, Vitaly Volpert2 1 2

Department of Mathematics, University of Toronto, Toronto, Ontario, M5S 2E4, Canada Institute Camille Jordan, UMR 5208 CNRS, University Lyon 1, Villeurbanne, 69622, France Submission Info Communicated by Valentin Afraimovich Received 1 February 2016 Accepted 17 February 2016 Available online 1 October 2016

Abstract The article is devoted to the proof of the existence of solutions of a system of integro-differential equations appearing in the case of anomalous diffusion when the negative Laplacian is raised to some fractional power. The argument relies on a fixed point technique. Solvability conditions for elliptic operators without Fredholm property in unbounded domains along with the Sobolev inequality for a fractional Laplace operator are being used.

Keywords Integro-differential equations Non Fredholm operators Sobolev spaces


1 Introduction In the present article we study the existence of stationary solutions of the following system of N ≥ 2 integro-differential equations ˆ ∂ um = −Dm (−Δ)s um + Km (x − y)gm (u(y,t))dy + fm (x), (1) ∂t R3 with 1 ≤ m ≤ N and 1/4 < s < 3/4, which appears in cell population dynamics. Here Dm are the diffusion coefficients and fm (x) denote the influxes of cells for different genotypes. Note that the restriction on the power s here comes from the solvability conditions of our problem. The space variable x corresponds to the cell genotype, um (x,t) denote the cell densities for various groups of cells as functions of their genotype and time, such that u(x,t) = (u1 (x,t), u2 (x,t), . . . , uN (x,t))T . The right side of this system of equations describes the evolution of cell densities due to cell proliferation, mutations and cell influx. In this context the anomalous diffusion terms correspond to the change of genotype via small random mutations, and the nonlocal terms describe large mutations. Here gm (u) denote the rates of cell birth which depend on u (density dependent proliferation), and the functions Km (x − y) express the proportions † Corresponding

author. Email address: [email protected], [email protected]


286

Vitali Vougalter, Vitaly Volpert / Discontinuity, Nonlinearity, and Complexity 5(3) (2016) 285–295

of newly born cells which change their genotype from y to x. We assume here that they dependent on the distance between the genotypes. The operator (−Δ)s , 1/4 < s < 3/4 in system (1) represents a particular case of the anomalous diffusion actively studied in relation with various applications in plasma physics and turbulence [1, 2], surface diffusion [3, 4], semiconductors [5] and so on. The physical meaning of the anomalous diffusion is that the random process occurs with longer jumps in comparison with normal diffusion. The moments of jump length distribution are finite in the case of normal diffusion, but this is not the case for the anomalous diffusion. The operator (−Δ)s , 1/4 < s < 3/4 is defined by virtue of the spectral calculus. A similar problem in the case of the standard Laplacian in the diffusion term was treated recently in [6]. Stationary solutions of (1) will occur when ∂ um /∂ t = 0 for all 1 ≤ m ≤ N. We set all Dm = 1 and prove the existence of solutions of the system ˆ 1 3 s Km (x − y)gm (u(y))dy + fm (x) = 0, −(−Δ) um + 1 the dimension of its kernel and the codimension of its image are not finite. The present article is devoted to the studies of some properties of such operators. Note that elliptic equations with non Fredholm operators were treated actively in recent years. Approaches in weighted Sobolev and Hölder spaces were developed in [7–11]. The non Fredholm Schrödinger type operators were studied with the methods of the spectral and the scattering theory in [12–14]. The Laplacian with drift from the point of view of non Fredholm operators was treated in [15] and linearized Cahn-Hilliard equations in [16] and [17]. Nonlinear non Fredholm elliptic problems were studied in [18] and [19]. Important applications to the theory of reaction-diffusion equations were developed in [20, 21]. Non Fredholm operators arise also when studying wave systems with an infinite number of localized traveling waves (see [22]). In particular, when a = 0 the operator A is Fredholm in some properly chosen weighted spaces (see [7–10], [11]). However, the case of a = 0 is significantly different and the approach developed in these articles cannot be applied. Front propagation problems with anomalous diffusion were treated actively in recent years (see e.g. [23, 24]). Let us set Km (x) = εm Km (x) for 1 ≤ m ≤ N with εm ≥ 0, such that

ε := max εm 1≤m≤N

and suppose that the following assumption is satisfied. Assumption 1. Let 1 ≤ m ≤ N and consider 1/4 < s < 3/4. Let fm (x) : R3 → R, such that fm (x) ∈ L1 (R3 ) and (−Δ)1−s fm (x) ∈ L2 (R3 ). For some 1 ≤ m ≤ N, fm (x) is nontrivial. Assume also that Km (x) : R3 → R, such that Km (x) ∈ L1 (R3 ) and (−Δ)1−s Km (x) ∈ L2 (R3 ). Furthermore, K 2 :=

N

∑ Km (x)2L (R ) > 0 1

3

m=1

and Q2 :=

N

∑ (−Δ)1−s Km (x)2L (R ) > 0. 2

m=1

3


287

We choose the space dimension d = 3, which is related to the solvability conditions for the linear Poisson type equation (24) established in Lemma 5 below. From the point of view of applications, the space dimension is not restricted to d = 3 because the space variable corresponds to the cell genotype but not to the usual physical space. Let us use the Sobolev inequality for the fractional Laplace operator (see e.g. Lemma 2.2 of [25], also [26]) fm (x)

6

L 4s−1 (R3 )

3 1 < 1−s < 4 4

≤ cs (−Δ)1−s fm (x)L2 (R3 ) ,

(4)

along with the assumption above and the standard interpolation argument. Hence, fm (x) ∈ L2 (R3 ),

1 ≤ m ≤ N.

