Structural Distinction in ODE and PDE Chaos:Lorenz vs Kuramoto--Sivashinsky Equation

TL;DR

This paper compares chaos in the Kuramoto-Sivashinsky equation and Lorenz systems, highlighting their structural differences and the limitations of low-dimensional reductions in capturing infinite-dimensional chaos.

Contribution

It demonstrates the fundamental differences in chaos between infinite-dimensional PDEs and finite-dimensional ODE systems through numerical analysis.

Findings

KS equation exhibits intrinsic spatio-temporal chaos.

Lorenz system shows low-dimensional temporal chaos.

Low-dimensional reductions may not capture infinite-dimensional chaos structures.

Abstract

We study the nature of chaos in finite and infinite dimensional systems through a comparison between the Kuramoto Sivashinsky (KS) equation, the Lorenz system, and a Lorenz type reduction of the KS equation proposed by Wilczak. Numerical simulations of the KS equation reveal intrinsic spatio temporal chaos, with disorder evolving simultaneously in space and time. In contrast, the Lorenz system and the Wilczak reduction exhibit low dimensional temporal chaos lacking spatial complexity. Lyapunov exponent analysis highlights the finite-dimensional convergence properties of the reduced systems and underscores the fundamentally different dynamical nature of chaos in the KS equation. In particular, we demonstrate that low-dimensional reductions may reproduce transient chaotic signatures but do not necessarily retain the structural properties of infinite-dimensional dissipative systems.