Numerical Computation of the Stable and Unstable Manifolds of Invariant Tori

Details Numerical Computation of the Stable and Unstable Manifolds of Invariant Tori Numerical Computation of the Stable and Unstable Manifolds of Invariant Tori

2005-04-26

arxiv.org/abs/nlin/0504054v1

Chaos 16: 023129 (2006)

Nonlinear Sciences

Chaotic Dynamics

laTeX with 16 figures

Scientific paper

10.1063/1.2200159

We develop an iterative technique for computing the unstable and stable eigenfunctions of the invariant tori of diffeomorphisms. Using the approach of Jorba, the linearized equations are rewritten as a generalized eigenvalue problem. Casting the system in this light allows us to take advantage of the speed of eigenvalue solvers and create an efficient method for finding the first order approximations to the invariant manifolds of the torus. We present a numerical scheme based on the power method that can be used to determine the behavior normal to such tori, and give some examples of the application of the method. We confirm the qualitative conclusions of the Melnikov calculations of Lomel\'i and Meiss (2003) for a volume-preserving mapping.

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