Discontinuity, Nonlinearity, and Complexity

Discontinuity, Nonlinearity, and Complexity 5(4) (2016) 437–454

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

Relaxation Oscillations and Chaos in a Duffing Type Equation: A Case Study L. Lerman1†, A. Kazakov2,1 , N. Kulagin3 1 Institute

of Information Technology, Mathematics & Mechanics, Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod, 603950, Russia 2 National Research University Higher School of Economics, Nizhny Novgorod, 603155, Russia 3 Moscow Aviation Institute (MAI), Moscow, Russia Submission Info Communicated by A.C.J. Luo Received 1 April 2016 Accepted 15 June 2016 Available online 1 January 2017 Keywords System with slow varying parameter Adiabatic invariant Chaos Relaxation oscillation Reversibility Symmetric orbit

Abstract Results of numerical simulations of a Duffing type Hamiltonian system with a slow periodically varying parameter are presented. Using theory of adiabatic invariants, reversibility of the system and theory of symplectic maps, along with thorough numerical experiments, we present many details of the orbit behavior for the system. In particular, we found many symmetric mixed mode periodic orbits, both being hyperbolic and elliptic, the regions with a perpetual adiabatic invariant and chaotic regions. For the latter region we present details of chaotic behavior: calculation of homoclinic tangles and Lyapunov exponents.

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

1 Introduction Slow-fast systems model different phenomena in many branches of science and their study is a rather developed part of the theory of dynamical systems and asymptotic analysis [1–3]. Here many tools collaborate to get a more or less detailed picture of dynamics. One of the first and most elaborated theory when applying to Hamiltonian systems is the adiabatic theory [3–5] which gives an approximate description of the orbit behavior in large regions of the phase space. For the dissipative systems the so-called geometric theory of slow-fast systems initiated by the work of Fenichel [6] is important. This technique is mainly applicable when somebody is interested in the orbit behavior near the sets made up of the hyperbolic equilibria or periodic orbits of the fast systems generated by a slow-fast system at some its limit. But when this set contains nonhyperbolic equilibria then other tools should be applied. As such, the blow-up methods are used here [7–9]. Also, many efforts were spent to study using other tools and numerically the chaotic orbit behavior in the stochastic regions near separatrix sets (see, for instance, [10–16]). † Corresponding author. Email address: [email protected]

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

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L. Lerman, A. Kazakov, N. Kulagin / Discontinuity, Nonlinearity, and Complexity 5(4) (2016) 437–454

In recent paper [17] it was shown that blow-up methods can also be applied for studying slow fast Hamiltonian systems. In particular, it was shown there that for the case of one slow and one fast degrees of freedom the orbit behavior near a generic point on a singular curve of the slow manifold can be reduced in the principal approximation to the study of Painlevé-I (for the case of a fold) or to the Painlevé-II equation (for the case of a cusp). For particular cases it was known before [18–20]. An interesting problem is to understand the orbit behavior of a slow fast Hamiltonian system in a vicinity of such point and to connect this with the observed relaxation behavior. The most simple situation is met here when studying a nonautonomous Hamiltonian system with slow varying parameters to which a system with two degrees of freedom, one fast and one slow ones, can be often reduced. In this paper we study, as a representative example, a 2π -periodic nonautonomous differential system of the Duffing type in the phase space R2 × S1 = {(x, y, θ )}, θ (mod 2π ) x˙ = y = Hy , y˙ = − sin θ − x cos θ − x3 = −Hx , θ˙ = ε .

(1)

First two equations at ε > 0 and θ = ε t + θ0 give a periodic nonautonomous Hamiltonian system of period 2π /ε with the Hamiltonian y2 x4 x2 H = + + cos(ε t + θ0 ) + x sin(ε t + θ0 ). (2) 2 4 2 When the parameter ε is small, this system is slow fast with the slow varying variable θ and two fast variables (x, y). In a sense, it is a prototype of any Hamiltonian system in one degree of freedom with slow varying parameters that were by the subject of many investigations [4, 5, 12, 21–23]. We have deliberately chosen a system which on the one hand is very simple from the point of view of its fast dynamics, but from the other hand it does change its phase portrait passing through generic possible codimension-1 bifurcations. Nonetheless, the system is not chosen by chance, it appears in a slow fast Hamiltonian system with one fast and one slow degrees of freedom, when its 2-dimensional slow manifold has a cusp point w.r.t the projection of the slow manifold onto the space of slow variables. The fast systems near this point depends on two parameters (= slow variables) and on the corresponding leaf of the fast variables the fast system has a degenerate equilibrium of the type degenerate saddle or degenerate center. This equilibrium just corresponds to the cusp point on the slow manifold. Such equilibria are of codimension 2 generically. If one goes slowly around this specific point in the parameter plane (slow variables) in time, then one gets in the main approximation a system coinciding with that with Hamiltonian (2).

