1. LMS Durham Symposium Dynamical Systems and Statistical Mechanics 3 – 13 July 2006 Locating periodic orbits in high-dimensional systems by stabilising transformations Ruslan L Davidchack Jonathan J Crofts Department of Mathematics University of Leicester

2. LMS Durham Symposium Dynamical Systems and Statistical Mechanics 3 – 13 July 2006 Outline The problem of locating periodic orbits Stabilising transformations approach Application to low-dimensional maps Extension to higher-dimensional system Locating periodic orbits in high-dimensional flows Kuramoto-Sivashinsky equation Periodic and relative periodic orbits

3. LMS Durham Symposium Dynamical Systems and Statistical Mechanics 3 – 13 July 2006 The problem of locating periodic orbits Finding periodic orbits of a map is equivalent to finding zeros of a function For discrete systems, the period p = 1, 2, 3, … For flows: period is unknown. Typically, the number of periodic orbits grows exponentially with p. Systematic detection of POs of increasing period requires a reliable recipe for selecting useful seeds and an efficient iterative scheme. When the fixed point is stable, simply iterating the map will find it. For unstable orbits the general strategy is to construct an alternative map (e.g. Newton-Raphson method) where the POs become stable. The seeds are obtained from close returns or the knowledge of the structure of the system.

4. LMS Durham Symposium Dynamical Systems and Statistical Mechanics 3 – 13 July 2006 Stabilising transformations To locate fixed points of, integrate the associated flow: where C is an constant orthogonal matrix. The map and flow  have identical sets of fixed points. We can select such C that an unstable fixed point of becomes a stable fixed point of . The goal is to find a (small) set such that every unstable fixed point of can be stabilised by at least one.

5. LMS Durham Symposium Dynamical Systems and Statistical Mechanics 3 – 13 July 2006 “ Global” convergence of associated flow g(x)g(x) 1D case: q-Newton C = 1 C = −1 Associated flow with C = 1 ( −1 ) converges to roots with g´(x) 0 )

6. LMS Durham Symposium Dynamical Systems and Statistical Mechanics 3 – 13 July 2006 Stabilising transformations Conjecture: ( Schmelcher and Diakonos, 1997 ) Any unstable fixed point of can be stabilised by at least one C from the set of all orthogonal matrices with only non-zero entries. C = SP, where and P is a permutation matrix. Total number of such matrices: M = 2 n n! The conjecture is easily verified for and appears to be true for n > 2.

7. LMS Durham Symposium Dynamical Systems and Statistical Mechanics 3 – 13 July 2006 Convergence basins of associated flows for Ikeda map: p = 5 – sinks – saddles

8. LMS Durham Symposium Dynamical Systems and Statistical Mechanics 3 – 13 July 2006 Integrating the associated flow Schmelcher and Diakonos use the Euler method A modified semi-implicit Euler method (RLD and Y.-C. Lai, 1999) where  is a parameter, s n = |g(x n )|, and G n = Dg(x n ) is the Jacobian Standard (variable order, variable step-size) methods for solving stiff ODEs, e. g. ode15s in Matlab lsoda from netlib.org (odepack)

9. LMS Durham Symposium Dynamical Systems and Statistical Mechanics 3 – 13 July 2006 Seeding with periodic orbits When the goal is to locate complete sets of periodic orbits for each p, the traditional seeding schemes are not satisfactory Use already detected periodic orbits as seeds to locate new ones: For simple maps (Hénon, Ikeda) it is sufficient to use orbits of period p – 1 to locate orbits of period p In more complex cases (e.g. n > 2 ) it is necessary to use orbits of period q > p to locate complete sets of orbits for each p Scalability: The seeding scheme can be set up and tuned at small periods and then applied to larger periods

10. LMS Durham Symposium Dynamical Systems and Statistical Mechanics 3 – 13 July 2006 Periodic orbits in the Ikeda attractor pnpnp NpNp p –1 lnN p 10464830.6180 11768370.6118 121101383 0.6027 131942 523 0.6026 143174 511 0.6010 155668 517 0.6033 1695015 327 0.6023 171 64627 983 0.6023 182 79950 667 0.6018 194 88492 797 0.6020 208 404168 575 0.6018 2114 700308 777 0.60192 2225 550562 939 0.60186 2344 6561 027 089 0.60184 2478 0771 875 231 0.60184 (RLD Lai Bollt Dhamala, 2000)

11. LMS Durham Symposium Dynamical Systems and Statistical Mechanics 3 – 13 July 2006 Extension to higher-dimensional systems The use of SD stabilising transformations becomes inefficient for larger n due their rapidly growing number ( M = 2 n n! ) Even though in practice it appears that only a small subset of these transformations is necessary to stabilise all periodic orbits of a given system, it is not clear how to identify this subset a priori To efficiently extend this approach to higher dimensions, we need to find a smaller set of stabilising transformations

12. LMS Durham Symposium Dynamical Systems and Statistical Mechanics 3 – 13 July 2006 Understanding stabilising transformations For n = 2: Eigenvalues of C s,  G ( G = Dg(x*) ): x* is stabilised if s = sign det G and |  −  | <  /2 Re – solid Im – dashed s = −sign det G s = sign det G

