The mechanism of the hard appearance of a two-frequency oscillation mode in the case of Andronov-Hopf reverse bifurcation
type: Item
instance of: scholarly article
title: The mechanism of the hard appearance of a two-frequency oscillation mode in the case of Andronov-Hopf reverse bifurcation
zbMATH Open document ID: 0731.58050
DOI: 10.1016/0021-8928(89)90128-7
author: L. P. Vozovoj
author: Valentin Afraimovich
published in: Journal of Applied Mathematics and Mechanics
publication date: 1989-01-01T00:00:00Z
review text: This paper deals with the sudden birth of a two-frequency periodic solution in an m-dimensional autonomous ordinary differential equation, \(m>2\), when an equation parameter crosses through a critical value. This parameter value corresponds to a bifurcation leading to the disappearing of a saddle limit cycle, when it merges into a node equilibrium point (Andronov-Hopf reverse bifurcation). After the bifurcation, locally it remains only the equilibrium point which has turned into a saddle. When \(m=3\), the unstable invariant manifold \(W^ u\) of this saddle point is a two-dimensional one with unstable focus type trajectories. The stable invariant manifold \(W^ S\) has the dimension one. In this situation a phase trajectory coming from a neighborhood of \(W^ u\) can be injected in a neighborhood of \(W^ S\), this giving rise to a stable closed trajectory in the three-dimensional phase space. The projection of this trajectory on a plane is a curve with a certain number of loops related to the rank of a higher harmonic resonance. When the phase trajectory is injected in \(W^ S\), and so returns to the equilibrium point, a limit case, corresponding to infinitely many loops, is obtained, and one has a homoclinic situation. Sufficient conditions of existence of such periodic solutions are given for \(m=3\), and \(m=4\).
Mathematics Subject Classification ID: 37G99
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 37C70
zbMATH DE Number: 4209158
zbMATH Keywords: nonlinear oscillations
zbMATH Keywords: dynamical systems
zbMATH Keywords: flows
zbMATH Keywords: periodic solution
zbMATH Keywords: bifurcation
reviewed by: Christian Mira
MaRDI profile type: Publication
full work available at URL: 0021-8928(89)90128-7
OpenAlex ID: W2046240510
cites work: A CONTRIBUTION TO THE PROBLEM OF THE STRUCTURE OF AN EXTENDED NEIGHBORHOOD OF A ROUGH EQUILIBRIUM STATE OF SADDLE-FOCUS TYPE
cites work: Q4061289
description: scientific article; zbMATH DE number 4209158