Embedding of global attractors and their dynamics

TL;DR

This paper demonstrates that the complex dynamics of global attractors in dissipative PDEs can be embedded into finite-dimensional ODEs using shape theory and cellularity, preserving key dynamical features.

Contribution

It introduces a method to embed global attractors of dissipative PDEs into finite-dimensional ODEs with controlled approximation, leveraging shape theory and Assouad dimension.

Findings

Existence of finite-dimensional ODEs reproducing PDE attractor dynamics

Approximation of attractors by homeomorphic images in Euclidean space

Preservation of dynamical properties in the embedding

Abstract

Using shape theory and the concept of cellularity, we show that if $A$ is the global attractor associated with a dissipative partial differential equation in a real Hilbert space $H$ and the set $A-A$ has finite Assouad dimension $d$, then there is an ordinary differential equation in ${\mathbb R}^{m+1}$, with $m >d$, that has unique solutions and reproduces the dynamics on $A$. Moreover, the dynamical system generated by this new ordinary differential equation has a global attractor $X$ arbitrarily close to $LA$, where $L$ is a homeomorphism from $A$ into ${\mathbb R}^{m+1}$.