Is the Hénon map chaotic Zbigniew Galias Department of Electrical Engineering AGH University of Science and Technology, Poland, galias@agh.edu.pl International Workshop on Complex Networks and Applications May 30th, 2014, Melbourne
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.
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 0.4 0.2 0 0.2 0.4 1.5 1 0.5 0 0.5 1 1.5 x
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.
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.
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.
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.
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 1 + 1 > 0, and a 0 a 1 + 1 > 0.
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.
Exhaustive search, details and results, b = 0.3 a b n par n init n skip n sink n win 1 1 10 6 1000 2 10 6 44 16 2 2 10 7 1000 10 6 850 63 3 1. 4001 3999 0.3 2 10 7 1000 4 10 6 862 73 4 8 10 7 1000 10 6 3396 101 5 2 10 8 1000 10 6 8502 134 6 1 10 6 1000 10 5 3 1 7 1. 40001 39999 0.3 2 10 7 1000 10 6 79 9 8 2 10 8 1000 10 6 730 18 9 2 10 7 1000 10 6 4 1 10 1. 400001 399999 0.3 2 10 8 1000 10 5 8 2 11 2 10 8 1000 10 6 53 8 12 1. 4000001 3999999 0.3 1 10 6 2000 10 5 0 0 13 2 10 8 1000 10 6 9 2 14 1. 400000025 399999975 0.3 1 108 1000 10 7 0 0 15 1. 40000001 39999999 0.3 2 108 1000 10 6 0 0 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.
Exhaustive search, details and results a b n par n init n skip n sink n win 1 2 10 7 1000 10 6 15989 82 2 1.4 0. 3001 99 2 10 7 1000 4 10 6 16001 92 3 2 10 8 1000 10 6 159835 185 4 1.4 0. 30001 999 2 10 8 1000 10 6 8649 39 5 1.4 0. 300001 2 10 7 1000 10 6 1 1 6 9999 2 10 8 1000 10 6 2 1 7 1.4 0. 3000001 99999 2 10 8 1000 10 6 8 2 8 1.4 0. 300000025 9999975 1 10 8 1000 10 7 123 3 9 1.4 0. 30000001 999999 2 10 8 1000 10 6 0 0 10 1. 4001 3999 0.3001 2 10 7 1000 10 6 54810 79 11 2 10 8 1000 10 6 548148 153 12 1. 4001 3999 0.99 2 10 7 1000 10 6 246 41 13 2 10 8 1000 10 6 2459 128 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.
Sinks close to (a, b) = (1, 4.0.3) (a, b) p w λ 1 r ε d (1.3999769098975, 0.3) 18 1.56 10 9 1.9683 10 5 1.4607 10 9 2.3090 10 5 (1.4, 0.30009066023) 19 8.19 10 8 1.0812 10 5 8.0639 10 8 9.0660 10 5 (1.40003418556, 0.3001) 19 7.70 10 8 1.0815 10 5 6.7199 10 8 1.0568 10 4 (1.39998447659, 0.3) 20 4.30 10 10 5.9049 10 6 4.0765 10 10 1.5523 10 5 (1.39990560396, 0.3001) 20 6.79 10 10 5.9246 10 6 5.8702 10 10 1.3752 10 4 (1.40004308355, 0.3001) 20 8.19 10 10 5.9246 10 6 5.8702 10 10 1.0889 10 4 (1.40002435, 0.3) 21 1.10 10 10 3.2342 10 6 1.1377 10 10 2.43 10 5 (1.4, 0.993238744) 21 2.14 10 10 3.2266 10 6 1.9659 10 10 6.7613 10 5 (1.399994921843, 0.3) 22 1.78 10 11 1.7715 10 6 1.8374 10 11 5.0782 10 6 (1.4, 0.993671494) 23 1.25 10 11 9.6792 10 7 1.60 10 11 6.3285 10 5 (1.4000128910375, 0.3) 24 6.06 10 12 5.3144 10 7 6.1534 10 12 1.2891 10 5 (1.400004161333, 0.3) 25 1.30 10 12 2.9108 10 7 1.1926 10 12 4.1613 10 6 (1.40001070815, 0.3) 26 2.53 10 12 1.5943 10 7 2.47 10 12 1.0708 10 5 (1.399977420742, 0.3) 7.63 10 13 8.7325 10 8 6.9015 10 13 2.2579 10 5 (1.4, 0.99999774905) 28 1.24 10 13 4.7830 10 8 1.1146 10 13 2.2530 10 8 (1.40001197567, 0.3) 4.96 10 12 2.6197 10 8 4.32 10 12 1.1976 10 5 (1.399999189126, 0.3) 30 1.56 10 13 1.4349 10 8 1.3376 10 13 8.1721 10 7 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).
