Validated computation of heteroclinic sets

Authors: Jason D. Mireles James and Maciej Capinski.

Abstract:In this work we develop a method for computing mathematically rigorous enclo- sures of some one dimensional manifolds of heteroclinic orbits for nonlinear maps. Our method exploits a rigorous curve following argument build on high order Taylor approximation of the local stable/unstable manifolds. The curve following argument is a uniform interval Newton method ap- plied on short line segments. The definition of the heteroclinic sets involve compositions of the map and we use a Lohner representation to overcome the accumulation of round off errors. Our argument requires precise control over the local unstable and stable manifolds so that we must first obtain validated a-posteriori error bounds on the truncation errors associated with the manifold approxi- mations. We illustrate the utility of our method by proving some computer assisted theorems about heteroclinic invariant sets for a volume preserving map.

  • "Validated computation of heteroclinic arcs"

  • Computer Assisted Proof Codes: (codes require the IntLab package)

  • Programs for validated computatoin of the local stable/unstable manifold parameterizations are found here.

  • Compressed Compressed folder containing all programs for validated computation of heteroclinic sets used in the paper.