Effects of Discrete Time Delays and Parameters Variation on Dynamical Systems

Abstract

Delay Differential Equations (DDE's) have received considerable attention in recent years. While most of these articles focused on the effects of the time delays on the stability of the equilibrium points and on the bifurcation that they may raised, very few papers address the key roles that system parameters play on if and how the discrete delays induce stability changes of the equilibria and produce bifurcations near such equilibria. In this article we focus on that question in a general setting, that is, if you have a system of DDE's with one or multiple discrete time delays, what are the results of changing the system parameters values on the effects of the discrete time delays on the dynamic of the system. We present general results for one equation with one and two delays and study a specific example of one equation with one delay. We then establish the procedure for n equations with multiple delays and do a specific example for two equations with two delays. We compute the steady states and analyze their stability as both chosen bifurcation parameters, the discrete time delay ...

Keywords

• Delay differential equations
• bifurcation
• predator-prey