Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

A two-parameter geometrical criteria for delay differential equations

Pages: 397 - 413, Volume 9, Issue 2, March 2008      doi:10.3934/dcdsb.2008.9.397

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Suqi Ma - Department of Mathematics, China Agricultural University, Beijing 100083, China (email)
Zhaosheng Feng - Department of Mathematics, University of Texas-Pan American, Edinburg, TX 78541, United States (email)
Qishao Lu - School of Science, Beijing University of Aeronautics and Astronautics, Beijing 100083, China (email)

Abstract: In some cases of delay differential equations (DDEs), a delay-dependant coefficient is incorporated into models which takes the form of a function of delay quantity. This brings forth frequent stability-switch phenomena. A geometrical stability criterion is developed on the two-parameter plane for analyzing Hopf bifurcations of equilibria. It is shown that the increasing direction of parameter $\sigma$ would confirm bifurcation directions (from stable one to unstable one, or whereas) at the critical delay values. These lead to the definite partition of stable and unstable regions on the $(\sigma-\tau)$ plane. Several examples are given to illustrate how to use this method to detect both Hopf and double Hopf bifurcations.

Keywords:  delay differential equations, delay dependent parameters, geometrical criterion, stability, Hopf bifurcation, double Hopf bifurcation.
Mathematics Subject Classification:  Primary: 34C05, 34C14, 34C20; Secondary: 35B40.

Received: August 2007;      Revised: November 2007;      Available Online: December 2007.