Journal
JOURNAL OF DIFFERENTIAL EQUATIONS
Volume 210, Issue 1, Pages 65-86Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jde.2004.10.016
Keywords
singular perturbation; delayed loss of stability; Hopf bifurcation; degenerate linear part
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We consider the so-called delayed loss of stability phenomenon for singularly perturbed systems of differential equations in case that the associated autonomous system with a scalar parameter undergoes the Hopf bifurcation at the zero equilibrium point. It is assumed that the linearization of the associated system is independent of the parameter and the next terms in the expansion of the right-hand parts at zero are positive homogeneous of order alpha > 1. Simple formulas are presented to estimate the asymptotic delay for the delayed loss of stability phenomenon. More precisely, we suggest sufficient conditions which ensure that zeros of a simple function psi defined by the positive homogeneous nonlinear terms are the Hopf bifurcation points of the associated system, the sign of psi at other points determines stability of the zero equilibrium, and the asymptotic delay equals the distance between the bifurcation point and a zero of some primitive of psi. (C) 2004 Elsevier Inc. All rights reserved.
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