American Institute of Mathematical Sciences

Advanced
March  2008, 9(2): 397-413. doi: 10.3934/dcdsb.2008.9.397

A two-parameter geometrical criteria for delay differential equations

Suqi Ma 1, , Zhaosheng Feng 2, and  Qishao Lu 3,

1. 

Department of Mathematics, China Agricultural University, Beijing 100083, China
2. 

Department of Mathematics, University of Texas-Pan American, Edinburg, TX 78541
3. 

School of Science, Beijing University of Aeronautics and Astronautics, Beijing 100083

Received  August 2007 Revised  November 2007 Published  December 2007

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: stability, Hopf bifurcation, double Hopf bifurcation., delay differential equations, delay dependent parameters, geometrical criterion.
Mathematics Subject Classification: Primary: 34C05, 34C14, 34C20; Secondary: 35B4.
Citation: Suqi Ma, Zhaosheng Feng, Qishao Lu. A two-parameter geometrical criteria for delay differential equations. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 397-413. doi: 10.3934/dcdsb.2008.9.397
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