Rigorous A Posteriori Computation of (Un)Stable Manifolds and Connecting Orbits for Analytic Maps.

Jason D. Mireles-James
Konstantin Mischaikow

Abstract:This work is concerned with high order polynomial approximation of stable and unstable manifolds for analytic discrete time dynamical systems. We develop a-posteriori theorems for these polynomial approximations which allow us to obtain rigorous bounds on the truncation errors via a computer assisted argument. Moreover we represent the truncation error as an an- alytic function, so that that the derivatives of the truncation error can be bound using classical estimates of complex analysis. As an application of these ideas we combine the approximate mani- folds and rigorous bounds with a standard Newton-Kantorovich argument in order to obtain a kind of analytic-shadowing result for connecting orbits between fixed points of discrete time dynamical systems. Examples of the manifold computation are given for invariant manifolds which have dimen- sion between two and ten. Examples of the a-posteriori error bounds and the analytic shadowing argument for connecting orbits are given for dynamical systems in dimension three and six.

  • "Rigorous A Posteriori Computation of (Un)Stable Manifolds and Connecting Orbits for Analytic Maps."
    SIAM Journal on Applied Dynamical Systems, Vol. 12, No. 2, pp. 957-1006 (2013).

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

  • Compressed Folder Containing all the MatLab/IntLab scripts used in the paper.