Discontinuity, Nonlinearity, and Complexity

Discontinuity, Nonlinearity, and Complexity 3(1) (2014) 1-6

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

How the minimal Poincaré return time depends on the size of a return region in a linear circle map N. Semenova, E. Rybalova, V. Anishchenko† Saratov State University, Saratov, 410012, Russia Submission Info Communicated by Referees Received DAY MON YEAR Accepted DAY MON YEAR Available online DAY MON YEAR

Abstract It is found that the step function of dependence of the minimal Poincaré return time on the size of a return region τinf (ε) for the linear circle map with an arbitrary rotation number can be approximated analytically. All analytical results are confirmed by numerical simulation. ©2014 L&H Scientific Publishing, LLC. All rights reserved.

Keywords Circle map Poincaré recurrence Afraimovich-Pesin dimension rotation number Diophantine number Fibonacci stairs

1 Introduction Poincaré recurrence is one of the fundamental features occurring in the time evolution of dynamical systems. Almost every trajectory in the phase space of a system with a fixed measure returns in the vicinity of an initial state. H. Poincaré called such a trajectory as Poisson-stable [1]. If the system demonstrates chaotic behaviour, then a sequence of Poincaré recurrences is random and thus can be described by using statistical methods. There are two approaches for analysing of Poincaré recurrences, namely, local and global ones. Unlike the classical local approach, in which Poincareé recurrences are calculated in a ε-vicinity of the initial state, the global approach deals with the recurrence characteristics for the whole set. The main characteristic of Poincaré recurrences in the global approach is the recurrence time dimension which is called the Afraimovich–Pesin dimension (AP dimension) [2, 3]. The return time statistics in the global approach depends on the topological entropy hT . The case of mixing sets (hT > 0) has been studied analytically [2–4] and the results have been confirmed by numerical simulations [5–7]. If hT = 0, then the behaviour is ergodic and without mixing. Such a system can be exemplified by the circle map: Θn+1 = Θn + ∆ + K sin Θn mod 2π. (1) † Corresponding

author. Email address: [email protected] ISSN 2164-6376, eISSN 2164-6414/$-see front materials © 2014 L&H Scientific Publishing, LLC. All rights reserved.

2

N. Semenova, E. Rybalova, V. Anishchenko/ Discontinuity, Nonlinearity, and Complexity Vol-number (year) page1–page2

The variable Θn , which can take a value from zero to 2π, characterizes the rotation angle of the point on a circle with radius 1. The sign mod 2π means that the 2π-fold part of the phase variable is discarded. K is the parameter of nonlinearity, ∆ sets the rotation number [8, 9] and is the fixed shift on the circle. The map (1) simulates the dynamics of two-frequency quasiperiodic oscillations in the Poincaré section of a two-dimensional torus [10–12]. If K = 0, the map is linear: Θn+1 = Θn + ∆ mod 2π

(2)

The analytical results for the linear circle map [3] has been extensively confirmed in numerical experiments in the works [7, 13, 14]. A new dependence of the minimal return time τinf (ε) on the vicinity size ε has been found. In [13, 14] we call it the “Fibonacci Stairs”. It has been proven that this dependence has a universal geometry for the golden and silver rotation numbers, i.e., the height and the length of each step D are D = ln δ , where δ is the rotation number [13]. For another irrational rotation numbers this universal feature of τinf (ε)dependence does not occur [14]. The aim of the present work is to analyse analytically and numerically the geometrical features of the Fibonacci stairs in a general case of any rotation numbers.

2 System under study In this work we analyse the particular case of the linear shift on the circle (K = 0 in (1)). The rotation number δ is the main characteristic which enables one to diagnose periodic and quasiperiodic regimes. In general, the rotation number is defined as follows: Θn − Θ0 δ (∆, K) = lim , (3) n→∞ 2πn where Θn is the rotation angle of the circle map (1) or (2). Now we take into account the 2π-fold part. Θ0 is the initial angle, n is the number of iterations. Thus, the rotation number is the mean rotation angle Θ after one iteration of the map. In the linear case, K = 0, the rotation number depends only on the parameter ∆ as follows: δ = ∆/2π.

