On the sighting of unicorns: a variational approach to computing invariant sets in dynamical systems

TL;DR

This paper introduces a variational method for approximating invariant sets in dynamical systems by minimizing the distance between finite point sets and their images, effectively capturing complex invariant structures.

Contribution

It presents a novel variational approach for computing invariant sets of arbitrary topology and stability, including extensions with a Lennard-Jones potential for improved distribution.

Findings

Successfully converges to complex invariant sets in experiments

Handles invariant sets of arbitrary topology and stability

Enhanced distribution of points with Lennard-Jones potential

Abstract

We propose to compute approximations to general invariant sets in dynamical systems by minimizing the distance between an appropriately selected finite set of points and its image under the dynamics. We demonstrate, through computational experiments that this approach can successfully converge to approximations of (maximal) invariant sets of arbitrary topology, dimension and stability as, e.g., saddle type invariant sets with complicated dynamics. We further propose to extend this approach by adding a Lennard-Jones type potential term to the objective function which yields more evenly distributed approximating finite point sets and perform corresponding numerical experiments.