The CAPD library has been used in several articles in which chaotic dynamics, bifurcations, heteroclinic/homoclinic solutions and periodic orbits were studied.
 The existence of simple choreographies for the nbody problem
 Homoclinic and heteroclinic solutions
 Chaotic dynamics for various ODE's
 Invariant curves through the KAM theory
 Cocoon bifurcations
 Rigorous verification of period doubling bifurcations for ODE's
 Rigorous numerics for homoclinic tangencies
 Uniformly hyperbolic attractors for Poincare maps
 Rigorous numerics for dissipative PDE's
 Normally Hyperbolic Invariant Manifolds
 Dynamics of the universal areapreserving map associated with period doubling (written by Tomas Johnson)
Dynamics of the universal areapreserving map associated with period doubling
It is known that the famous FeigenbaumCoulletTresser period doubling universality has a counterpart for areapreserving maps of R^{2}. A renormalization approach has been used in (Eckmann et al, Mem. Amer. Math. Soc 289, 1982) in a computerassisted proof of existence of a "universal" areapreserving map F^{*}  a map with orbits of all binary periods 2^{k}, k ∊ N. We consider maps in some neighbourhood of F^{*} and study their dynamics. We first demonstrate that the map F^{*} admits a "biinfinite heteroclinic tangle": a sequence of periodic points {z_{k}}, k ∊ Z, lim_{k→∞} z_{k} = 0, lim_{k→∞} z_{k} = ∞, 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 twosided 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 ≥ dim_{H}(C_{F}) ≥ 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 C^{∞}_{F} such that the Lyapunov exponents of F ∣_{C∞F} are zero, and whose Hausdorff dimension satisfies dim_{H}(C^{∞}_{F}) < 0.5324. We also show that there exists a submanifold, W_{ω}, of finite codimension in the renormalization local stable manifold, such that for all F ∊ W_{ω} the set C^{∞}_{F} is "weakly rigid": the dynamics of any two maps in this submanifold, restricted to the stable set C^{∞}_{F}, is conjugated by a biLipschitz transformation that preserves the Hausdorff dimension. The heteroclinic connections of the universal areapreserving 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. References:
