Revealing the intrinsic geometry of finite dimensional invariant sets of infinite dimensional dynamical systems

TL;DR

This paper introduces a method combining embedding techniques with diffusion maps to uncover the intrinsic geometry of finite dimensional invariant sets in infinite dimensional dynamical systems, demonstrated on specific equations.

Contribution

It presents a novel approach that integrates diffusion maps with existing embedding techniques to reveal the geometry of invariant sets.

Findings

Successfully applied to the unstable manifold of Kuramoto--Sivashinsky equation

Effectively characterized the attractor of Mackey-Glass delay differential equation

Demonstrated the method's ability to uncover intrinsic geometric structures

Abstract

Embedding techniques allow the approximations of finite dimensional attractors and manifolds of infinite dimensional dynamical systems via subdivision and continuation methods. These approximations give a topological one-to-one image of the original set. In order to additionally reveal their geometry we use diffusion mapst o find intrinsic coordinates. We illustrate our results on the unstable manifold of the one-dimensional Kuramoto--Sivashinsky equation, as well as for the attractor of the Mackey-Glass delay differential equation.