(4)

1 From a physical standpoint, the rotation number characterizes the ratio of independent frequencies δ = ω ω2 for two-frequency quasiperiodic behavior (see, for instance, [12, 15]). Rational rotation numbers correspond to a periodic sequence Θn = Θn+q , where q is the period of motion. In the case of irrational rotation numbers, this sequence Θn covers the circle uniformly as n → ∞. It corresponds to the two-frequency quasiperiodic regime with an irrational ratio of ω1 and ω2 frequencies.

3 Poincaré recurrences in the linear circle map In the global approach, the whole set of phase trajectories of a dynamical system is covered with cubes (or balls) of size ε 1. For each covering element ξ j ( j = 1, 2, ...m) a minimal return time τinf (ξ j ) of the trajectory to the ξ j neighbourhood is calculated. Then the mean minimal return time hτinf (ε)i is found over the whole set of covering elements ξ j . The map (2) produces the set {Θn , mod 2π}. This set is an example of the simplest minimal set with irrational rotation number δ , for which theory of Poincaré recurrences has been fully developed. In this work we use the following main theoretical results. It was proved [2, 3] that for the linear circle map (2) d

hτinf (ε)i ∼ ε ν(δ )

or (5)

lnhτinf (ε)i ∼

d − ν(δ )

ln ε, ε 1, d = 1,


3

where hτinf (ε)i is the mean minimal return time which is found over the whole set of covering elements of size ε, ν(δ ) is the maximal rate of Diophantine approximations of an irrational number δ over all possible pairs of p and q, and d is the fractal dimension of the set, which is equal to one. In general for ergodic sets with zero topological entropy it was proved [2, 3] that lnhτinf (ε)i ∼ −

d ln ε, ε 1, αc

(6)

where αc is the Afraimovich–Pesin dimension. Comparing (5) and (6) one can obtain that αc = ν(δ ) for the circle shift (2). In such a way the AP dimension is equal to the rate of Diophantine approximations ν(δ ). For Diophantine irrational numbers ν(δ ) = 1 and thus αc = 1. For an irrational rotation number the probability distribution p(Θ) is uniform in the interval [0; 2π). This implies that in this case the local and global approaches can give equivalent results. Hence we can calculate τinf (ε) instead of hτinf (ε)i. The theoretical results corroborated for the circle√map have been confirmed by numerical simulation in our paper [13] for the case of the golden ratio δ = 21 ( 5 − 1). The universal dependence of τinf (ε) which we referred√to as the “Fibonacci Stairs” has been found. This dependence is shown in Fig. 1 for the golden ratio δ = 21 ( 5 − 1). We have established that the “Fibonacci Stairs” has several features which are as follows.

377

ln((ε))

6

233 144

5

89

D

4

55 34 21

3

13

D

8

2 −5

−4

−3

−2

−1

ln(ε) Fig. 1 “Fibonacci Stairs”: Dependence of the minimal return time on the vicinity size for the circle map (2) with √ δ = 21 ( 5 − 1) [13]

1. When ε decreases, the sequence of τinf (ε) values grows and strictly corresponds to the basic Fibonacci series {Fi } (as indicated in Fig. 1). Each minimal return time which relates to each ith step of the Fibonacci stairs corresponds to the denominator qi of the ith convergent for the fraction pi /qi . For the golden ratio qi = Fi . 2. When ε is varied within any of the stair steps, three return times τ1 < τ2 < τ3 can be distinguished. Additionally, τ1 = τinf . This property follows from Slater’s theorem [15]. 3. The length and height of the steps in Fig.1 depend on the rotation number as D = − ln δ . √ The silver ratio corresponds to the rotation number δ = 2 − 1. In this case we obtain the same step function with all the features of “Fibonacci Stairs”. However, when ε decreases, the sequence of values τinf (ε) strictly

4


√ corresponds to the basic Pell series. In general cases, for example, δ = 3 2, δ = e or δ = lg 5, the first two universal properties given above are preserved but the third one is violated. We note that in the first property, the sequence of τinf (ε) for each indicated δ still looks like as a step function but is described by a different law (series) with no special name. Thus, the Afraimovich–Pesin dimension cannot be calculated for the system (2) using (5) because of unpredictability of the next step emergence. In the present work we try to predict analytically the ε values, which correspond to the emergence of new steps, by using the rotation number value and the stepwise configuration of the function τinf (ε). √ This enables one to calculate the exact value of AP dimension for any Diophantine rotation numbers as δ = 3 2, δ = e or δ = lg 5.