Sinks close to (a, b) = (1, 4.0.3), cont. (a, b) p w λ 1 r ε d (1.4, 0.9989925114) 31 4.56 10 13 7.8551 10 9 3.9939 10 13 1.0075 10 5 (1.399985536811, 0.3) 32 1.84 10 12 4.3047 10 9 1.1926 10 12 1.4463 10 5 (1.3999994869436, 0.3) 33 1.02 10 12 2.3578 10 9 6.9015 10 13 5.1306 10 7 (1.4, 0.99986087693) 34 1.43 10 13 1.13 10 9 1.3376 10 13 1.3912 10 6 (1.39993062514792, 0.3) 35 6.63 10 14 7.0733 10 10 4.4794 10 14 6.9375 10 5 (1.40000755949567, 0.3) 36 6.57 10 13 3.8742 10 10 4.79 10 13 7.5595 10 6 (1.399988818205, 0.3) 37 1.55 10 12 2.1220 10 10 9.9382 10 13 1.1182 10 5 (1.4, 0.99955934368) 38 2.96 10 14 1.1619 10 10 2.1602 10 14 4.4066 10 6 (1.4, 0.999808226) 39 1.09 10 13 6.3652 10 11 1.1146 10 13 1.9702 10 6 (1.400020074, 0.3001) 40 1.48 10 12 3.5101 10 11 8.2818 10 13 1.0418 10 4 (1.4, 0.996374713205) 41 4.63 10 13 1.9051 10 11 3.9939 10 13 3.6253 10 5 (1.400053043104, 0.99) 42 2.57 10 13 1.0387 10 11 2.3113 10 13 1.1320 10 4 (1.4, 0.30008613807) 43 2.91 10 13 5.7648 10 12 2.7736 10 13 8.6138 10 5 (1.399990561167, 0.99) 47 1.97 10 13 5.1162 10 13 1.9261 10 13 1.0044 10 4 (1.399930212636, 0.3001) 56 6.06 10 13 2.3091 10 15 5.7512 10 13 1.2194 10 4 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).
Trajectories for a = 1.399999486944, b = 0.3 (x 0, y 0 ) = (0.1, 0.1) 10000 iterations, 5 10 9 iterations skipped 6 10 9 iterations skipped chaotic transient period 33 sink y y 0.4 0.2 0 0.2 0.4 0.2 0 0.2 0.4 0.4 x x 1.5 1 0.5 0 0.5 1 1.5 1.5 1 0.5 0 0.5 1 1.5
Trajectories for a = 1.4, b = 0.99999774905 (x 0, y 0 ) = (0, 0), 10000 points after skipping 10 10 iterations (blue dots), period 28 sink after skipping 1.05 10 10 iterations (red circles). 0.4 y 0.2 0 0.2 0.4 1.5 1 0.5 0 0.5 1 1.5 What is claimed to be a chaotic trajectory, might in fact be a transient to a periodic steady state. x
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 100000 random initial conditions. for (a, b) = (1.3999769102, 0.3) (period 18 sink): n conv (0.5) 3.35 10 6, n conv (0.9) 1.11 10 7, for (a, b) = (1.399999486944, 0.3) (period 33 sink): n conv (0.5) 1.66 10 9, n conv (0.9) 5.51 10 9, (a, b) = (1.4, 0.99999774905) (period 28 sink): n conv (0.5) 1.44 10 10, n conv (0.9) 4.74 10 10.
