On the global dynamics of nonlinear autonomous differential delay equations

Abstract

In this section we use the prototype equation

$$\mathop x\limits^. (t) = f(x(t - 1)),$$

((1.1))

with a smooth function f: ℝ→ℝ, in order to illustrate basic results on the long-term behaviour of solutions and on the organization of the phase space. We assume that f satisfies the condition

$$xf{\text{(}}x{\text{)}} \ne {\text{0,}}\,\,{\text{for }}x \ne 0$$

((NF))

for negative feedback and that/is bounded from above or from below.