2 The model pecularities Sometimes it is convenient to consider this system as autonomous one. System (1) is reversible w.r.t. involution L of the phase space acting as L(x, y, θ ) = (−x, y, 2π − θ ). This means that if (x(t), y(t), θ (t)) is its solution, then (x1 (t), y1 (t), θ1 (t)) = (−x(−t), y(−t), 2π − θ (−t)) is the solution as well. The set of fixed points of the involution Fix(L) consists of two disjoint lines: x = 0, θ = 0 and x = 0, θ = π . As is known [24, 25], any orbit of a reversible system that intersect Fix(L) at exactly two points is symmetric periodic. We use this property to search for symmetric periodic orbits geometrically and numerically. For our case symmetric periodic orbits can be of three types: 1. orbits that intersect at one of its point the line x = 0, θ = 0 (mod 2π ) and at another point the line x = 0, θ = π (mod 2π ); using the symmetry L we conclude that such orbits go around the circle S1 odd times before their closing, in particular, one-round symmetric periodic orbits belong to this type; 2. orbits that intersect at both points the line x = 0, θ = 0 (mod 2π ), such orbits go around the circle even times before their closing;


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3. orbits that intersect at both points the line x = 0, θ = π (mod 2π ), such orbits also go around the circle even times before their closing. Generically, these symmetric periodic orbits differ from each other. System (1) for ε > 0 has a global cross-section, as such any plane θ = θ0 can be taken. Thus the Poincaré map P is well defined on such a plane, this map is symplectic (area-preserving w.r.t. 2-form dx ∧ dy), since this map is generated by the nonautonomous periodic Hamiltonian system in its period 2π /ε . The map depends on the parameter ε , Pε , but the limit ε → +0 is singular, since the transition time 2π /ε goes to infinity and it is unclear which dynamical structures of Pε survive at this limit. For small positive ε the system is a slow fast one. For such a system it is useful to investigate the dynamics near its slow manifold (if it exists). This manifold is defined as the set of all equilibria for fast systems for all θ . Recall that the fast systems are obtained at the limit ε = 0 in (1). In fact, it is a one parameter family of Hamiltonian systems in one degree of freedom, the individual system is given, if one fixes a parameter θ = θ0 . In the phase space of the full system R2 × S1 the slow curve is made of these equilibria when parameter θ0 varies on the circle θ0 ∈ [0, 2π ]. For the system under consideration the slow curve is a smooth closed curve given by solutions of equations y = 0, x3 + x cos θ + sin θ = 0. In dependence on θ0 , solutions of this system consist generically either of three points or one point with two intermediate sections at angles θ = θ∗ , θ = 2π − θ∗ , where there are two equilibria. Here the angle π /2 < θ∗ < π is defined as follows. Double roots (in x) of the cubic equation x3 + x cos θ + sin θ = 0 arise when the derivative in x is also vanishes: 3x2 + cos θ = 0. From these two equations one can exclude θ since from two equations we derive sin θ = 2x3 , − cos θ = 3x2 > 0, and come to the equation 4σ 3 + 9σ 2 − 1 = 0, σ = x2 . The root under search should satisfy inequality 0 < σ < 1/3, this gives a unique root σ∗ ∼ 0.312. Thus, we have cos θ∗ = −3σ∗ , π /2 < θ∗ < π and the related x∗ > 0. The second related pair (−x∗ , 2π − θ∗ ) is given by symmetry. The specific section θ = θ∗ contains a disruption point (x∗ , θ∗ ) on the slow curve where two of three intersection points existing for θ∗ < θ < 2π − θ∗ coalesce at one point when decreasing θ . The similar situation for other two intersection points occurs near the second disruption point by the symmetry when increasing θ near 2π − θ∗ . These two specific sections θ = θ∗ and 2π − θ∗ divide the closed slow curve into four segments being each the graph of a function x = xi (θ ), y = 0, i = 1 − 4. Near the disruption point on section θ = θ∗ the slow curve has a representation y = 0, θ − θ∗ = a(x − x∗ )2 + · · · , a > 0. Indeed, at the point (x∗ , θ∗ ) the derivative in θ of the cubic function is −2σ∗2 − 3σ < 0, thus its solution near this point is given as θ − θ∗ = r(x − x∗ ), r(0) = 0, r′ (0) = 0, a = r′′ (0) = 6x∗ /(x∗ sin θ∗ − cos θ∗ ) > 0. For the second disruption point we have similar representation, but the second derivative is negative, since −x∗ < 0. The whole picture of the fast phase portraits is presented schematically in Fig. 1, the phase portrait depends on the section θ = θ0 chosen. There are three significantly different types of phase portraits for such a system. One of them is the phase portrait of a nonlinear oscillator. Such a system has a unique equilibrium, a center, enclosed by periodic orbits of different periods. This orbit behavior takes place for |θ | < θ∗ (mod 2π ). The second type system occurs on intermediate sections |θ | = θ∗ , or what is the same, on sections θ = θ∗ and θ = 2π − θ∗ . Here one more equilibrium appears (disappears) on the x-axis. This additional equilibrium is parabolic with the double zero eigenvalue and 2-dimensional Jordan box of the linearized system at the equilibrium. The parabolic equilibrium possesses a unique symmetric (w.r.t. the symmetry (x, y) → (x, −y)) homoclinic orbit, orbits inside of the homoclinic orbit are periodic ones and they shrink to the center equilibrium as a value of the Hamiltonian changes in one direction but they expand to the homoclinic loop as the value of the Hamiltonian changes in another direction. All orbits outside of the loop are also periodic and tend to infinity as the value of the Hamiltonian for this θ increases to infinity. The representation for the homoclinic solution of the fast Hamiltonian system on the section θ = θ∗ has the form x(t) = x∗

2x2∗ t 2 − 3 16x2∗ t 1 , y(t) = x , x2∗ = − cos θ∗ ∗ 2 2 2 2 2 2x∗ t + 1 (2x∗ t + 1) 3

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Fig. 1 Fast systems for different θ .

it is symmetric w.r.t. the involution (x, y) → (x, −y). The action I∗ corresponding to this solution (= the area inside it divided at 2π ) is equal to 4x3∗ . This will be useful for the further purposes. Also we present expressions for periodic orbits of the fast system on the section θ0 = π . Periodic solutions inside the negative loop (x < 0) are given as follows [26]: x(τ ) = −x1 dn(Kτ /π ), τ = ω t, ω = √ 2 1 + 4C 2 √ k = , 1 + 1 + 4C

π x1 √ , 2K

x1 =

p

√ 1 + 1 + 4C, y(τ ) =

k2 x21 √ sn(Kτ /π )cn(Kτ /π ), 2

here K is the complete elliptic integral of the first kind with parameter k. This solution is defined by an elliptic integral which is derived from the first equation using the Hamiltonian at θ = π ˆx −x1

ds q

(x21 − s2 )(s2 − x22 )

√ , x22 = 1 − 1 + 4C, −x1 ≤ s ≤ −x2 .

