Abstract
Delay Differential Equations (DDE's) have received considerable attention in recent years. В 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, В very few papers address the key roles that В system parameters В play В on if and how В 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 В 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 ...
Highlights
It is well known, that the values of the parameters play a crucial role in the behavior of dynamical systems and that changes in the values can change the behavior significantly
Consider the one dimensional delay differential equation with the time delay τ, and the parameter μ as bifurcation parameters: Published papers have shown that the incorporation of discrete time delays can highly impact the dynamics of the system, since they can switch the stability of a steady state point, and can cause the system to go through a Hopf bifurcation near that steady state point (Culshaw[6], Gakkhar[7], Bellen[3])
We extend our analysis to a system of ndelay differential equations with multiple discrete time delays τ1, τ2, ..., τk, and a local bifurcation parameter μ
Summary
That the values of the parameters play a crucial role in the behavior of dynamical systems and that changes in the values can change the behavior significantly. It has been shown by many researchers (Perelson[1],Allen[2],Bellen[3]) that there is a need to incorporate discrete time delays in dynamical systems (biological systems, physical systems,...) as studied. Models that incorporate such delays are referred to as delay differential equations (DDE’s). While most of these research papers focus on issue of the stability changes caused by the delay(s), the main motivation of this paper is to study how a local bifurcation parameter of the system may affect the changes in stability caused by the delay (s)
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