On periodic perturbations of asymmetric Duffing-Van-der-Pol equation

TL;DR

This paper investigates the effects of periodic perturbations on an asymmetric Duffing-Van-der-Pol equation near a homoclinic figure-eight, analyzing solution behaviors, limit cycles, resonance zones, and separatrix dynamics with analytical and numerical methods.

Contribution

It provides a detailed analytical study of the perturbed asymmetric Duffing-Van-der-Pol equation near a homoclinic figure-eight, including limit cycle existence and resonance zone analysis.

Findings

Behavior of solutions outside the figure-eight neighborhood analyzed.

Limit cycles for the autonomous case are characterized.

Resonance zones for the nonautonomous case are identified.

Abstract

Time-periodic perturbations of an asymmetric Duffing-Van-der-Pol equation close to an integrable equation with a homoclinic "figure-eight" of a saddle are considered. The behavior of solutions outside the neighborhood of "figure-eight" is studied analytically. The problem of limit cycles for an autonomous equation is solved and resonance zones for a nonautonomous equation are analyzed. The behavior of the separatrices of a fixed saddle point of the Poincare map in the small neighborhood of the unperturbed "figure-eight" is ascertained. The results obtained are illustrated by numerical computations.