We use the Sobolev spaces H 2s (R3 ) := {φ (x) : R3 → R | φ (x) ∈ L2 (R3 ), (−Δ)s φ ∈ L2 (R3 )}, equipped with the norm

0 0 is the constant of the embedding. When all the nonnegative parameters εm vanish, we arrive at the linear Poisson type equations (8) (−Δ)s um = fm (x), 1 ≤ m ≤ N. By means of part 1) of Lemma 5 below along with Assumption 1 each equation (8) admits a unique solution u0,m (x) ∈ H 2s (R3 ),

3 1 1/4 to be able to use the Sobolev type inequality (4). By virtue of Assumption 1, using that −Δum (x) = (−Δ)1−s fm (x) ∈ L2 (R3 ), we obtain for the unique solution of the linear problem (8) that u0,m (x) ∈ H 2 (R3 ). Therefore, u0 (x) := (u0,1 (x), u0,2 (x), . . . , u0,N (x))T ∈ H 2 (R3 , RN ). Let us look for the resulting solution of the nonlinear system (2) as u(x) = u0 (x) + u p (x),

(9)

288

where


u p (x) := (u p,1 (x), u p,2 (x), . . . , u p,N (x))T .

Evidently, we derive the perturbative system of equations ˆ s Km (x − y)gm (u0 (y) + u p (y))dy, (−Δ) u p,m (x) = εm

1 3 0 as well, otherwise the functions gm (z) will be constants in the ball I, such that a2 will vanish. For instance, gm (z) = z2 , z ∈ RN clearly satisfy this assumption above. Let us introduce the operator Tg , such that u = Tg v, where u is a solution of system (12). Our main statement is as follows. Theorem 3. Let Assumptions 1 and 2 be fulfilled. Then system (12) defines the map Tg : Bρ → Bρ , which is a strict contraction for all 0 < ε < ε ∗ for a certain ε ∗ > 0. The unique fixed point u p (x) of this map Tg is the only solution of system (10) in Bρ . Apparently, the resulting solution of system (2) given by (9) will be nontrivial because for some 1 ≤ m ≤ N the source term fm (x) is nontrivial and all gm (0) = 0 as assumed. We will make use of the following elementary lemma.


289

Lemma 4. For R ∈ (0, +∞) consider the function

ϕ (R) := α R3−4s +

β , R4s

1 3 0.

It attains the minimal value at R∗ = (4β s/α (3 − 4s))1/3 , which is given by 4s

4s

4s

4s

ϕ (R∗ ) = 3(3 − 4s) 3 −1 (4s)− 3 α 3 β 1− 3 . We proceed to the proof of our main proposition. 2 The existence of the perturbed solution Proof of Theorem 3. Let us choose an arbitrary v(x) ∈ Bρ and denote the terms involved in the integral expression in the right side of system (12) as Gm (x) := gm (u0 (x) + v(x)),

1 ≤ m ≤ N.

We apply the standard Fourier transform (25) to both sides of system (12), which gives us 3

um (p) = εm (2π ) 2

K m (p)Gm (p) , 2s |p|

1 ≤ m ≤ N.

Hence for the norm we obtain um 2L2 (R3 ) = (2π )3 εm2

ˆ R3

2 2 |K m (p)| |Gm (p)| d p. 4s |p|

(14)

As distinct from works [18] and [19] containing the standard Laplace operator in the diffusion term, here we do not try to control the norms K m (p) ∞ 3. |p|2s L (R ) Let us estimate the right side of (14) applying inequality (26) with R ∈ (0, +∞) as (2π )3 εm2

ˆ

ˆ 2 2 2 2 |K |K m (p)| |Gm (p)| m (p)| |Gm (p)| 3 2 d p + (2 π ) ε dp m |p|4s |p|4s |p|≤R |p|>R

≤ εm2 Km 2L1 (R3 ) {

1 1 R3−4s 2 + 4s Gm (x)2L2 (R3 ) }. G (x) 1 (R3 ) m L 2 2π 3 − 4s R

Due to the fact that v(x) ∈ Bρ , we get u0 + vL2 (R3 ,RN ) ≤ u0 H 2 (R3 ,RN ) + 1. The Sobolev embedding (7) yields |u0 + v| ≤ ce u0 H 2 (R3 ,RN ) + ce . By virtue of the representation ˆ Gm (x) =

1 0

∇gm (t(u0 (x) + v(x))).(u0 (x) + v(x))dt,

1 ≤ m ≤ N,

(15)

