Computing periodic orbits using the anti-integrable limit

Abstract Chaotic dynamics can be effectively studied by continuation from an anti-integrable limit. Using the Henon map as an example, we obtain a simple analytical bound on the domain of existence of the horseshoe that is equivalent to the well-known bound of Devaney and Nitecki. We also reformulate the popular method for finding periodic orbits introduced by Biham and Wenzel. Near an anti-integrable limit, we show that this method is guaranteed to converge. This formulation puts the choice of symbolic dynamics, required for the algorithm, on a firm foundation.

[1]  Y. Lai,et al.  Characterization of blowout bifurcation by unstable periodic orbits , 1997 .

[2]  James D. Meiss,et al.  Cantori for symplectic maps near the anti-integrable limit , 1992 .

[3]  Ott,et al.  Optimal periodic orbits of chaotic systems occur at low period. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[4]  M. Hirsch,et al.  Differential Equations, Dynamical Systems, and Linear Algebra , 1974 .

[5]  Antonio Politi,et al.  On the topology of the Hénon map , 1990 .

[6]  Wenzel,et al.  Characterization of unstable periodic orbits in chaotic attractors and repellers. , 1989, Physical review letters.

[7]  Biham,et al.  Unstable periodic orbits in the stadium billiard. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[8]  Michael J. Davis,et al.  Statistical rate theory for transient chemical species: classical lifetimes from periodic orbits , 1990 .

[9]  Unstable periodic orbits and characterization of the spatial chaos in a nonlinear monatomic chain at the T=0 first-order phase-transition point. , 1995, Physical review. B, Condensed matter.

[10]  Wenzel,et al.  Unstable periodic orbits and the symbolic dynamics of the complex Hénon map. , 1990, Physical review. A, Atomic, molecular, and optical physics.

[11]  Type-I intermittency for the Hénon-map family. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[12]  Wenzel,et al.  Periodic orbits in the dissipative standard map. , 1991, Physical review. A, Atomic, molecular, and optical physics.

[13]  Robert L. Devaney,et al.  Shift automorphisms in the Hénon mapping , 1979, Hamiltonian Dynamical Systems.

[14]  Remarks on the symbolic dynamics for the Hénon map , 1992 .

[15]  Politi,et al.  Towards a statistical mechanics of spatiotemporal chaos. , 1992, Physical review letters.

[16]  H. Fang,et al.  Dynamics for a two-dimensional antisymmetric map , 1994 .

[17]  P. Grassberger,et al.  On the symbolic dynamics of the Henon map , 1989 .

[18]  S. Aubry.  Anti-integrability in dynamic and variational problems , 1995 .

[19]  Robert S. MacKay,et al.  Markov shifts in the He´non family , 1991 .

[20]  S. Aubry,et al.  Chaotic trajectories in the standard map. The concept of anti-integrability , 1990 .