In addition, we present the expressions for homoclinic solutions on the sections θ = θ0 for 2π − θ∗ < θ0 < θ∗ . Between two specific sections θ∗ < θ0 < 2π − θ∗ fast systems have three equilibria, a saddle with two homoclinic loops and two centers inside of the each loop, other orbits are periodic. Denote the equilibria as (x′e , y), (xs , 0), (x′′e , y), x′e < xs < x′′e . For Hamiltonian (2) let us denote Vθ (x) the potential, Vθ = x4 /4 + x2 cos θ /2 + x sin θ . Then the polynomial Vθ (x) −Vθ (xs ) has the double root xs , since V ′ (xs ) = Hx (xs , 0) = 0. Expressing y from the equation H = H(xs , 0) = V (xs ) and using the first equation in (1), we get a differential equation p 1 x˙ = √ (x − xs ) (x − x1 )(x2 − x), 2


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where x1 , x2 are two remaining simple roots for the polynomial Vθ (x) − Vθ (xs ). Integration of this differential equation gives the following representations for its solutions, the left loop and right loops 1 − tanh2 (α t/2) xs − x1 , r= , 2 x2 − xs 1 + r tanh (α t/2) p 1 − tanh2 (α t/2) , α = (xs − x1 )(x2 − xs )/2. x(t) = xs + (x2 − xs ) 1 + r−1 tanh2 (α t/2)

x(t) = xs + (x1 − xs )

(3)

The area inside of the left homoclinic loop is monotonically increases from zero till the value 8π x3∗ , when θ0 increases from θ∗ till 2π − θ∗ . The area inside of the right homoclinic loop decreases from 8π x3∗ till zero on the same segment of θ . Saddle equilibria of the fast systems make up the middle piece of the slow curve. Thus, it is a hyperbolic invariant curve of the system at ε = 0. When approaching the specific section θ = 2π − θ∗ as θ increases, two equilibria of the fast system, the saddle and the left center, coalesce and then disappear through a parabolic equilibrium. For the system with small positive ε the orbits which start inside the small loop close to the left center move slowly in θ -direction. This is accompanied when crossing the section θ = 2π − θ∗ by the sharp transition from small amplitude fast oscillations near the piece of the slow curve to the fast oscillations of the large amplitude connected with going around near the former degenerate homoclinic orbit of the fast system. After that these fast large-amplitude oscillations are continued along some tube composed from periodic orbits of the fast systems due to an approximate preservation of a related adiabatic invariant [3] between sections θ = −θ∗ and θ = θ∗ . This tube is the surface of the constant value 4x3∗ of this adiabatic invariant. Numerical simulations with this system show several characteristic features in the orbit behavior and will be presented in the next sections.

3 Known results To substantiate further simulations recall some relevant known rigorous results. For the case of one fast and any number of slow degrees of freedom a slow fast Hamiltonian system can have a slow manifold which is generically filled with either center equilibria or saddle equilibria. For the former case the related part of the slow manifold was called in [27] (see also [28]) that near an almost elliptic slow manifold of an analytic slowfast Hamiltonian system with one fast and k slow degrees of freedom the Hamiltonian of the system can be transformed by an analytic symplectic ε -dependent transformation to the form where fast variables (x, y) enter to the transformed Hamiltonian only in the combination I = (x2 + y2 )/2 up to an exponentially small error in ε . For our case this theorem reads as follows: for those pieces of the slow curve where variables x, y can be expressed from the equations Hy = 0, Hx = 0 as functions of θ : x = f (θ ), y = g(θ ), and related equilibria of the fast system are centers, the Hamiltonian can be transformed by an analytic symplectic coordinate change Φ to the form (we preserve the same notation for new coordinates) H ◦ Φ = h(I, ε t) + R(x, y, ε t), I =

x2 + y2 , |R| ≤ C exp[−B/ε ]. 2

(4)

The needed transformation is given first by the shift X = x − f (θ ), Y = y − g(θ ), and after that using the procedure developed by Neishtadt [29]. For our case the study is reduced to the theorem in [27], if one introduces a new Hamiltonian Hˆ = ρ + H(x, y, θ ), considering (x, y) and (θ , ρ ) as conjugated variables w.r.t the singular symplectic 2-form dy ∧ dx + ε −1 d ρ ∧ d θ . Then the system is reduced to the autonomous slow-fast system with two degrees of freedom, and results of [27] on the existence of almost invariant elliptic slow manifold are applicable.

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a) hyperbolic orbit

b)its projection on (x, y)-plane

Fig. 2 Relaxation symmetric periodic orbits (RSPO).

Fig. 3 The image and preimage of symmetry line x = 0, θ = 0 on θ = π in T = ±π /ε

This theorem says that the motion near the related pieces of the slow curve looks as fast rotations with small amplitudes around the curve. This indeed can be seen on Fig. 2 below. Another relevant result is due to Fenichel [30, 31]. It describes the behavior near that hyperbolic piece of the slow curve for which fast systems have saddle equilibrium points. For small ε > 0 near this piece there exists a true invariant smooth slow curve being for our case an orbit segment in θ : |θ − π | ≤ T1 < π − θ∗ of the flow with a hyperbolic nearby behavior. The drawback of this result is in the fact that many such orbits exist, since only finite segments of the orbits stay in the neighborhood of the slow curve: they leave the neighborhood in both directions in time through the incoming and outcoming boundary parts of the neighborhood.

