Is the Hénon map chaotic

Transcription

1 Is the Hénon map chaotic Zbigniew Galias Department of Electrical Engineering AGH University of Science and Technology, Poland, International Workshop on Complex Networks and Applications May 30th, 2014, Melbourne

2 Plan of the talk The Hénon map. Motivation for finding sinks close to the classical case. Tools locating sinks: exhaustive search, proving the existence of sinks: interval Newton method, finding regions of existence of sinks: continuation method. alternative method to find sinks: locate all periodic orbits for given parameter values and continue towards sinks. Search results. Conclusions.

3 The Hénon map The Hénon map: h(x, y) = (1 + y ax 2, bx). Classical parameter values: a = 1.4, b = 0.3. Example trajectory y x

4 Is the Hénon attractor chaotic? The Hénon is one of the best known examples of chaotic dynamical systems. In many applications it is quietly assumed that the Hénon map is chaotic trajectories are aperiodic and sensitive to initial conditions. The Hénon attractor may be chaotic. There is a set of parameters (near b = 0) with positive Lebesgue measure for which the map has a strange attractor. This set is believed to be densely filled with regions, where the attractor is periodic. In a region where this is true, given (a, b), it is impossible to prove that the attractor is chaotic. The Hénon attractor may be periodic What we observe in computer simulations may be a transient behaviour to a periodic steady state, or a rounding error artifact, or a periodic orbit with a very long period. It is theoretically possible to prove the existence of a periodic attractor.

5 Goal of this study Ultimate goal: prove that the Hénon map supports a sink for the classical parameter values. Practical goal: find parameter values close to the classical ones for which there exists a sink.

6 Locating sinks in the parameter space Method to find a sink: follow a trajectory and monitor whether it converges to a periodic orbit. First, a number of iterates are computed in the hope that a trajectory reaches a steady-state. The number of iterations which are discarded is usually chosen by trial-and-error (it depends on Lyapunov exponents of the attractor, the size and shape of its basin of attraction, etc.). Next, we take the current iterate as the new initial point and check if the trajectory periodically returns very close to this point. To locate sinks in the parameter space perform exhaustive search in some cases it may by sufficient to search along straight lines.

7 The existence of a periodic orbit Interval methods provide simple tests for the existence and uniqueness of zeros of a map within a given set. To investigate zeros of F in the interval vector v one evaluates an interval operator, for example the interval Newton operator: N(v) = ˆv F (v) 1 F(ˆv), where ˆv v, and F (v) is an interval matrix containing the Jacobian matrices F (v) for all v v. Theorem: if N(v) v, then F has exactly one zero in v. To study the existence of period p orbits of h, we construct the map F defined by [F(v)] k = z (k+1) mod p h(z k ) for 0 k < p, v is a zero of F if and only if z 0 is a fixed point of h p.

8 Stability of a periodic orbit If eigenvalues λ i of the Jacobian matrix J = (h p ) (z 0 ) = h (z p 1 ) h (z 1 ) h (z 0 ) lie within the unit circle, i.e. λ i < 1, then the orbit v = (z 0, z 1,..., z p 1 ) is asymptotically stable. If at least one eigenvalue lies outside the unit circle ( λ i > 1) then the orbit is unstable. Verification that a periodic orbit is a sink: Compute rigorous enclosures of eigenvalues and verify that they are enclosed in the unit circle, or Use the Jury criterion: the second-order polynomial λ 2 + a 1 λ + a 0 has all zeros within the unit circle if and only if a 0 < 1, a 0 + a > 0, and a 0 a > 0.

9 Immediate basin of attraction We say that a point z belongs to the immediate basin of attraction B ε (A) of the attractor A if its trajectory converges to the attractor and does not escape further than ε from it: B ε (A) = {z : d(h n (z), A) ε n 0 and lim n d(h n (z), A) = 0}, where d(z, A) denotes the distance between the point z and the set A (ε for example 1% of the attractor size). the minimum immediate basin radius r ε (A) of the attractor, is the largest number such that all points lying closer than r ε from the attractor belong to the immediate basin of attraction r ε (A) = sup{r : x B ε (A) for all d(x, A) r}. r ε (A) and convergence times. r ε (A) and the arithmetic precision.

10 Exhaustive search, details and results, b = 0.3 a b n par n init n skip n sink n win n par the number of parameter values, n init the number of random initial points and, n skip the number of iterations skipped n sink the numbers of parameter values with a sink found, n win the number of periodic windows found.

11 Exhaustive search, details and results a b n par n init n skip n sink n win n par the number of parameter values, n init the number of random initial points and, n skip the number of iterations skipped n sink the numbers of parameter values with a sink found, n win the number of periodic windows found.

12 Sinks close to (a, b) = (1, 4.0.3) (a, b) p w λ 1 r ε d ( , 0.3) (1.4, ) ( , ) ( , 0.3) ( , ) ( , ) ( , 0.3) (1.4, ) ( , 0.3) (1.4, ) ( , 0.3) ( , 0.3) ( , 0.3) ( , 0.3) (1.4, ) ( , 0.3) ( , 0.3) p period, w window width, r ε the minimum immediate basin radius, λ 1 the largest Lyapunov exponent, d distance from (1, 4.0.3).

13 Sinks close to (a, b) = (1, 4.0.3), cont. (a, b) p w λ 1 r ε d (1.4, ) ( , 0.3) ( , 0.3) (1.4, ) ( , 0.3) ( , 0.3) ( , 0.3) (1.4, ) (1.4, ) ( , ) (1.4, ) ( , 0.99) (1.4, ) ( , 0.99) ( , ) p period, w window width, r ε the minimum immediate basin radius, λ 1 the largest Lyapunov exponent, d distance from (1, 4.0.3).

