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arXiv · 2604.13099

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

Sumita Datta
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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.

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Sumita Datta. 2026-04-10. Melnikov Analysis of Deterministic and Stochastic Manifold Splitting in the Kuramoto--Sivashinsky Equation. https://arxiv.org/abs/2604.13099

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