4 Symmetric periodic orbits and relaxation symmetric periodic orbits In this section we present a method for finding symmetric periodic orbits. This method allows one to search not only elliptic or hyperbolic orbits but also parabolic periodic orbits from which, through a bifurcation, one can


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localize elliptic periodic orbits with very small regions of their stability (elliptic islands for the related Poincaré map). The method was first presented in [32] but we give here its description, for completeness. We consider only the case of S-reversible area-preserving maps, f ◦ S = S ◦ f −1 for which the involution S has a smooth line of fixed points, Fix(S). For our case these maps are Poincaré maps on the sections θ = 0 or θ = π . The map is reversible w.r.t. the involution S that is inherited by the involution L of our system on the sections. For the diffeomorphisms we consider these lines are given as x = 0 on the related section. Theorem 1. Suppose that f is a C2 -smooth area-preserving diffeomorphism that is reversible w.r.t. a smooth involution S, and the fixed points set Fix(S) of the involution is a smooth curve. If ξ = Fix(S) ∩ f p (Fix(S)) is a point of transversal intersection of these two curves, then ξ is a point on either an elliptic or a hyperbolic period-2p orbit, while if ξ is a point of quadratic tangency, it is a parabolic period-2p orbit. Proof. Since ξ ∈ Fix(S) ∩ f p (Fix(S)), then ξ = S(ξ ) and there is a point η ∈ Fix(S) such that f p (η ) = ξ . Consider first p = 1. Then we have f 2 (η ) = f ( f (η )) = f (ξ ) = f (S(ξ )) = S( f −1 (ξ )) = S(η ) = η . Similarly, one has f 2 (ξ ) = ξ . By induction, the same is true for any p ∈ Z. Below we work with p = 1 to facilitate calculations. According to the Bochner-Montgomery theorem [33] we can take two symplectic charts: V near η with coordinates (x, y) and U near ξ with coordinates (u, v) such that in V the involution S becomes S(x, y) = (x, −y), and similarly in U it becomes S(u, v) = (u, −v). Moreover, f |V = f1 : V → U is written as follows (we assume with no loss of generality that ξ and η have zero coordinates in the related charts) u x F (x, y) =A + 1 v y G1 (x, y) where A is a constant matrix and F1 and G1 are O2 (x, y). Similarly f |U = f2 : U → V has the form x u F2 (u, v) =B + . y v G2 (u, v) Note that in both cases, du ∧ dv = dx ∧ dy by the area preservation. If ξ is the point of transverse intersection of f1 (Fix(S)) and Fix(S), then two vectors (a11 , a21 )⊤ and (1, 0)⊤ are transverse, i.e., a21 6= 0. In this case, when −1 < a12 a21 < 0, the point η is elliptic (its eigenvalues satisfy |λ1,2 | = 1), while if a12 a21 > 0 it is an orientable saddle, and if a12 a21 < −1 it is a non-orientable saddle. The tangency of D f1 (FixS) and FixS at ξ implies a21 = 0 and area preservation gives a22 = a−1 11 . The reversibility written in both coordinate charts provides the following relations for direct and inverse maps f1 ◦S = S ◦ f2−1 , f2 ◦ S = S ◦ f1−1 , or in coordinate form: x a −a12 u F2 (u, −v) f1−1 : = 22 + , v y 0 a11 −G2 (u, −v) and f2−1

u a11 a12 x F2 (x, y) + , : = v a21 a22 y G2 (x, y)

from where we get relations: a11 = b22 , a12 = b12 , a22 = b11 , b21 = 0, U2 (x, y) = F1 (x, −y), V2 (x, y) = −G1 (x, −y), U1 (u, v) = F2 (u, −v), V1 (u, v) = −G2 (u, −v), here U1 ,V1 , U2 ,V2 are nonlinear terms of the inverse maps f1−1 , f2−1 . Denote below for brevity a11 = α , a12 = β , then a22 = α −1 . The quadratic tangency of f1 (FixS) and FixS at ξ implies ∂ 2 G1 /∂ x2 6= 0 at (0, 0). The map f 2 near a 2-periodic point η has the form f2 ◦ f1 . Hence, the linear part of this map has the matrix 1 2β /α , γ = 2β /α 6= 0. 0 1

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Let us notice that for the map f 2 near the point η to guarantee its fixed point be parabolic (not more higher degenerate) we need only to check that in the local coordinates x1 = x + γ y + p(x, y), y1 = y + q(x, y), dx1 ∧ dy1 = dx ∧ dy the inequality ∂ 2 q/∂ x2 6= 0 at the fixed point holds. For our case this quantity is the following 2 ∂ 2 G1 ∂ 2q 2 ∂ V1 (0, 0) = α (0, 0) − α (0, 0). ∂ x2 ∂ x2 ∂ u2

From identities derived from the representation for f1 and f2 = S ◦ f1−1 ◦ S we get

