The CAPD library has been used in several articles in which chaotic dynamics, bifurcations, heteroclinic/homoclinic solutions and periodic orbits were studied.

Dynamics of the universal area-preserving map associated with period doubling

It is known that the famous Feigenbaum-Coullet-Tresser period doubling universality has a counterpart for area-preserving maps of R2. A renormalization approach has been used in (Eckmann et al, Mem. Amer. Math. Soc 289, 1982) in a computer-assisted proof of existence of a "universal" area-preserving map F* - a map with orbits of all binary periods 2k, k ∊ N. We consider maps in some neighbourhood of F* and study their dynamics.

We first demonstrate that the map F* admits a "bi-infinite heteroclinic tangle": a sequence of periodic points {zk}, k ∊ Z, limk→∞ |zk| = 0,   limk→-∞ |zk| = ∞, whose stable and unstable manifolds intersect transversally; and, for any N ∊ N, a compact invariant set on which F* is homeomorphic to a topological Markov chain on the space of all two-sided sequences composed of N symbols.

A corollary of these results is the existence of unbounded and oscillating orbits.

We also show that the third iterate for all maps close to F* admits a horseshoe. We use distortion tools to provide rigorous bounds on the Hausdorff dimension of the associated locally maximal invariant hyperbolic set: 0.7673 ≥ dimH(CF) ≥ ε * e-7499, where ε≈0.00013.

We consider infinitely renormalizable maps - maps on the renormalization stable manifold in some neighborhood of F*. For all such infinitely renormalizable maps in a neighborhood of the fixed point F* we prove the existence of a "stable" invariant Cantor set CF such that the Lyapunov exponents of F ∣CF are zero, and whose Hausdorff dimension satisfies dimH(CF) < 0.5324. We also show that there exists a submanifold, Wω, of finite codimension in the renormalization local stable manifold, such that for all F ∊ Wω the set CF is "weakly rigid": the dynamics of any two maps in this submanifold, restricted to the stable set CF, is conjugated by a bi-Lipschitz transformation that preserves the Hausdorff dimension.

The heteroclinic connections of the universal area-preserving map are proved using covering relations and cone conditions, provided by the CAPD library. The hyperbolic sets are constructed using covering relations computed using the CAPD library together with cone fields. To compute the distortion estimates, tight enclosures of the Markov partition of a dynamically defined Cantor set are needed. These enclosures are estimated from enclosures of invariant manifolds that are also computed using covering relations and cone conditions as provided by the CAPD library. The stable set is constructed as a Hausdorff limit of the hyperbolic sets for the third iterate.

  • D. Gaidashev, T. Johnson, Dynamics of the Universal Area-Preserving Map Associated with Period Doubling: Hyperbolic Sets,Nonlinearity 22(10) (2009), 2487-2521.
  • D. Gaidashev, T. Johnson, Dynamics of the Universal Area-Preserving Map Associated with Period Doubling: Stable Sets, J. Mod. Dyn. 3(4) (2009), 1-39.