# American Institute of Mathematical Sciences

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

## A two-parameter geometrical criteria for delay differential equations

 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.
Citation: Suqi Ma, Zhaosheng Feng, Qishao Lu. A two-parameter geometrical criteria for delay differential equations. Discrete and Continuous Dynamical Systems - B, 2008, 9 (2) : 397-413. doi: 10.3934/dcdsb.2008.9.397
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