∂ 2V 1 ∂ 2 G1 (0, 0) = − (0, 0), ∂ u2 α ∂ x2 therefore we come to 2 ∂ 2 G1 ∂ 2 G1 ∂ 2q 2 ∂ V1 (0, 0) = α (0, 0) − α (0, 0) = 2 α (0, 0) 6= 0 ∂ x2 ∂ x2 ∂ u2 ∂ x2

due to the quadratic tangency of Fix(S) and f (Fix(S)) at ξ . In order to use this tool for finding periodic orbits, we need to search intersection points of symmetry line x = 0, θ = π with the image of the symmetry line x = 0, θ = 0 under the flow map in the passage time T = π /ε . These points give traces of symmetric periodic orbits of the type 1 mentioned above, they go around the circle one time. If we search for the intersection of the symmetry line x = 0, θ = 0 with its image in time T = 2π /ε , then we get type 2 symmetric periodic orbits, they go around the circle 2 times before closing. The same will occur, if one search the intersection points of the symmetry line x = 0, θ = π with its image in time T = 2π /ε . In fact, there are many such symmetric periodic orbits. The related results obtained by the numerical calculation of the flow orbits in time T = π /ε or T = 2π /ε are shown on Fig. 5. As we shall see, these orbits are closely connected with the dynamics in the chaotic region. One type of symmetric periodic orbits (SPO) is those which will be called relaxation symmetric periodic orbits (RSPO). They are similar to mixed mode oscillation orbits found in dissipative systems [2]. These are SPO which have on its period both the segments of small oscillations near an elliptic part of slow curve and segments of fast oscillations with large amplitudes. Such orbits can be seen on Fig. 2 and Fig. 8(c), one of which is hyperbolic and another one is elliptic. For example, at ε = 0.0499542 an elliptic RSPO cuts the section θ = 0 at the point (0, 0.000039). Its unfolding is plotted on Fig. 2. The reason of their existence is very transparent. Indeed, take a small segment of the symmetry line x = 0, θ = 0 |y| ≤ δ , and iterate it till the section θ = π . We will get a curvilinear segment of an almost same length, due to preservation of adiabatic invariant I near the related piece of slow curve between points (0, 0, 0) and (−1, 0, π ). The central point (0, 0, 0) of the segment is mapped to a point near (−1, 0, π ) (see blue line on Fig. 4). Let us iterate this curve further till it returns to the section θ = 2π . The curve extends around former separatrix of the parabolic point after passing the disruption point on the section 2π − θ∗ and we get as a result a curve on the section θ = 2π which make one and a half rounds in the polar angle ϕ (see Fig. 4). Thus this curve intersects symmetry line at least two times but in fact this curve acquires several folds. Hence, varying ε one can achieve the tangency of the curve and symmetry line. This guarantees the existence of elliptic periodic orbits by a small variation of ε . The smaller ε is the more long curve becomes and it makes more revolutions along ϕ -coordinate and simultaneously acquires the more and more folds. This gives a mechanism of the multiplication of symmetric periodic orbits. Moreover, they approach close to the origin on the section θ = 0 (see, Fig. 5b).


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Fig. 4 Intersections of a small symmetry line segment x = θ = 0 (blue line is its trace on the section θ = π .)

a)

b)

Fig. 5 Intersections of symmetry lines: (a) 0 → π and (b) π → 2π .

5 Three regions with the different behavior Based on the previous studies [3, 4, 10–14, 14, 15, 21, 22], we will distinguish three regions in the phase space R2 × S1 which we call the adiabatic region, the chaotic region and the transition region. Under the adiabatic region we will understand such that the system (1) possesses a perpetual adiabatic invariant. Recall some relevant results. As is known, the following theorem holds [3]. Theorem 2. For a smooth Hamiltonian system with one degree of freedom slow periodically varying in time the action I of the fast system is the perpetual adiabatic invariant in the region where all orbits of fast systems are periodic, if some nondegeneracy condition holds. For the system we study the adiabatic region is distinguished by the condition that for all θ ∈ S1 we choose for fast systems on the related plane the regions being out of separatrices of the saddle and parabolic equilibrium points. This is done for values |θ | ≥ θ∗ . For values of θ where |θ | ≤ θ∗ the fast system has the only equilibrium,

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the center, and we distinguish the region where the value of action I is greater than I∗ = 4x3∗ (see above). In such a region we introduce the action-angle variables (I, ϕ ) for any θ . This is done in the usual way [34] by means of the equations (1). The curve H = C on the θ -plane consists of one oval due to our assumption about the region out of separatrices. It is a periodic orbit of the related fast system. In virtue of the reversibility of a fast system w.r.t. the involution (x, y) → (x, −y), this oval is a symmetric curve relative to x-axis and intersects it at two points x1 (C, θ ) < x2 (C, θ ), being the roots of the polynomial H(x, 0) −C. To construct the action-angle variables we search according to [34], a canonical transformation (x, y) → (I, ϕ ) that satisfies two conditions ˛ 1)I = I(H);

2)

d ϕ = 2π ,

(5)

Mh

here Mh is the curve H(x, y, θ ) = C on the related θ -plane. The change of variables has the form

ϕ=

∂S ∂S ∂S , y= , H(x, ) = h(I), ∂I ∂x ∂x

with the generating function S(I, x) of the canonical transformation (we omit here for brevity the dependence on θ ). If after the transformation Hamiltonian depends only on I and function h(I) has the inverse one (for instance, if h′ (I) 6= 0), then for a fixed I we get a closed curve and differential dS = SI dI + Sx dx of the function S equals dSI=const = ydx. Integrating dS along the curve gives in a neighborhood of x0 the generating function ´x S(I, x) = ydx. The complete variation of S x0

∆S =

˛

ydx Mh

when going around the curve equals to the area bounded by the curve Mh , thus this function is multi-valued. But its derivative in x is the single-valued function though the function ϕ = ∂∂ SI has an increment by d∆S/dI when a complete route around the curve is done. In order this increment would be 2π one needs the equality 2π = d∆S/dI to hold, from which one gets ∆S = 2π I. Therefore, the action I has to be equal to the area bounded by the curve divided at 2π . The fast system after the transformation casts  ∂H    I˙ = 0 = ∂ϕ (6)  ∂ H   ϕ˙ = ω (I) = − ∂I ´x2 In accordance to [34], the action variable is sought as I(x) = π1 ydx. From the Hamiltonian (2) we express x1

1 p y = ±√ 4C − x4 − 2x2 cos θ − 4x sin θ , 2 then one has 1 I(x; θ ,C) = √ π 2

x2ˆ(θ ,C)

x1 (θ ,C)

p

4C − x4 − 2x2 cos θ − 4x sin θ dx.