4 Fibonacci Stairs approximation The structure of the “Fibonacci Stairs” is closely related to the theory of convergents and continued fractions. An irrational number is a real number which cannot be written as a fraction p/q, where p and q are natural numbers, 1, 2, . . .. In the general case, an irrational rotation number can be presented in the form of a continued fraction [16]: δ = a0 +

1 a1 + a

2+

1

(7)

1 1 a3 + ...

This produces a sequence of approximation coefficients {ai }, i ≥ 0. The notation [a0 ; a1 , a2 , a3 , . . .] is an infinite continued fraction representation of the irrational number. The irrational rotation number δ can be approximated by the fraction of two integers pi /qi . This is the method of rational approximations. The ith convergent of the continued fraction δ = [a0 ; a1 , a2 , a3 . . . ] is a finite continued fraction [a0 ; a1 , a2 , . . . , ai ], which value is equal to the rational number pi /qi . The increasing sequences of numerators {pi } and denominators {qi } are called continuants of the ith convergent (7) and can be found using fundamental recurrence formulas: p−1 = 1, p0 = a0 , pi = ai pi−1 + pi−2 ,

(8)

q−1 = 0, q0 = 1, qi = ai qi−1 + qi−2 , where {ai } are natural coefficients of the continued fraction, pi , qi are numerators and denominators of the convergent. It has been found [7,13] that for any rotation number, the dependence τinf (ε) is a step function and each value τinfi , which corresponds to the ith step, is equal to the denominators of the ith convergent pi /qi of the rotation number δ . Using the equality τinfi (ε) = qi we obtain the minimal vicinity size which corresponds to this return time τinfi . As noted in the Introduction, after one iteration of the linear circle map (2) the position of the point on the circle changes by 2πδ . The expression τinf (ε) = qi means that the point returns in the neighbourhood of its initial state after qi iterations, shifting by 2πδ qi . During these iterations the point can make several complete circles and appear to the left or right of the initial state. To take this fact into account we introduce the modulus and subtract the convergent numerator pi which defines the number of complete circles. Thus, the return in the neighbourhood of the initial state x0 takes place at the distance of 2π|δ qi − pi | from the point x0 [17]. Let us consider the case when we start not from the point x0 but from the right boundary of its neighborgood, i.e., from the point x00 = x0 + εc /2. The return in εc after the minimal number of iterations qi happens near the left boundary of this neighbourhood, i.e., at the point x0 − εc /2. In such a case, as mentioned above, the point shifts by 2π|δ qi − pi | from the initial position x00 (see Fig. 2). This means that x00 − 2π|δ qi − pi | = x0 − εc /2

(9)


5

Fig. 2 Schematic representation of the neighbourhood ε and initial and return points on the circle

x0 + εc /2 − 2π|δ qi − pi | = x0 − εc /2

(10)

This enables one to derive the expression for calculating the value εi , which corresponds to the left boundary of the stairs step with the minimal return time τinf = qi for any irrational rotation number δ : εi = εc = 2π|δ qi − pi |,

(11)

where δ is the rotation number, qi is convergent denominator, pi is the convergent numerator. As discussed above, the golden (silver) ratio represents a special case. The universal feature of the staircase dependence is due to the fact that numerators and denominators of convergents have the same definition rules and are elements of the Fibonacci (Pell) series. Thus, for the golden ratio, (11) can be rewritten as follows; εi = 2π|δ Fi − Fi−1 |,

or (12)

εi ≈

2π Li ,

√ where Fi is the ith Fibonacci number, δ = ( 5 − 1)/2 is the golden ratio, and Li is the ith Lucas number. A more detailed description is given in Appendix 1. Following the same motivation, for the silver ratio we can find εi = 2π|δ Pi − Pi−1 | or (13) εi ≈

2π Qi ,

√ where Pi is the ith Pell number, δ = 2 − 1 is the silver ratio, and Qi is the ith Pell-Lucas number. We confirm our analytical results (11)–(13) by√ numerical simulation for the golden and silver ratios (Fig. 3) as well as for more complex Diophantine numbers 3 2, e, and lg(5) which correspond to the absence of universal geometry of the step dependence (Fig. 4). Using (11) we can find the dependence of each step length Di on its number in general (see Appendix 2).

