Sjoerd M. Verduyn Lunel

The prototype equation for delayed negative feedback: periodic solutions

In this chapter we study the nonlinear equation n n

Semiflows for nonlinear systems

dot x(t) = f(x(t - alpha ))

Behaviour near a hyperbolic equilibrium

n n(1.1) n nwhere f: ℝ→ℝ is a continuous function. The positive parameter α is called the time lag. Equation (1.1) is the prototype for nonlinear delayed feedback. We shall see that the lag α can cause much more complex behaviour of solutions than in the ODE case α=0 where only monotone solutions converging to equilibria or infinity are possible. In particular, we shall prove the existence of periodic solutions (Section 5).

Time-dependent linear systems

When a linear delay system is subject to external “forcing”, it can be described by the inhomogeneous equation n n

Completeness or small solutions

dot x(t) = {langlezeta,{x_t}rangle_n} + f(t)

Linear RFDE as bounded perturbations

n n(1.1) n nwith f: ℝ → ℂ n a given (continuous) function describing the influence of the outside world. As in the foregoing chapters we shall rewrite (1.1) in the abstract form n n