Oscillations near separatrix for perturbed Duffing equation

TL;DR

This paper develops an asymptotic solution near the separatrix for a perturbed Duffing oscillator, revealing instability through a separatrix map applicable at any perturbation order.

Contribution

It introduces a general separatrix map for the perturbed Duffing oscillator, providing insights into the complex dynamics and instability near the separatrix.

Findings

Constructed an asymptotic solution close to the separatrix.

Derived a separatrix map valid for any perturbation order.

Showed the instability of motion near the separatrix in the perturbed system.

Abstract

A periodic perturbation generates a complicated dynamics close to separatrices and saddle points. We construct an asymptotic solution which is close to the separatrix for the unperturbed Duffing's oscillator over a long time. This solution is defined by a separatrix map. This map is obtained for any order of the perturbation parameter. Properties of this map show an instability of a motion for the perturbed system.