Melnikov Analysis of Deterministic and Stochastic Manifold Splitting in the Kuramoto--Sivashinsky Equation

TL;DR

This paper develops a Melnikov framework for the Kuramoto--Sivashinsky equation to analyze how deterministic and stochastic forces cause invariant manifold splitting, linking geometric theory to chaos in PDEs.

Contribution

It introduces a Melnikov functional for the KS equation, capturing manifold splitting under weak forcing, both deterministic and stochastic, in an infinite-dimensional setting.

Findings

Periodic forcing causes phase-dependent manifold intersections.

Stochastic forcing results in random manifold splitting with quantifiable variance.

The framework links invariant manifold theory to spatiotemporal chaos in PDEs.

Abstract

We develop a Melnikov framework for the Kuramoto Sivashinsky (KS) equation under weak deterministic and stochastic forcing. By treating KS as an infinite dimensional dynamical system, we derive a Melnikov functional that measures splitting of stable and unstable manifolds of a homoclinic orbit. Periodic forcing leads to phase dependent transverse intersections, while stochastic forcing produces random manifold splitting characterized by a variance determined by the adjoint solution. This provides a geometric mechanism linking invariant manifold theory to spatiotemporal chaos in dissipative partial differential equations.