The effect of oscillations in the dynamics of differential equations with delay

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:335

Article

Qesmi, R. ; Hbid, M. L.

关键词: differential equation;    delay;    integral manifold;    bifurcation;   

DOI  :  10.1016/j.jmaa.2007.01.078

来源: Elsevier

【 摘 要 】

The purpose of this paper is to study the dynamic behavior of delay differential equations of the form x(t) = f(x(t-1); a(epsilon sin (vt), epsilon cos(vt)); alpha), epsilon, v, alpha epsilon R, provided that a and f meet some hypotheses. By augmenting the above equation, the explicit time-dependent terms are replaced by state-dependent terms. The augmented system is autonomous and has a pair of purely imaginary and simple zero eigenvalues. Applying the center manifold reduction, the existence of an attractive integral manifold with periodic structure for the original equation is shown. Furthermore, we give a description of the flow on the obtained manifold. This allows us to determine the sufficient conditions for existence of saddle-node bifurcation. To illustrate our results, we consider an autonomous equation perturbed by a periodic function. (c) 2007 Elsevier Inc. All rights reserved.

【 授权许可】

Free