Computing the Invariant Circle and the Foliation by Stable Manifolds for a 2-D Map by the Parameterization Method: Numerical Implementation and Results

TL;DR

This paper introduces a quadratic convergent algorithm based on the parameterization method for computing invariant circles and stable manifolds in 2D maps, with extensions to 3D models, revealing new dynamical phenomena.

Contribution

It provides a novel, efficient numerical algorithm for invariant circle computation applicable to various internal dynamics, including phase-locking, and extends the method to three dimensions.

Findings

Successfully computed invariant circles and stable manifolds in standard models.

Uncovered a bundle merging scenario where hyperbolicity is lost.

Extended the algorithm to 3D and demonstrated on the 3D-FAF map.

Abstract

We present and implement an algorithm for computing the invariant circle and the corresponding stable manifolds for 2-dimensional maps. The algorithm is based on the parameterization method, and it is backed up by an a-posteriori theorem established in [YdlL21]. The algorithm works irrespective of whether the internal dynamics in the invariant circle is a rotation or it is phase-locked. The algorithm converges quadratically and the number of operations and memory requirements for each step of the iteration is linear with respect to the size of the discretization. We also report on the result of running the implementation in some standard models to uncover new phenomena. In particular, we explored a bundle merging scenario in which the invariant circle loses hyperbolicity because the angle between the stable directions and the tangent becomes zero even if the rates of contraction are separated. We also discuss and implement a generalization of the algorithm to 3 dimensions, and implement it on the 3-dimensional Fattened Arnold Family (3D-FAF) map with non-resonant eigenvalues and present numerical results.