Melnikov Analysis of Deterministic and Stochastic Manifold Splitting in the Kuramoto--Sivashinsky Equation
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866918446906736640 |
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| author | Datta, Sumita |
| author_facet | Datta, Sumita |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_13099 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Melnikov Analysis of Deterministic and Stochastic Manifold Splitting in the Kuramoto--Sivashinsky Equation Datta, Sumita Dynamical Systems Other Condensed Matter 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. |
| title | Melnikov Analysis of Deterministic and Stochastic Manifold Splitting in the Kuramoto--Sivashinsky Equation |
| topic | Dynamical Systems Other Condensed Matter |
| url | https://arxiv.org/abs/2604.13099 |