Homoclinic tangles, phase locking, and chaos in a two-frequency perturbation of Duffing's equation
type: Item
instance of: scholarly article
title: Homoclinic tangles, phase locking, and chaos in a two-frequency perturbation of Duffing's equation
zbMATH Open document ID: 0939.34039
published in: Journal of Nonlinear Science
publication date: 2000-06-28T00:00:00Z
review text: The perturbation of the Duffing equation (with a saddle at the origin) with two-frequency forcing with small dissipation is studied. The phase equations describing two-frequency phase dependence do not depend on the system variables, and their dynamics is chosen to be very simple: the flow on the torus is decoupled into a rigid rotation in one angular variable and a one-dimensional flow on the circle that can be either periodic or has two equilibria, stable and unstable, that appear via tangent bifurcation. Without perturbation the two-dimensional system has a saddle with two homoclinic orbits to it, after switching the perturbation instead of equilibrium one has a saddle invariant 2-torus whose 3-dimensional stable and unstable manifolds can intersect forming homoclinic tangle. This intersection is detected by Melnikov's method. In dependence on the type of the flow on the phase torus different conclusions about the behavior of orbits are derived.
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 37J40
Mathematics Subject Classification ID: 70K44
Mathematics Subject Classification ID: 34D10
Mathematics Subject Classification ID: 37G15
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 37L05
zbMATH DE Number: 1247986
zbMATH Keywords: chaos
zbMATH Keywords: homoclinic
zbMATH Keywords: phase locking
zbMATH Keywords: multi-frequency perturbation
zbMATH Keywords: Melnikov method
zbMATH Keywords: Bernoulli shift
author: Kazuyuki Yagasaki
reviewed by: L. M. Lerman
MaRDI profile type: Publication
full work available at URL: s003329900066
OpenAlex ID: W2053133372
DOI: 10.1007/S003329900066
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description: scientific article; zbMATH DE number 1247986