14 Trajectories for a = , b = 0.3 (x 0, y 0 ) = (0.1, 0.1) iterations, iterations skipped iterations skipped chaotic transient period 33 sink y y x x

15 Trajectories for a = 1.4, b = (x 0, y 0 ) = (0, 0), points after skipping iterations (blue dots), period 28 sink after skipping iterations (red circles). 0.4 y What is claimed to be a chaotic trajectory, might in fact be a transient to a periodic steady state. x

16 Convergence times Convergence time depends on the initial point. n conv (p) the number of iterations required to converge to the sink with probability p. Statistics based on random initial conditions. for (a, b) = ( , 0.3) (period 18 sink): n conv (0.5) , n conv (0.9) , for (a, b) = ( , 0.3) (period 33 sink): n conv (0.5) , n conv (0.9) , (a, b) = (1.4, ) (period 28 sink): n conv (0.5) , n conv (0.9)

17 Convergence times, examples The number N k of random initial points with the convergence time in the interval [2 k 1, 2 k ) (a, b) = ( , 0.3), period-18 sink (+ ), (a, b) = ( , 0.3), period-33 sink, ( ), (a, b) = (1.4, ) period-28 sink, (+) N k p=18 p=33 p= k

18 Finding sink existence regions When a point (a, b) in the parameter space with a sink is found one may use the continuation method to find a connected region in the parameter space for which this sink exists. The simplest version is to select grid points (a + i a, b + j b), i, j Z in the parameter space and continue to neighboring grid points from the set of active grid points. The position of the sink for a close test point can be easily found using the standard Newton method started at the position of the orbit for the current point (positions of periodic orbits change continuously with the parameters).

19 Finding borders of the sink existence regions The border of the region is defined by conditions the periodic orbit exists, one of the eigenvalues of the Jacobian matrix has the absolute value 1. Procedure to find the border: Continue along a straight line, starting from the point for which the existence of the sink has been verified. This, combined with the bisection method gives us two points belonging to the border of the existence region. For each of the two points use the simplex continuation method in two directions a sequence of triangles is constructed such that each triangle has non-empty intersection with the border. Corners of the triangles are located on a regular grid.

20 Better approximation of the position of periodic orbits Assume that for the current point (a, b) in the parameter space the position of the sink is x = (x 0, x 1,..., x p 1 ). In the simplest approach one may use x = x as a guess of the position of the sink for the test point (a + a, b + b). Let F(a, b, x) = 0 be the equation defining position of periodic orbit. From the implicit function theorem, it follows that if the matrix F x is invertible, then the partial derivatives x/ a of the solution x(a, b) of F(x) = 0 can be obtained by solving the linear equation: F x x a + F a = 0, Similarly one can find x/ b. approximation of the position of the orbit for the test point (ã, b) = (a + a, b + b) can be constructed as x = x + x x a + a b b.

21 Method for narrow regions Observation: most regions locally resemble narrow stripes. Observation: the direction of the stripe locally agrees with the direction in which the maximum eigenvalue of the orbit is constant. Let λ be the eigenvalue of the Jacobian matrix with a larger absolute value. For the current point (a, b) we compute derivatives λ/ a, λ/ b (automatic differentiation, x/ a, x/ b), Close to the point (a, b) we have λ(a + a, b + b) λ(a, b) + λ λ a + a b b. If we continue in the directions ±( λ/ a, λ/ b), we reach the borders of the existence region. If we continue in the directions ±( λ/ b, λ/ a) we move along the existence region. With this approach we can move much further in one step.

22 Continuation procedure to find the existence regions b period period a

23 Sink existence regions, [1.3999, ] [0.99, ] b a

24 Sinks for (a, b) [ , ] [0.999, ] b a

25 Number of sink regions found 25 n The number of primary regions ( ), and the total number of regions (+ ) found for each period p. The total number of sink regions found: 461. p

26 Widths of sink regions 10 7 w Widths w of primary (+ ) and secondary ( ) existence regions versus period p p

27 Eigenvalues across the existence region 10 0 λ Eigenvalues of the Jacobian matrix across the existence region of one of the period-19 sinks, t = 0 and t = 1 correspond to borders of the existence region. t

28 Eigenvalues at the spine locus 10 4 λ p The absolute value of the larger eigenvalue versus period p, Explanation: λ 1 λ 2 = ( b) p, log λ 1 = 0.5p log b, b 0.3.

29 The minimum immediate basin radius 10 7 r The minimum immediate basin radius r ε versus period p for primary (+ ) and secondary ( ) sinks p

30 Period-35 swallowtail existence region b b a a Border of the period-35 complex existence region found using (a) the continuation method designed for narrow stripes and (b) grid continuation method

31 Period-39 and period-41 swallowtail existence regions Period-39 Period-41 b b a a

32 Conclusions An exhaustive search for sink regions close to (1.4, 0.3) have been carried out. Several parameter values in a neighborhood of (1.4, 0.3) for which a sink exists have been found. It has been shown that close to the classical case, the regions of existence of sinks are very narrow and finding them is not a trivial numerical task. We presented examples confirming that in some cases where there appears to be a strange attractor, the true underlying dynamics is in fact governed by a periodic sink. Using the continuation method, the regions of existence of sinks have been found, which made it possible to move closer to the classical case. It was explained why it is difficult to observe low period sinks in simulations in spite of very long computation times.