Effects of time delay on stability for simple linear DDEs

We illustrate the influence of time delay on the stability of steady state for some simple linear delay differential equations. We prove that for one DDE with one discrete delay the only stability switch that can occur while delay is increasing is from the stable case to the unstable one. We also justify that in the case of several discrete delays if the steady state is unstable in the case without delays then it remains unstable for positive delays and we give an example showing that for two discrete delays while enlarging them, we can have the following stability switch: stable - unstable - stable - unstable. We also present an example of a system of two linear DDEs with one discrete delay tau, such that the steady state is unstable for tau = 0 and is stable for some tau>0.