290


where the dot denotes the scalar product of two vectors in RN , we arrive at |Gm (x)| ≤ sup |∇gm (z)||u0 (x) + v(x)| ≤ a1 |u0 (x) + v(x)|, z∈I

with the ball I defined in (13). Thus Gm (x)L2 (R3 ) ≤ a1 u0 + vL2 (R3 ,RN ) ≤ a1 (u0 H 2 (R3 ,RN ) + 1). Apparently, for t ∈ [0, 1] and 1 ≤ j ≤ N, we have ˆ t ∂ gm ∂ gm (t(u0 (x) + v(x))) = ∇ (τ (u0 (x) + v(x))).(u0 (x) + v(x))d τ . ∂zj ∂zj 0 This implies |

∂ gm ∂ gm (t(u0 (x) + v(x)))| ≤ sup |∇ ||u0 (x) + v(x)| = a2,m, j |u0 (x) + v(x)|. ∂zj ∂zj z∈I

We use the Schwarz inequality to estimate N

|Gm (x)| ≤ |u0 (x) + v(x)| ∑ a2,m, j |u0, j (x) + v j (x)| ≤ a2 |u0 (x) + v(x)|2 , j=1

such that for 1 ≤ m ≤ N Gm (x)L1 (R3 ) ≤ a2 u0 + v2L2 (R3 ,RN ) ≤ a2 (u0 H 2 (R3 ,RN ) + 1)2 .

(16)

Thus we obtain the upper bound for the right side of (15) as a22 (u0 H 2 (R3 ,RN ) + 1)2 3−4s 2 2 R εm Km L1 (R3 ) (u0 H 2 (R3 ,RN ) + 1) { + 2π 2 (3 − 4s) 2

a21 }, R4s

with R ∈ (0, +∞). By virtue of Lemma 4, we obtain the minimal value of the expression above. Thus, 8s

u2L2 (R3 ,RN )

2

2+ 8s 3

2

≤ ε K (u0 H 2 (R3 ,RN ) + 1)

Evidently, (12) gives us for 1 ≤ m ≤ N 1−s

−Δum (x) = εm (−Δ)

2− 8s 3

3a23 a1

4s

8s

(3 − 4s)s 3 π 3 24s

.

(17)

ˆ R3

Km (x − y)Gm (y)dy.

By means of (26) along with (16) Δum 2L2 (R3 ) ≤ ε 2 Gm 2L1 (R3 ) (−Δ)1−s Km 2L2 (R3 ) ≤ ε 2 a22 (u0 H 2 (R3 ,RN ) + 1)4 (−Δ)1−s Km 2L2 (R3 ) , such that

N

∑ Δum 2L (R ) ≤ ε 2a22(u0 H (R ,R 2

m=1

3

2

3

N)

+ 1)4 Q2 .

(18)

Hence, by virtue of the definition of the norm (6) along with inequalities (17) and (18), we obtain the estimate from above for uH 2 (R3 ,RN ) as

ε (u0 H 2 (R3 ,RN ) + 1)2 a2 × [K 2 (

a2 (u0 H 2 (R3 ,RN ) + 1) 8s −2 1 3 )3 + Q2 ] 2 ≤ ρ 4s 8s 4s a1 (3 − 4s)s 3 π 3 2


291

for all ε > 0 small enough. Hence, u(x) ∈ Bρ as well. If for some v(x) ∈ Bρ there exist two solutions u1,2 (x) ∈ Bρ of system (12), their difference w(x) := u1 (x) − u2 (x) ∈ L2 (R3 , RN ) satisfies (−Δ)s wm = 0,

1 ≤ m ≤ N.

Since the operator (−Δ)s considered in R3 does not have nontrivial square integrable zero modes, w(x) = 0 a.e. in the whole space. Therefore, system (12) defines a map Tg : Bρ → Bρ for all ε > 0 sufficiently small. Our goal is to prove that such map is a strict contraction. We choose arbitrarily v1,2 (x) ∈ Bρ . By means of the argument above u1,2 = Tg v1,2 ∈ Bρ as well. By virtue of system (12), we have ˆ Km (x − y)gm (u0 (y) + v1 (y))dy, 1 ≤ m ≤ N, (19) (−Δ)s u1,m (x) = εm (−Δ)s u2,m (x) = εm

ˆ

R3

R3

Km (x − y)gm (u0 (y) + v2 (y))dy,

1 ≤ m ≤ N,

with 1/4 < s < 3/4. Let us denote G1,m (x) := gm (u0 (x) + v1 (x)),

G2,m (x) := gm (u0 (x) + v2 (x)),

1≤m≤N

and apply the standard Fourier transform (25) to both sides of systems (19) and (20). This gives us 3

u 1,m (p) = εm (2π ) 2

K m (p)G1,m (p) , |p|2s

Clearly, u1,m − u2,m 2L2 (R3 )

=

εm2 (2π )3

ˆ R3

3

u 2,m (p) = εm (2π ) 2

K m (p)G2,m (p) . |p|2s

2 2 |K m (p)| |G1,m (p) − G2,m (p)| d p. |p|4s

Apparently, such expression can be bounded above using (26) by

εm2 Km 2L1 (R3 ) {

G1,m (x) − G2,m (x)2L1 (R3 ) R3−4s G1,m (x) − G2,m (x)2L2 (R3 ) + }, 2π 2 3 − 4s R4s

with R ∈ (0, +∞). For t ∈ [0, 1] and 1 ≤ m ≤ N, let us use the representation ˆ 1 ∇gm (u0 (x) + tv1 (x) + (1 − t)v2 (x)).(v1 (x) − v2 (x))dt. G1,m (x) − G2,m (x) = 0