6


For the golden and silver ratios, the length of stairs steps is constant and independent of the step number but is defined by the rotation number (see Appendix 2): Di = const = ln δ .

(14)

Figure 5 illustrates dependences of the step length on the step number for the golden ratio (the universal √ geometry is valid) and for two different values of the rotation number, namely, δ = 3 2 and lg 5 (no universal geometry is observed). Figure 5,a corresponds to the golden ratio and shows that all step lengths are equal. Figures 5,b and c indicate that the universal geometry fails for the other rotation numbers. As can be seen from Fig. 5, analytical and numerical results are in full agreement. This means that the universal geometry can be obtained only in cases of the golden and silver ratios for which the step length (Fig. 5,a) does not depend on the step number.

(a)

(b)

δ=(51/2−1)/2

12

12 Fibonacci Stairs εi=2π |δFi − Fi−1| εi=2π/Li

8

Stairs εi=2π |δPi − Pi−1| εi=2π/Qi

10

ln (τinf)

10

ln (τinf)

δ=21/2−1

6 4 2

8 6 4 2

0

0 −8

−6

−4

ln ε; ln εi

−2

0

−8

−6

−4

−2

0

ln ε; ln εi

√ √ Fig. 3 Dependences ln τinf (ln ε) for (a) the golden ratio (δ = ( 5 − 1)/2) and (b) the silver ratio (δ = 2 − 1) are indicated by solid lines, dashed lines with plus points and circle points represent the corresponding approximations using (12) for the golden ratio and (13) for the silver ratio

7


(a) 10

Stairs εi=2π |δqi − pi|

ln (τinf)

8

6 4 2

6 4 2

0

0 −8

−6

−4

−2

0

ln ε; ln εi

(c)

−8

6 4 2

−2

0

−2

0


8

ln (τinf)

8

−4

δ=lg5

10


−6

ln ε; ln εi

(d)

δ=e

10

ln (τinf)

δ=71/3

10


8

ln (τinf)

(b)

δ=21/3

6 4 2

0

0 −8

−6

−4

ln ε; ln εi

−2

0

−8

−6

−4

ln ε; ln εi

√ √ Fig. 4 Dependences ln τinf (ln ε) for four values of the rotation number: (a) δ = 3 2, (b) δ = 3 7, (c) δ = e and (d) δ = lg(5) (solid lines). Dashed lines with circle points show the corresponding approximations using (11)

8


(a)

(b)

δ=(51/2−1)/2

0.6

3

Numerical simulation Analytics

0.55


2.5

0.5

Di

δ=21/3

2

Di

0.45

1.5

0.4

1

0.35

0.5

0.3

0 2

4

6

8 10 12 14 16 18 20

1

2

3

i

4

5

6

7

8

9

i

(c)

δ=lg5

3


2.5 2

Di

1.5 1 0.5 0 1

2

3

4

5

6

7

i Fig. 5 Dependences of the step length on the step number for three values of the rotation number: (a) the golden ratio, (b) √ δ = 3 2, and (c) δ = lg 5


9

5 Conclusion We have shown that the dependence τinf (ε) has a step structure for any irrational rotation number. The values τinfi , which corresponds to the ith step, are equal to denominators of the ith convergents pi /qi of the rotation number δ . Using τinf (εi ) = qi we find the minimal vicinity size which corresponds to the left boundary of the step (11). Correctness of this approximation is confirmed by numerical simulation not only for the golden and silver ratios, for which the dependence τinf (ε) is named the “Fibonacci Stairs” and has several features, but also √ √ 3 3 for more complex values of the rotation number, namely, algebraic ( 2 and 7) and transcendental (e and lg 5). Using the rotation number value and the step form of the function τinf (ε) one can predict the critical values εi which correspond to the emergence of new steps. This enables one to calculate the Afraimovich-Pesin dimen√ 3 sion for Diophantine rotation numbers, for example, for δ = 2, δ = e or δ = lg(5).

