Bifurcations in a Mathieu equation with cubic nonlinearities. II
type: Item
instance of: scholarly article
title: Bifurcations in a Mathieu equation with cubic nonlinearities. II
zbMATH Open document ID: 1043.34040
DOI: 10.1016/S1007-5704(02)00018-7
author: Leslie Ng
author: Richard Rand
published in: Communications in Nonlinear Science and Numerical Simulation
publication date: 2003-04-28T00:00:00Z
review text: This paper is a continuation of a previous publication dealing with solution properties of the nonautonomous Mathieu equation with a nonlinear damping of cubic type. The study, made in the neighborhood of the 2:1 resonance, is based on the analytical averaging method supposing that the forcing periodic term and the nonlinearity are sufficiently small. It was shown that a degenerate bifurcation point exists in the resulting slow flow and some of the bifurcations near this point were looked at. The introduction reminds these results with comments on a bifurcation diagram in a parameter plane, and the phase plane portraits related to the autonomous ODE (variable being amplitude and phase) obtained from the averaging method using a second-order approximation. In particular, the parameter plane represents a heteroclinic bifurcation curve, a Poincaré-Andronov bifurcation curve (here, wrongly called ``Hopf''), and a curve related to the creation of a limit cycle at infinity. With respect to the previous publication, the authors complete the former bifurcation diagram by additional bifurcations in the neighborhood of the degenerate bifurcation. Moreover, an analytic approximation to the heteroclinic bifurcation on the above parameter plane is given. This permits to refine the structure of the bifurcation plane with a new drawing of the heteroclinic and Poincaré-Andronov bifurcation curves, more the curves related to type 1 and type 2 saddle-node, and degenerated Poincaré-Andronov bifurcations. The last part is devoted to an approximation to saddle-node bifurcation of periodic orbits and degenerate Poincaré-Andronov bifurcation. This is made using a bifurcation and continuation software which generates numerically the corresponding diagrams. An analytic expression for the Poincaré-Andronov bifurcation is given and appears in agreement with the numerical results of the software.
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C15
Mathematics Subject Classification ID: 34C29
zbMATH DE Number: 1901810
zbMATH Keywords: nonautonomous ordinary differential equations
zbMATH Keywords: Mathieu equation
zbMATH Keywords: nonlinear dynamics
zbMATH Keywords: Bifurcations
zbMATH Keywords: analytical methods for nonlinear ordinary differential equations
zbMATH Keywords: averaging method
reviewed by: Christian Mira
describes a project that uses: AUTO
MaRDI profile type: Publication
cites work: Nonlinear Mathieu equation and coupled resonance mechanism
cites work: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields
cites work: THE NONLINEAR MATHIEU EQUATION
cites work: Bifurcations in a Mathieu equation with cubic nonlinearities
cites work: Q4715203
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cites work: Computation of normal forms via a perturbation technique
full work available at URL: s1007-5704(02)00018-7
OpenAlex ID: W2096312336
description: scientific article; zbMATH DE number 1901810