Continuation and bifurcation analysis of delay differential equations

Mathematical modeling with delay differential equations (DDEs) is widely used in various application areas of science and engineering (e.g., in semiconductor lasers with delayed feedback, high-speed machining, communication networks, and control systems) and in the life sciences (e.g., in population dynamics, epidemiology, immunology, and physiology). Delay equations have an infinite-dimensional state space because their solution is unique only when an initial function is specified on a time interval of length equal to the largest delay. Consequently, analytical calculations are more difficult than for ordinary differential equations andnumerical methods are generally the only way to achieve a complete analysis, prediction and control of systems with time delays. Delay differential equations are a special type of functional differential equation (FDE). In FDEs the time evolution of the state variable can depend on the past in an arbitrary way as long as the dependence is a bounded function of the past. However, DDEs impose a constraint on this dependence, namely that the evolution depends only on certain past values of the state at discrete times. (We do not consider here the case of distributed delay.) The delays can be constant or state dependent. The equations can also involve delayed values of the derivative of the state, which leads to equations of neutral type. In this chapter we mainly discuss the simplest case, namely a finite number of constant delays. Specifically, we consider a nonlinear system of DDEs with constant delays τj > 0, j = 1, . . . ,m, of the form x′(t) = f(x(t), x(t− τ1), x(t− τ2), . . . , x(t− τm), η), (1)