Since v2 (x) + t(v1 (x) − v2 (x))H 2 (R3 ,RN ) ≤ tv1 (x)H 2 (R3 ,RN ) + (1 − t)v2 (x)H 2 (R3 ,RN ) ≤ ρ , we have v2 (x) + t(v1 (x) − v2 (x)) ∈ Bρ . We estimate |G1,m (x) − G2,m (x)| ≤ sup |∇gm (z)||v1 (x) − v2 (x)| = a1,m |v1 (x) − v2 (x)|. z∈I

Hence G1,m (x) − G2,m (x)L2 (R3 ) ≤ a1,m v1 − v2 L2 (R3 ,RN ) ≤ a1,m v1 − v2 H 2 (R3 ,RN ) . Evidently, for 1 ≤ m, j ≤ N we have

∂ gm (u0 (x) + tv1 (x) + (1 − t)v2 (x)) ∂zj ˆ 1 ∂ gm ∇ (τ [u0 (x) + tv1 (x) + (1 − t)v2 (x)]).[u0 (x) + tv1 (x) + (1 − t)v2 (x)]d τ , = ∂zj 0

(20)

292


such that

∂ gm (u0 (x) + tv1 (x) + (1 − t)v2 (x))| ∂zj ∂ gm |(|u0 (x)| + t|v1 (x)| + (1 − t)|v2 (x)|), ≤ sup |∇ ∂zj z∈I |

t ∈ [0, 1].

By means of the Schwarz inequality we obtain the upper bound for |G1,m (x) − G2,m (x)| as N

1

1

∑ a2,m, j |v1, j (x) − v2, j (x)|(|u0 (x)| + 2 |v1 (x)| + 2 |v2 (x)|)

j=1

1 1 ≤ a2,m |v1 (x) − v2 (x)|(|u0 (x)| + |v1 (x)| + |v2 (x)|). 2 2 By virtue of the Schwarz inequality the norm G1,m (x) − G2,m (x)L1 (R3 ) 1 1 ≤ a2,m v1 − v2 L2 (R3 ,RN ) (u0 L2 (R3 ,RN ) + v1 L2 (R3 ,RN ) + v2 L2 (R3 ,RN ) ) 2 2 ≤ a2 v1 − v2 H 2 (R3 ,RN ) (u0 H 2 (R3 ,RN ) + 1).

(21)

Hence we obtain the upper bound for the norm u1 (x) − u2 (x)2L2 (R3 ,RN ) given by 3−4s a1 2 a2 2 2R + (u + 1) }. 2 3 N 0 H (R ,R ) 2π 2 3 − 4s R4s Lemma 4 gives us the minimum of the expression above over R ∈ (0, +∞). Thus we arrive at the estimate from above for u1 (x) − u2 (x)2L2 (R3 ,RN ) as

ε 2 K 2 v1 − v2 2H 2 (R3 ,RN ) {

2

ε K

2

v1 − v2 2H 2 (R3 ,RN )

2− 8s 3

3a1

(3 − 4s)24s s

Formulas (19) and (20) yield 1−s

(−Δ)(u1,m (x) − u2,m (x)) = εm (−Δ)

4s 3

[

a2 (u0 H 2 (R3 ,RN ) + 1) 8s ]3. π

(22)

ˆ R3

Km (x − y)[G1,m (y) − G2,m (y)]dy.

By virtue of inequalities (26) and (21) we derive Δ(u1,m − u2,m )2L2 (R3 ) ≤ ε 2 (−Δ)1−s Km 2L2 (R3 ) G1,m − G2,m 2L1 (R3 ) ≤ ε 2 a22 v1 − v2 2H 2 (R3 ,RN ) (u0 H 2 (R3 ,RN ) + 1)2 (−Δ)1−s Km 2L2 (R3 ) . Therefore, N

∑ Δ(u1,m − u2,m )2L (R ) 2

≤

3

m=1 ε 2 a22 v1 − v2 2H 2 (R3 ,RN ) (u0 H 2 (R3 ,RN ) + 1)2 Q2 .

(23)

Inequalities (22) and (23) yield u1 − u2 H 2 (R3 ,RN ) ≤ ε a2 (u0 H 2 (R3 ,RN ) + 1) × {

3K

2 4s 3

(3 − 4s)24s s π

8s 3

[

a2 (u0 H 2 (R3 ,RN ) + 1) 8s −2 1 ] 3 + Q2 } 2 v1 − v2 H 2 (R3 ,RN ) . a1

Thus, the map Tg : Bρ → Bρ defined by system (12) is a strict contraction for all values of ε > 0 sufficiently small. Its unique fixed point u p (x) is the only solution of system (10) in the ball Bρ . The resulting u(x) ∈ H 2 (R3 , RN ) given by (9) is a solution of system (2).


293

3 Auxiliary results We recall the solvability conditions for the linear Poisson type equation with a square integrable right side (−Δ)s φ = f (x),

x ∈ R3 ,

0 < s < 1,

easily obtained in [29] by using the standard Fourier transform ˆ 1 f (x)e−ipx dx. f (p) := 3 3 2 (2π ) R

(24)

(25)

Evidently, we have the estimate for it as f (p)L∞ (R3 ) ≤

1 3

(2π ) 2

f (x)L1 (R3 ) .