(7)

This integral is elliptic, it can be transformed to the normal form by some transformation [35]. Denote P4 the polynomial in x under the square root P4 (x; θ ,C) = −x4 − 2x2 cos θ − 4x sin θ + 4C.

(8)


447

P4 has two real and two imaginary roots, since we assumed that in the region where we work the level H = C consists of the only closed curve. Such closed curve is symmetric w.r.t. x-axis, hence P4 indeed has two real roots and can be represented as P4 (x; θ ,C) = −(x − x1 )(x − x2 )(x2 + ax + b),

where a2 − 4b < 0.

(9)

The values of roots x1 (θ ,C) < x2 (θ ,C) depend on a θ -section chosen and a value of C. The elliptic integral can be transformed in such a way that the polynomial P4 will acquire the Legendre form 2 (α − s2 )(β 2 + s2 ) that is attained by the substitution x = (ps + q)/(s + 1), where p(θ ,C), q(θ ,C), p > q, are two real roots of the quadratic polynomial z2 − (p + q)z + pq with positive discriminant [36]: p+q = 2

x1 x2 − b ax1 x2 + b(x1 + x2 ) , pq = − . a + x1 + x2 a + x1 + x2

For P4 we have the equality x1 + x2 = a, since coefficient before x3 vanishes. Thus these formulas can be simplified x1 x2 − b x1 x2 + b p+q = , pq = − . x1 + x2 2 In case if x1 + x2 = 0, then a = 0 and P4 already has the needed form. After the change of variables the integral takes the form p−q I=A √ π 2

ˆα p

−α

(α 2 − s2 )(β 2 + s2 ) ds. (s + 1)4

Here constants α , β , A are the following p p (q − x1 )(x2 − q) 2 q2 + aq + b 2 2 + ap + b (p − x )(p − x ). , β = α = p , A = 1 2 (x1 − p)(x2 − p) p2 + ap + b The elliptic integral J is calculated as J=

ˆα

−α

where J1 = 0, J2 = − J3 = − R=

(α 2 − s2 )(β 2 + s2 )) ds = J1 + J2 + J3 + R(G1 + G2 ), (s + 1)4

4α 4 β 4 − 4α 2 β 4 + 4α 4 β 2 + 2α 2 β 2 + 3β 4 + 3α 4 K(iα /β ), 3β (1 − α 2 )2 (1 + β 2 )2

β 4 + 10α 2 β 2 − 2β 2 + α 4 + 2α 2 + 2α 2 β 4 − 2α 4 β 2 β [K(iα /β ) − E(iα /β )], 3(1 − α 2 )2 (1 + β 2 )2

(1 + α 2 β 2 )(α 2 + β 2 )2 , 2(1 − α 2 )2 (1 + β 2 )2

G1 = 0, G2 = (2/β )Π(α 2 , iα /β ), here K, E, Π are complete elliptic integrals of the first, second, and third kinds, see [35–37]. In the action-angle variables the Hamiltonian takes the form H(I, θ ) with parameter θ , it does not depend on the angle variable ϕ . In order the theorem on the perpetual adiabatic invariant would be valid, the following conditions of “nonlinearity” have to be satisfied [3]. To express it, consider an analytic Hamiltonian H(I, θ ),

448


θ = ε t, written in the action-angle variables (I, ϕ ). Suppose the frequency ω (I, θ ) = HI (I, θ ) 6= 0 in some domain, and its mean value in θ , ˆ 2π 1 ω¯ = ω (I, θ )d θ 2π 0 satisfies the inequality ω¯ ′ (I) 6≡ 0. Then I is the perpetual adiabatic invariant. For the case under study after introducing the action variable I = I(C, θ ) = I(H, θ ) the Hamiltonian H(I, θ ) is the inverse function of I. Thus, we need to require that IC = 1/HI 6= 0 in the region under consideration. Then the needed condition of nonlinearity casts as follows d dI

ˆ2π

dθ 6= 0. IC (C, θ )

0

We checked this condition numerically and almost everywhere it is satisfied. In the adiabatic region the dynamics of the system is KAM-like type: the related Poincaré map on the cross-section θ = 0 in any compact invariant subregion possesses an almost full measure set filled with invariant KAM curves interspersed with thin stochastic regions near resonant periodic orbits existing due to the resonances between frequencies of the integrable adiabatic system and the fast frequency 2π /ε . The picture reminds a usual behavior in the KAM region presented in many papers. The chaotic region is that where a stochastic orbit behavior was observed. It contains the slow curve of the system (1) and captures some its neighborhood. The behavior in this region will be discussed below.