6 Acknowledgements This work was partly supported by the RFBR (Grant No. 15-02-02288).

7 Appendix 7.1

Golden ratio

√ In the case of the golden ratio (δ = ( 5 − 1)/2), denominators and numerators of the convergents of δ can be found as qi = Fi and pi = Fi−1 , where {Fi } is the Fibonacci sequence. The golden ratio is a special case when numerators and denominators have the same determination rule and are elements of the same sequence. Thus, we can simplify the expression (11). Each ith Fibonacci number is defined by the following recurrence relation: Fi = Fi−1 + Fi−2 .

(15)

with the set values F0 = 1, F1 = 1. The ith Fibonacci number can be also found using Binet’s formula:

Fi =

ϕ i − (−ϕ)−i , 2ϕ − 1

(16)

√ where ϕ = ( 5 + 1)/2 is the root of the equation ϕ 2 − ϕ − 1 = 0 and depends on δ as δ = ϕ −1

ϕ = 1+δ.

(17)

The value of ε which corresponds to the emergence of a new stairs step and relates to the left boundary of the step with the minimal return time τinf = Fi is εi = 2π|δ Fi − Fi−1 |

(18)

10


Using (16) and (17) this expression can be rewritten as follows: εi = 2π|(ϕ − 1)Fi − Fi−1 | = 2π|ϕFi − (Fi + Fi−1 )| = i −i ϕ i+1 −(−ϕ)−i−1 = 2π|ϕFi − Fi+1 | = 2π ϕ −(−ϕ) − = 2ϕ−1 2ϕ−1 i+1 −i+1 −ϕ i+1 +(−ϕ)−i−1 = 2π ϕ +(−ϕ) 2ϕ−1 = −i−1 −i+1 +ϕ −i−1 ) −i−1 × = 2π (−1) = 2π (−1) (ϕ 2ϕ−1 2

(19)

2

2

ϕ +1 −i−1 ϕ +1 ϕ +1 −i−1 × 2ϕ−1 ϕ = 2π 2ϕ−1 ϕ = 2πϕ −i 2ϕ 2 −ϕ = 2

−ϕ−1+ϕ+2 −i = = 2πϕ −i 2ϕϕ2 −ϕ−ϕ−2+2+ϕ = 2πϕ −i ϕ+2 ϕ+2 = 2πϕ

=

2πϕ −i (ϕ i +(−ϕ)−i ) ϕ i +(−ϕ)−i

=

2π

ϕ i +(−ϕ)−i

1 + (−1)−i ϕ −2i .

Since ϕ > 1, the second term between the brackets tends to zero when i → ∞. Thus, lim εi (τinf = Fi ) =

i→∞

2π ϕ i + (−ϕ)−i

=

2π , Li

(20)

where Li is the ith Lucas number. It is defined by the same recurrence relation as the Fibonacci numbers (15) but with another set values L0 = 2, L1 = 1. The Lucas numbers can be approximately defined by the following formula: Li = ϕ i + (−ϕ)−i . (21) 7.2

Calculation of the step length

The size of the neighbourhood εi , which corresponds to the left boundary of a step with the minimal return time τinf = qi for any irrational rotation number δ can be found as follows: εLi = εi = 2π|δ qi − pi | .

(22)

Similarly we can obtain the value εi−1 . Since the dependence τinf (ε) is a step-like function, εi−1 is simultaneously the left boundary of the step with the minimal return time τinf = qi−1 and the right boundary of the step with τinf = qi : εRi = εi−1 = 2π|δ qi−1 − pi−1 | (23) Hence, the length of the ith step of the dependence ln τinf (ln ε) can be calculated as follows: Di = ln εLi − ln εRi = ln εεRiLi = 2π|δ qi −pi | |δ qi −pi | = ln 2π|δ qi−1 −pi−1 | = ln |δ qi−1 −pi−1 | =

(24)

−pi /qi | i = ln qi−1q|δi |δ−p ≈ ln qqi−1 . i−1 /qi−1 |

Thus, in general the length of stairs steps depends on denominators of convergents of rotation numbers. For the golden ratio, the denominators and numerators of the convergents are related to the Fibonacci series as pi−1 = qi = Fi . It follows that for the golden ratio, qi pi−1 Di ≈ ln ≈ ≈ ln δ . (25) qi−1 qi−1


11

The same motivation can be used for the silver ratio. In this case, the numerators and denominators are connected with the Pell series: pi−1 = qi = Pi . For this reason the step lengths for the golden and silver ratios are constant and independent of the step number. They are defined only by the rotation number ln δ .

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