(26)

f (x)g(x)dx, ¯

(27)

Let us denote the inner product as ˆ ( f (x), g(x))L2 (R3 ) :=

R3

with a slight abuse of notations when the functions involved in (27) fail to be square integrable, like for instance the ones involved in orthogonality relation (28) below. Indeed, if f (x) ∈ L1 (R3 ) and g(x) ∈ L∞ (R3 ) , then the integral in the right side of (27) makes sense. The technical result easily derived in [29] by virtue of ( 25) is formulated as follows. Lemma 5. Let f (x) ∈ L2 (R3 ). 1) When 0 < s < 34 and additionally f (x) ∈ L1 (R3 ), problem (24) has a unique solution φ (x) ∈ H 2s (R3 ). 2) When 34 ≤ s < 1 and in addition |x| f (x) ∈ L1 (R3 ), equation (24) admits a unique solution φ (x) ∈ H 2s (R3 ) if and only if the orthogonality condition (28) ( f (x), 1)L2 (R3 ) = 0 holds. Note that for the lower values of the power of the negative Laplacian 0 < s < 34 under the assumptions stated above no orthogonality relations are needed to solve the linear Poisson type equation (24) in H 2s (R3 ). 4 Discussion We will conclude this article with a brief discussion of biological interpretations of the results obtained above. All tissues and organs in a biological organism are characterized by cell distribution with respect to their genotype. Without mutations all cells would have the same genotype. By means of mutations, the genotype changes and represents a certain distribution around its principal value. Stationary solutions of such system give a stationary cell distribution with respect to the genotype. Existence of these stationary distributions is an important property of biological organisms which allows their existence as steady state systems. Existence of stationary solutions is established in the spaces of integrable functions decaying at infinity. Biologically this means that cell distribution with respect to the genotype decays as the distance from the main genotype increases. The results of the article show what conditions should be imposed on cell proliferation, mutations and influx to get such distributions. In the context of the population dynamics, this result applies also to biological species where individuals are distributed around some average genotype. In such case, existence of stationary solutions corresponds to the existence of biological species [30].

294


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9(3), 5–25.

295



Existence of Semi Linear Impulsive Neutral Evolution Inclusions With Infinite Delay in Frechet Spaces Dimplekumar N. Chalishajar†, K. Karthikeyan, A. Anguraj Department of Applied Mathematics, Virginia Military Institute (VMI), 431, Mallory Hall, Lexington, VA 24450, USA Department of Mathematics, KSR College of Technology, Tiruchengode-637215, Tamilnadu, India Department of Mathematics, PSG College of Arts and Science, Coimbatore- 641 014, Tamil Nadu, India Submission Info Communicated by Valentin Afraimovich Received 17 February 2016 Accepted 7 March 2016 Available online 1 October 2016 Keywords Impulsive differential inclusions Fixed point Frechet spaces nonlinear alternative due to Frigon

Abstract In this paper, sufficient conditions are given to investigate the existence of mild solutions on a semi-infinite interval for first order semi linear impulsive neutral functional differential evolution inclusions with infinite delay using a recently developed nonlinear alternative for contractive multivalued maps in Frechet spaces due to Frigon combined with semigroup theory. The existence result has been proved without assumption of compactness of the semigroup. We study a new phase space for impulsive system with infinite delay.


1 Introduction In recent years, impulsive differential and partial differential equations have become important in mathematical models of real phenomena relating to biological and medical domains. In these models, the investigated simulating processes and phenomena are often subject to short-term perturbations whose duration is negligible in comparison with the duration of the process. Consequently, it is natural to assume that these perturbations act instantaneously, that is, in the form of impulses. The theory of impulsive differential equations has seen considerable development, see the monographs of Bainov and Semeonov [1], Lakshimikantham at.el. [2] and Perestyuk [3]. Simultaneously the theory of impulsive differential equations as much as neutral differential equations has been emerging as an important area of investigations in recent years, stimulated by their numerous applications to problems in physics, mechanics, electrical engineering, medicine biology, ecology, and so on. The impulsive differential systems can be used to model processes which are subject to abrupt changes, and which cannot be described by the classical differential systems, we refer to the readers [2]. Partial neutral integro-differential equation with infinite delay has been used for modeling the evolution of physical systems, in † Corresponding

author. Email address: [email protected], [email protected], karthi [email protected], [email protected]


298

Chalishajar, Karthikeyan, Anguraj / Discontinuity, Nonlinearity, and Complexity 5(3) (2016) 297–312