6 System for large actions The system under interest for large values of variables x, y is more hard for simulations. To facilitate them one may use the following considerations. The system admits the transformation to the action-angle variables (I, ϕ ) where the action I is the perpetual adiabatic invariant. To this end one needs to introduce these coordinates for a fast Hamiltonian system where θ is a parameter as we did in the previous section. In these coordinates the fast system takes the standard form I˙ = 0, ϕ˙ = −Hˆ I (I, θ ) 6= 0. But this change of variable is rather hard implement. Therefore we may use the idea proposed by A.M. Lyapunov in [38] when he studied a stability of degenerate equilibrium for the equation x¨ + x2n−1 = X (x, x). ˙ To display this more precisely, let us introduce the generalized polar coordinates. The related coordinate transformation is as follows: x = rC(ϕ ), y = r2 S(ϕ ). 3 For the case of the functions C, S of√ϕ are in fact√the elliptic Jacobi functions with the √ √ nonlinearity x periodic √ modulus k = 1/ 2: C(ϕ ) = cn(ϕ ; 1/ 2), S(ϕ ) = sn(ϕ ; 1/ 2)dn(ϕ ; 1/ 2), of the period 4K( 2/2) with K being the complete elliptic integral of the first kind [35, 37]. We omit writing k further. Using the standard formulae for elliptic functions (see, for instance, [35, 37]): cn4 ϕ + 2sn2 ϕ dn2 ϕ ≡ 1, cn′ ϕ = −snϕ dnϕ , sn′ ϕ = cnϕ dnϕ , dn′ ϕ = −(snϕ cnϕ )/2, we come to the following system

(sin θ + r cos θ cnϕ )snϕ dnϕ , r (sin θ + r cos θ cnϕ )cnϕ ϕ˙ = −r− r2 r˙ = −

One may also use a symplectic transformation with similar properties: x = (3r)1/3C(ϕ ), y = (3r)2/3 S(ϕ ), dx ∧ dy = dr ∧ d ϕ .

(10)


449

Then the equations cast as follows r˙ = −(3r)1/3 snϕ dnϕ sin θ + (3r)1/3 cnϕ cos θ ,

ϕ˙ = −

cnϕ sin θ + (3r)1/3 cn2 ϕ cos θ + 3r . (3r)2/3

We used system (10) for simulations at large values r ≥ 5. 7 Blow-up and Painlevé-I equation The fast system has two specific θ -sections which contain each a parabolic point of the fast system along with its homoclinic orbit. We would like to investigate the full system near these layers for small nonzero ε > 0. The first problem here is to describe the transition of orbits for the system with small ε > 0 through a small neighborhood of a former parabolic point. To that end, let us consider this problem separately for any slow varying Hamiltonian H(x, y, s) x˙ = Hy , y˙ = −Hx, s˙ = ε (11) which has at ε = 0 a parabolic point x = y = 0. Since the study is local, we do not require here H to be periodic in s. Asymptotic expansions for such transition solutions were presented in [39] for the so-called primary parametric resonance equation ε iU ′ + (|U |2 − t)U = 1, ε ≪ 1,

where U is a complex-valued function of t. This equation can be written in a Hamiltonian form w.r.t. real variables (u, v), U = u + iv and fast time t/ε = τ : 1 du dv λ (u2 + v2 ) + u − (u2 + v2 )2 , = λ v − v(u2 + v2 ) = Hv , = −1 − λ u + u(u2 + v2 ) = −Hu , H = dτ dτ 2 4 if λ = ετ = t considers as a parameter. The same Hamiltonian arises when studying a pendulum with a small slow varying periodic force near its 1:1 resonance of the center equilibrium [14]. After a passage to new variables (action-angle ones or symplectic polar coordinates) the same system appears in the first nonlinear approximation. The difference with the presented Hamiltonian is a small additional parameter in front of the linear √ term in u. For equation (11) the parabolic point for the frozen system (λ is a parameter) arises at λ∗ = 3 3 2/2. This parabolic point has a homoclinic loop enclosing a center equilibrium. The parabolic point breaks up into saddle and center for λ > λ∗ and disappears for λ < λ∗ . The center equilibrium inside of the former loop persists. Thus adding the equation λ˙ = ε we come to the same form of the Hamiltonian system. Now we add one more equation ε˙ = 0 to the system (11), then the extended system will have an equilibrium at the point (x, y, s, ε ) = (0, 0, 0, 0) (we preserve the old notations for variables to avoid extra letters). The linearization of the system at this equilibrium has a matrix being nothing else as 4-dimensional Jordan box. To study the solutions of this system near this equilibrium we, following the idea in [7, 8] (see also a close situation in [20]), blow up a neighborhood of this point by means of the coordinate change x = r2 X , y = r3Y, s = r4 Z, ε = r5 E.

(12)

After blowing up we get five variables instead of former four. So we can take different charts in dependence of what four variables are assumed to be independent in the related chart. In fact, the blowing-up means passing to the space S3 × R instead of a neighborhood of R4 , thus the equilibrium at the origin is blown up to a unit sphere (X ,Y, Z, E) ∈ S3 and r ≥ 0. Since we consider ε > 0, then E is non-negative E ≥ 0, hence (X ,Y, Z, E) vary on the half sphere being the 3-ball D3 . In fact, it is not convenient to work near the sphere but it is better tackled in affine coordinates on the related tangent planes. This will be present elsewhere.

450


Let us carry out the blow-up transformation for the initial system after the shift its disruption point to the origin (ξ = x − x∗ , y, u = θ − θ∗ , ε ) = (0, 0, 0, 0). The system casts in the form

ξ˙ = y, y˙ = a(u) + b(u)ξ − γξ 2 − ξ 3 , u˙ = ε , ε˙ = 0.

