A two-parameter geometrical criteria  for delay differential equations

Suqi Ma; Zhaosheng Feng; Qishao Lu

doi:10.3934/dcdsb.2008.9.397

Article Contents

2008, Volume 9, Issue 2: 397-413. Doi: 10.3934/dcdsb.2008.9.397

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A two-parameter geometrical criteria for delay differential equations

1.
Department of Mathematics, China Agricultural University, Beijing 100083
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: July 31, 2007

Revised: October 31, 2007

Published: February 2008

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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.

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Mathematics Subject Classification: Primary: 34C05, 34C14, 34C20; Secondary: 35B40.

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