which the response of the system depends not only on the current state, but also on the past history of the system, for instance, for the description of heat conduction in materials with fading memory, we refer to the papers of Gurtin and Pipkin [4], Nunziato [5], and the references therein related to this matter. Recently, several works reported existence results for mild solutions for impulsive neutral functional differential equations or inclusions, such as [6, 7] and references therein. However, the results obtained are only in connection with finite delay. Since many systems arising from realistic models heavily depend on histories (i.e., there is an effect of infinite delay on state equations), there is a real need to discuss partial functional differential systems with infinite delay, where numerous properties of their solutions are studied and detailed bibliographies are given. The literature related to first and second order nonlinear non autonomous neutral impulsive systems with or without state dependent delay is not vast. To the best of our knowledge, this has not been thoroughly explored and is one of the main motivations of this paper. When the delay is infinite, the notion of phase space plays an important role in the study of both qualitative and quantitative theory. A common choice is a seminormed space satisfying suitable axioms, introduced by Hale and Kato in [8]; see also Corduneanu and Lakshmikantham [9]; J. R. Graef [10] and S. Baghli and M. Benchohra [11, 12]. Unfortunately, we have not discovered a detailed treatment of a system involving infinite delay with impulse effects. Henderson and Ouahab [13] discussed existence results for non-densely defined semi-linear functional differential inclusions in Frechet spaces. Hernandz et al. [14] studied existence of solutions for impulsive partial neutral functional differential equations for first and second order systems with infinite delay. Recently, Arthi and Balachandran [15] proved controllability of the second order impulsive functional differential equations with state dependent delay using a fixed point approach and a cosine operator theory. It has been observed that the existence or controllability results proved by different authors are based on an axiomatic definition of the phase space given by Hale and Kato [8]. However, as remarked by Hino, Murakami, and Naito [16], it has come to our attention that these axioms for the phase space are not correct for the impulsive system with infinite delay (refer [17, 18]). This motivated us to generate a new phase space for the existence of a non-autonomous impulsive neutral inclusion with infinite delay. This direction is another focus of our paper and to the best of our knowledge, has not yet been considered in the literature. On the other hand, researchers have proved the controllability results using the compactness assumption of semigroups and the family of cosine operators. However, as remarked by R. Triggiani [19], if X is an infinite dimensional Banach space, then the linear control system is never exactly controllable on given interval if either B is compact or associated semigroup is compact. According to R. Triggiani [19], this is a typical case for most control systems governed by parabolic partial differential equations and hence the concept of exact controllability is very limited for many parabolic partial differential equations. Nowadays, researchers are driven to overcome this problem, refer to ( [15], [17], [18]). Very recently, Chalishajar and Acharya [18] studied the controllability of second order neutral functional differential inclusion, with infinite delay and impulse effect on unbounded domain, without compactness of the family of cosine operators. Ntouyas and O’Regan [20] gave some remarks on controllability of evolution equations in Banach paces and proved a result without compactness assumption. The rest of this paper is organized as follows: In Section 2 we introduce the system, recall some basic definitions, and preliminary facts which will be used throughout this paper. The existence theorems for semi linear impulsive neutral evolution inclusions with infinite delay, and their proofs are arranged in Section 3. Finally, in Section 4, an example is presented to illustrate the applications of the obtained results.


299

2 Preliminaries In this paper, we shall consider the existence of mild solutions for first order impulsive partial neutral functional evolution differential inclusions with infinite delay in a Banach space E d [y(t) − g(t, yt )] ∈ A(t)y(t) + F(t, yt ) dt t ∈ J = [0, +∞), t = tk , k = 1, 2, . . . Δy|t=tk = Ik (y(tk− )),

(1)

k = 1, 2, . . .

(2)

y0 = φ ∈ Bh

(3)

where F : J × Bh → P(E) is a multivalued map with nonempty compact values, P(E) is the family of all subsets of E, g : J × Bh → E and Ik : E → E, k = 1, 2, . . . are given functions, φ ∈ Bh are given functions and {A(t)}0≤t 0, φ (θ ) is bounded and measurable ˆ 0 h(s) sups≤θ ≤0|φ (θ )|ds < +∞}. function on [−r, 0] and −∞

Here, Bh endowed with the norm

ˆ

φ Bh =

0

−∞

h(s) sup |φ (θ )|ds, ∀φ ∈ Bh . s≤θ ≤0

Then it is easy to show that (Bh , .Bh ) is a Banach space. Lemma 1. Suppose y ∈ Bh ; then, for each t > 0, yt ∈ Bh . Moreover, l|y(t)| ≤ yt Bh ≤ l sup |y(s)| + y0 Bh , 0≤s≤t

where l :=

´0

−∞ h(s)ds

< +∞.

Proof: For any t ∈ [0, a], it is easy to see that, yt is bounded and measurable on [−a, 0] for a > 0, and ˆ 0 h(s) sup |yt (θ )|ds yt Bh = −∞

ˆ =

−∞

ˆ =

−t

−∞

ˆ =

−t

−∞

ˆ ≤

−t

−t

−∞

θ ∈[s,0]

ˆ

h(s) sup |y(t + θ )|ds + θ ∈[s,0]

θ1 ∈[t+s,t]

h(s)[ sup θ1 ∈[t+s,0]

h(s)

sup

θ1 ∈[t+s,0]

−t

ˆ

h(s) sup |y(θ1 )|ds +

0

0 −t

h(s) sup |y(t + θ )|ds θ ∈[s,0]

h(s) sup |y(θ1 )|ds θ1 ∈[t+s,t]

ˆ

|y(θ1 )| + sup |y(θ1 )|]ds + θ1 ∈[0,t]

|y(θ1 )|ds +

ˆ

0

−∞

0

−t

h(s) sup |y(θ1 )|ds

h(s)ds. sup |y(s)| s∈[0,t]

θ1 ∈[0,t]