13. LMS Durham Symposium Dynamical Systems and Statistical Mechanics 3 – 13 July 2006 Extension to higher-dimensional systems Observations: Properties of stabilising transformations depend essentially on the eigenvectors and signs (but not the magnitudes) of unstable eigenvalues of G Typically, eigenvectors of different periodic orbits are locally aligned A matrix C that stabilises a given fixed point of is likely to stabilise neighbouring fixed points (for all p ) Within the seeding with periodic orbits scheme it should be possible to construct stabilising transformations based on the knowledge of already detected orbits

14. LMS Durham Symposium Dynamical Systems and Statistical Mechanics 3 – 13 July 2006 Extension to higher-dimensional systems Note that C s,  =  = − Q T, where Q is the orthogonal matrix in the polar decomposition of G : G = QB, where B = (G T G) 1/2 As in the 2D case, a fixed point x* with G = Dg(x*) is stabilised when the matrix C is “close” to − Q T, where Q = (G T G) 1/2 G −1. This is a corollary of Lyapunov’s stability theorem ( Crofts and RLD, 2006 ) Orthogonal matrices A and B are ”close” if all eigenvalues of the product A T B have positive real parts (i.e. all principal rotations are smaller than  /2 )

15. LMS Durham Symposium Dynamical Systems and Statistical Mechanics 3 – 13 July 2006 New stabilising transformations Using a fixed point x * as seed, construct matrix C = − Q T which will stabilise all the fixed points in the neighbourhood of x * with similar invariant directions and signs of unstable eigenvalues. To stabilise fixed points with other signs, construct stabilising transformations as follows: where For each seed there are a total of 2 k transformations, where k is the number of unstable eigenvalues

16. LMS Durham Symposium Dynamical Systems and Statistical Mechanics 3 – 13 July 2006 Periodic orbits of kicked double rotor map pnpnp NpNp p –1 lnN p 112 2.4849 245102 2.3125 3152468 2.0495 45222 190 1.9229 52 20011 012 1.8613 69 82459 502 1.8323 746 900328 312 1.8145 8*229 0821 834 566 1.8028 Period p + 1 periodic orbit points were used as seeds to complete detection of period p * − incomplete set of orbits Dimension: n = 4

17. LMS Durham Symposium Dynamical Systems and Statistical Mechanics 3 – 13 July 2006 Periodic orbits of three coupled Hénon maps Dimension: n = 6 p BWBW-rSTMAX 18888 228 30000 434 40 50000 674 7274 728 8271 285286 9–636466 10–565563568 11–272277278 12–19721999 13*––1079– 14*––6599– 15*––5899– BW – Biham-Wenzel (Politi Torcini, 1992) BW-r – Biham-Wenzel (reduced) ST – stabilising transformations MAX – max. no. of detected orbits

18. LMS Durham Symposium Dynamical Systems and Statistical Mechanics 3 – 13 July 2006 Otherwise, we can work with the flow as well Given, the associated flow is defined as The dimension of the associated flow is n + 1 Locating periodic orbits in flows If a “good” Poincaré surface of section can be found, the problem is reduced to that for the maps

19. LMS Durham Symposium Dynamical Systems and Statistical Mechanics 3 – 13 July 2006 Stabilising within unstable subspace If we don’t have any periodic orbits to use as seeds, is it possible to construct stabilising transformations? C = I is always a good starting point, since it stabilises all periodic orbits with eigenvalues whose real parts are smaller than one. In chaotic flows with few unstable directions there is a fair fraction of such orbits. It is also possible to construct transformations within the unstable subspace only. If we can find the set of d vectors spanning the unstable subspace at the seed x 0, we can construct where C d is a matrix ( ) from the SD set

20. LMS Durham Symposium Dynamical Systems and Statistical Mechanics 3 – 13 July 2006 Stabilising within unstable subspace Traditionally, the Schur decomposition of is used to find U (e.g. for continuation problems, Lust Roose Spence Champneys 1995 ). It appears better to use SVD:

21. LMS Durham Symposium Dynamical Systems and Statistical Mechanics 3 – 13 July 2006 Locating periodic orbits of Kuramoto-Sivashinsky equation or in Fourier space: Restricting to odd solutions:

22. LMS Durham Symposium Dynamical Systems and Statistical Mechanics 3 – 13 July 2006 C d = I 3 C d = − I 3 L = 51.3; n = 31; d = 3 Locating periodic orbits of Kuramoto-Sivashinsky equation

23. LMS Durham Symposium Dynamical Systems and Statistical Mechanics 3 – 13 July 2006 Relative periodic orbits Associated flow is n + 2 dimensional

24. LMS Durham Symposium Dynamical Systems and Statistical Mechanics 3 – 13 July 2006 Locating relative periodic orbits of KS equation L = 22.0; n = 30; C = I

25. LMS Durham Symposium Dynamical Systems and Statistical Mechanics 3 – 13 July 2006 Conclusions Within the stabilising transformations approach it is possible to construct a scheme to systematically locate complete sets of periodic orbits in maps Extension to high-dimensional flows with small dimension of unstable manifold allowed us to find periodic and relative periodic orbits in the Kuramoto-Sivashinsky equation Construction of the systematic detection scheme for such flows is possible Thank you!