(13)

For the system under consideration the coefficients are the following: a(u) = (3x2∗ + 2x4∗ ) sin u − x3∗ u2 /2 + O(u4 ), b(u) = 2x3∗ sin u − 3x2∗ (u2 /2 + O(u4 )), γ = 3x∗ > 0, for the point x = x∗ > 0, y = 0, θ = θ∗ . We denote below a0 = a′ (0) = 3x2∗ + 2x4∗ > 0, b0 = b′ (0) = 2x3∗ > 0. For the second disruption point x = −x∗ , y = 0, θ = 2π − θ∗ they are γ = −3x∗ < 0, −b0 . At the beginning we shall do the blow up near the point x = x∗ > 0, y = 0, θ = θ∗ , here we take γ > 0, b0 > 0, and after that shall do the same near the second point x = −x∗ , y = 0, θ = 2π − θ∗ where we set −γ < 0 instead of γ and b0 < 0. We shall work only in a chart which is generated on the 4-plane E = 1 being tangent to the sphere at the point (0, 0, 0, 1), then one obtains x = r2 X , y = r3Y, s = r4 Z, ε = r5 ,

(14)

or, since ε˙ = 0 we consider r = ε 1/5 as a small parameter. The system in these variables takes the form X˙ = rY, Y˙ = r(a0 Z − γ X 2 + O(r2 )), Z˙ = r. After re-scaling time rt = τ , denoting ′ = d/d τ , setting r = 0 we get X ′ = Y, Y ′ = a0 Z − γ X 2, Z ′ = 1 > 0. This system describes the behavior of the blown-up system inside of the ball D3 . The system is equivalent to the well known Painlevé-I equation X ′′ = a0 τ − γ X 2 [40–42]. The standard form of the Painlevé-I equation is d 2W = 6W 2 − z, dz2 to which our equation can be transformed by a scaling of X and τ . When studying the system near the second disruption point (−x∗ , 0, 2π − θ∗ ) we need to change γ to −γ . Thus, Painlevé-I equation describes approximately the behavior of solutions of our system near the disruption point (x∗ , 0, θ∗ ). Hence, some known solutions of Painlevé-I equation have to play an essential role in the description of solutions of our system. Among them there is the so-called tritronquée solution found first by Boutroux [43] (see details in [44, 45]). This solution is characterized by the property that it is the only real solution of the Painlevé-I equation that is monotone in all its existence interval (it has a unique pole on the real line). For our case for system (1) at small ε > 0 this corresponds to its solution which passes near elliptic part of the slow curve and in the backward time direction it follows the stable separatrix of the former parabolic point at the distance O(ε 4/5 ) as ε → +0. The topological limit of this solution as ε → +0 is the curve made up of the elliptic part of the slow curve and the stable separatrices of the parabolic point. The role of this analog of the tritronquée solution is that it is just the orbit around which all close solutions make fast rotations when passing near a related piece of slow curve (the instant center of rotations). In fact, all four known types of solutions of the Painlevé-I equation [45] have analogs in the slow fast system near its disruption point. All this true in a neighborhood of the disruption point and will be presented elsewhere.

8 Stochastic region The simulations showed the existence of a stochastic region in the phase space. On the cross-section θ = 0 this region has the form of a disk filled with iterations of one orbit. The topological explanation of such the behavior


a)

451

b)

Fig. 6 (a) Homoclinic tangle in the stochastic region; (b) Stochastic region for the Poincaré map on θ = 0. Red points are iterations of multi-round elliptic island.

Fig. 7 The chart of Lyapunov’s exponents

is the presence of a number of saddle periodic orbits that exist in this region. Their separatrices intersect each other forming a tangle leading to the possibility a transition from a neighborhood of one saddle periodic orbit to another one. This is clear seen on Fig. 6. The existence of symmetric saddle fixed and periodic orbits can be explained by the reversibility of the flow. The related results were presented above. The complicated homoclinic tangle cannot explain the chaotic behavior of the system from the ergodic point of view: this set could be of a measure zero. Moreover, as was mentioned above, there are many elliptic orbits inside this stochastic region. So, what prevails is a very interesting and hard question [46, 47]. To give some insight, we performed a calculations of Lyapunov’s exponents. They appeared positive, see Fig. 7. Despite the slow fast character of the system under consideration, its behavior is similar to what was observed

452


in other numerical simulations of area preserving maps starting since the standard map see, for instance, recent [15, 32]). All these simulations show the presence of elliptic islands around elliptic periodic orbits within the mess of chaotic orbits. Though here it is more subtle task to find such orbits due to a relaxation nature of the system, we found such orbits using the technique exploiting the reversibility. The related orbits are shown on Fig. 8.

a)

b)

c)

Fig. 8 (a) Image of the fixed point line, near a tangency. Intersection with x = 0 corresponds to symmetric PO; (b) near a tangency of fixed point lines; (c) graph of the elliptic periodic orbit

9 Conclusions We study the model Duffing-like system being slow fast with the periodically slow varying parameter. The combination of rigorous methods along with the accurate numerical simulations allowed us to find some new periodic orbits (relaxation symmetric periodic orbits), to find regions in the phase space where the dynamics is of KAM type (where there exists a perpetual adiabatic invariant) and a region with the clearly observed stochastic behavior. We present some explanations of this behavior using the features of the system, in particular, its reversibility.

Acknowledgements Authors thank A.I. Neishtadt and P. Clarkson for useful discussions and explanations, and A. Gonchenko for a help in preparing figures. The research for this paper was supported by the following grants: the research of Sections 1 – 4 were supported by the Russian Foundation for Basic Research under the grant 14-01-00344 (N.K. and A.K.), the results from Sections 5 – 8 were supported by the Russian Science Foundation under the grant No. 14-41-00044. Also the results of L.L. were supported by the Russian Ministry of Science and Education (project 1.1410.2014/K, target part), results of A.K. were supported by the Basic Research Program at the National Research University Higher School of Economics (project 98) in 2016 and (partially) by the Dynasty Foundation. Numerical experiments were conducted using software package Computer Dynamics: Chaos.

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