300


ˆ ≤

−t

−∞

ˆ ≤

−∞

ˆ =

0

0

−∞

h(s) sup |y(θ1 )|ds + l. sup |y(s)| θ1 ∈[s,0]

s∈[0,t]

h(s) sup |y(θ1 )|ds + l sup |y(s)| θ1 ∈[s,0]

s∈[0,t]

h(s) sup |y0 (θ1 )|ds + l sup |y(s)| θ1 ∈[s,0]

s∈[0,t]

= l sup |y(s)| + y0 s∈[0,t]

Since φ ∈ Bh , then yt ∈ Bh . Moreover, ˆ 0 ˆ h(s) sup |yt (θ )|ds ≥ |yt (θ )| yt Bh = −∞

θ ∈[s,0]

0 −∞

h(s)ds = l|y(t)|

The proof is complete. 2 Above definition of phase space also satisfies the conditions given by Hale and Kato [8]. (A1) if x : (−∞, b] → X , b > 0, continuous on [0, b] and x0 ∈ Bh , then for every t ∈ [0, b] the following conditions hold: (a) xt is in Bh (b) x(t) ≤ Hxt Bh (c) xt Bh ≤ M(t)x0 Bh + K(t) sup {x(s) : 0 ≤ s ≤ t}, where H > 0 is a constant; K, M : [0, ∞) → [1, ∞), K is continuous, M is bounded and H, K, M are independent of x(·). (A2) For the functions x in (A1), xt is Bh valued continuous functions on [0, b]. (A3) The space Bh is complete. Next, we introduce definitions, notation and preliminary facts from multi-valued analysis, which are useful for the development of this paper (refer [21]). Let C([0, b], E) denote the Banach space of all continuous functions from [0, b] into E with the norm y∞ = sup{y(t) : 0 ≤ t ≤ b} and let L1 ([0, ∞), E) be the Banach space of measurable functions y : [0, ∞) → E, that are Lebesgue integrable with the norm ˆ ∞ y(t)dt for all y ∈ L1 ([0, ∞), E). yL1 = 0

Let X be a Frechet space with a family of semi-norms { · n }n∈N . Let Y ⊂ X , we say that Y is bounded if for every n ∈ N, there exists M¯n > 0 such that yn ≤ M¯n

for all

y ∈ Y.

To X we associate a sequence of Banach spaces {(X n , · n )} as follows: For every n ∈ N, we consider the equivalence relation ∼n defined by :x ∼n y if and only if x − yn = 0 for all x, y ∈ X . We denote X n = (X |∼n , · n ) the quotient space, the completion of X n with respect to · n . To every Y ⊂ X , we associate a sequence {Y n } of subsets Y n ⊂ X n as follows: For every x ∈ X , we denote [x]n the equivalence class of x of subset X n and we define Y n = {[x]n : x ∈ Y }. We denote Y¯n , int(Y n ) and ∂nY n , respectively, the closure, the interior, and the boundary of Y n with respect to · in X n . We assume that the family of semi-norms { · n } verifies: x1 ≤ x2 ≤ x3 ≤ · · ·

for every

x ∈ X.


301

Let (X , d) be a metric space. We use the following notations: Pcl (X ) := {Y ∈ P(X ) : Y closed}, Pb (X ) := {Y ∈ P(X ) : Y bounded} Pcv (X ) := {Y ∈ P(X ) : Y convex}, Pcp (X ) := {Y ∈ P(X ) : Y compact}. Consider Hd : P(X ) × P(X ) → R+ ∪ {∞}, given by Hd (A , B) := max{ sup d(a, B), sup d(A , b)}, a∈A

b∈B

where d(A , b) := infa∈A d(a, b), d(a, B) := inf b∈B d(a, b). Then (Pb,cl (X ), Hd ) is a metric space and (Pcl (X ), Hd ) is a generalized (complete) metric space (see [22]). Definition 1. We say that a family {A(t)}t≥0 generates a unique linear evolution system {U (t, s)}(t,s)∈Δ for Δ1 = {(t, s) ∈ J × J : 0 ≤ s ≤ t < +∞} satisfying the following properties: (1) U (t,t) = I where I is the identity operator in E, (2) U (t, s)U (s, τ ) = U (t, τ ) for 0 ≤ τ ≤ s ≤ t < +∞, (3) U (t, s) ∈ B(E), the space of bounded linear operators on E, where for every (t, s) ∈ Δ1 and for each y ∈ E, the mapping (t, s) → U (t, s)y is continuous. More details on evolution systems and their properties could be found in the books of Ahmed [23], Engel and Nagel [24], and Pazy [25]. Definition 2. A multivalued map G : J → Pcl (X ) is said to be measurable if for each x ∈ E, the function Y : J → X defined by Y (t) = d(x, G(t)) = inf{|x − z| : z ∈ G(t)} is measurable where d is the metric induced by the normed Banach space X . Definition 3. A function F : J × Bh → P(X ) is said to be an L1loc -Caratheodory multivalued map if it satisfies: (i) x → F(t, y) is continuous(with respect to the metric Hd ) for almost all t ∈ J; (ii) t → F(t, y) is measurable for each y ∈ Bh ; (iii) for every positive constant k there exists hk ∈ L1loc (J; R+ ) such that F(t, y) ≤ hk (t)

for all

yBh ≤ k

and for almost all t ∈ J.

