Exponentially small splitting: A direct approach

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2 Scopus citations


In this paper, we go beyond what was proposed in theory by Melnikov ([15]) to introduce a practical method to calculate the high order splitting distances of stable and unstable manifold in time-periodic equations. Not only we derive integral formula for splitting distances of all orders, but also we develop an analytic theory to evaluate the acquired multiple integrals. We reveal that the dominance of the exponentially small Poincaré/Melnikov function for equations of high frequency perturbation is caused by a certain symmetry embedded in the kernel functions of high order Melnikov integrals. This symmetry is beheld by many non-Hamiltonian equations.

Original languageEnglish (US)
Pages (from-to)954-1036
Number of pages83
JournalJournal of Differential Equations
Issue number1
StatePublished - Jun 15 2020

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics


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