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) = (1.3999769102, 0.3), period-18 sink (+ ), (a, b) = (1.3999994869436, 0.3), period-33 sink, ( ), (a, b) = (1.4, 0.99999774905) period-28 sink, (+) 10 5 10 4 N k p=18 p=33 p=28 10 3 10 2 10 1 10 0 5 10 15 20 25 30 35 40 k
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).
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.
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.
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.
Continuation procedure to find the existence regions 0.3001 b period 33 0.30005 period 18 0.3 0.995 a 0.99 1.3999 1.39995 1.4 1.40005 1.4001
Sink existence regions, [1.3999, 1.4001] [0.99, 0.3001] 0.3001 21 32 32 24 25 34 25 34 33 39 39 39 20 24 18 30 30 25 25 26 23 36 24 0.30005 0.3 0.995 20 26 22 24 34 30 30 24 47 35 22 32 32 30 30 4141 34 36 37 b 22 24 21 26 28 32 32 30 23 24 39 28 22 28 30 31 31 30 26 56 30 23 39 39 24 23 25 34 26 23 33 33 34 33 30 28 37 37 38 26 25 32 26 30 25 34 24 25 28 30 34 28 24 41 32 24 25 28 25 19 34 26 24 34 33 31 33 41 28 0.99 1.3999 1.39995 1.4 1.40005 1.4001 30 26 36 36 35 35 43 31 34 33 34 36 33 37 32 41 43 41 20 40 25 33 33 38 40 30 28 37 30 25 30 28 28 35 38 19 28 28 38 32 26 28 30 28 28 35 28 30 42 35 22 a
Sinks for (a, b) [1.39999, 1.40001] [0.999, 0.30001] b 0.30001 34 68 25 50 32 64 32 64 72 24 48 96 25 50 75 33 66 36 21 42 84 63 28 56 58 33 0.300005 28 56 84 33 30 60 30 60 39 33 66 22 44 88 66 88 39 39 30 60 34 33 0.3 26 52 58 87 0.9995 20 24 40 48 36 72 90 54 18 144 37 74 80 96 60 80 90 72 90 0.999 1.39999 1.399995 1.4 1.400005 1.40001 28 33 66 99 38 31 34 a
Number of sink regions found 25 n 20 15 10 5 0 15 20 25 30 35 40 45 50 55 60 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
Widths of sink regions 10 7 w 10 8 10 9 10 10 10 11 10 12 10 13 10 14 10 20 30 40 50 60 Widths w of primary (+ ) and secondary ( ) existence regions versus period p p
Eigenvalues across the existence region 10 0 λ 10 2 10 4 10 6 10 8 10 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 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
Eigenvalues at the spine locus 10 4 λ 1 10 6 10 8 10 10 10 12 10 14 10 16 p 10 20 30 40 50 60 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.
The minimum immediate basin radius 10 7 r 10 8 10 9 10 10 10 11 10 12 10 13 10 14 10 20 30 40 50 60 The minimum immediate basin radius r ε versus period p for primary (+ ) and secondary ( ) sinks p
Period-35 swallowtail existence region b b 0.3000992 0.3000992 0.30009915 0.30009915 0.3000991 a 0.3000991 1.400067 1.40006703 1.40006706 1.400067 1.40006703 1.40006706 a Border of the period-35 complex existence region found using (a) the continuation method designed for narrow stripes and (b) grid continuation method
Period-39 and period-41 swallowtail existence regions Period-39 Period-41 b b 0.300088865 0.9958 0.30008886 0.300088855 0.99578 0.30008885 a a 0.99576 1.399912526 1.399912530 1.399912534 1.399912538 1.3999884 1.3999887 1.399989
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.