A multivalued map G : X → P(X ) has convex(closed) values if G(x) is convex(closed) for all x ∈ X . We say that G is bounded on bounded sets if G(B) is bounded in X for each bounded set B of X , i.e., sup{sup{y : y ∈ G(x)}} < ∞. x∈B

Finally, we say that G has fixed point if there exists x ∈ X such that x ∈ G(x). For each y ∈ B∗ , let the set SF,y known as the set of selectors from F defined by SF,y = {v ∈ L1 (J; E) : v(t) ∈ F(t, yt ),

a.e.t ∈ J}.

For more details on multivalued maps we refer to the books of J. P. Aubin and A. Cellina [26], Deimling [27], Gorniewicz [28], Hu and Papageorgiou [29], and Tolstonogov [30]. Definition 4. A multivalued map F : X → P(X ) is called an admissible contraction with constant {kn }n∈N if for each n ∈ N there exists kn ∈ (0, 1) such that (i) Hd (F(x), F (y)) ≤ kn x − yn for all x, y ∈ X . (ii) For every x ∈ X and every ε ∈ (0, ∞)n , there exists y ∈ F(x) such that x − yn ≤ x − F(x)n + εn

for every

n ∈ N.

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The following nonlinear alternative will be used to prove our main result. Theorem 2. (Nonlinear Alternative of Frigon, [31,32]). Let X be a Frechet space and U an open neighborhood of the origin in X and let N : U¯ → P(X ) be an admissible multivalued contraction. Assume that N is bounded. Then one of the following statements holds: (C1) N has a fixed point; (C2) There exists λ ∈ [0, 1) and x ∈ ∂ U such that x ∈ λ N(x). 3 Existence results We consider the space PC = {y : (−∞, ∞) → E|y(tk− ) and y(t) = φ (t)

y(tk− ) exist with y(tk ) = y(tk− ),

for t ∈ (−∞, ∞),

yk ∈ C(Jk , E), k = 1, 2, 3, . . . }

where yk is the restriction of y to Jk = (tk ,tk + 1], k = 1, 2, 3, . . . Now we set B∗ = {y : (−∞, ∞) → E : y ∈ PC ∩ Bh }. Bk = {y ∈ B∗ : sup |y(t)| < ∞}, t∈Jk∗

Jk∗ = (−∞,tk ].

where

Let · k be the semi-norm in Bk defined by yk = y0 Bh + sup{|y(s)| : 0 ≤ s ≤ tk },

y ∈ Bk .

To prove our existence result for the impulsive neutral functional differential evolution problem with infinite delay (1) − −(3), firstly we define the mild solution. Definition 5. We say that the function y(·) : (−∞, +∞) → E is a mild solution of the evolution system (1) − (3) if y(t) = φ (t) for all t ∈ (−∞, 0], Δy|t=tk = Ik (y(tk− )), k = 1, 2, . . . and the restriction of y(·) to the interval J is continuous and there exists f (·) ∈ L1 (J; E) : f (t) ∈ F(t, yt ) a.e in J such that y satisfies the following integral equation: ˆ t (4) y(t) = U (t, 0) φ (0) − g(0, φ ) + g(t, yt ) + U (t, s)A(s)g(s, ys )ds 0 ˆ t + U (t, s) f (s)ds + ∑ U (t,tk )Ik (y(tk− )), for each t ∈ [0, +∞). 0

0 0 such that A−1 (t)B(E) ≤ M0 (H5) There exists constant dk > 0,

k = 1, 2, . . . such that

¯ ≤ dk x − x ¯ Ik (x) − Ik (x) (H6) There exists a constant 0 < L
0 such that ¯ s, ¯ φ¯) ≤ L∗ (|s − s| ¯ + φ − φ¯Bh ) for all s, s¯∈ J A(s)g(s, φ ) − A(s)g(

and

φ , φ¯ ∈ Bh .

For every n ∈ N, let us take here l¯n (t) = M1 Kn [L∗ + ln (t)] for the family of semi-norm { · n }n∈N . In what follows we fix τ > 1 and assume [M0 L∗ Kn +

m 1 + M1 ∑ dk ] < 1 τ k=1

Theorem 3. Suppose that hypotheses (H1)—(H8) are satisfied. Moreover ˆ n ˆ +∞ M1 Kn ds > max(L, p(s))ds for each s + ψ (s) 1 − M0 LKn 0 δn

n∈N

(5)

with

δn = (Kn M1 H + Mn )φ Bh +

Kn [(M1 + 1)M0 L + M1Ln 1 − M0 LKn m

+M0 L[M1 (Kn H + 1) + Mn ]φ Bh + M1 ∑ ck ]. k=1

Then the impulsive neutral evolution problem (1) − (3) has a mild solution. Proof: We transform problem (1) − (3) into a fixed point problem. Consider an operator N : B∗ → B∗ defined by ⎧ ⎪ ⎨φ (t) if t ≤ 0 ´ N(y) = h ∈ B∗ : h(t) = U (t, 0)[φ (0) − g(0, φ )] + g(t, yt ) + 0t U (t, s)A(s)g(s, ys )ds ⎪ ⎩ ´t + 0 U (t, s) f (s)ds + ∑0