Singularly Perturbed Differential-Difference Equations

Abstract

A delay equation

$$ v\dot x\left( x \right) = - x\left( t \right) + f\left( {x\left( t \right)} \right),x\left( t \right):{R^ + } \to R, v >0, $$

(3.1)

and some its generalizations have recently become a matter of interest in the theory of differential equations. These equations are mathematical models of a number of real phenomena (Heiden and Mackey (1982)). The outward simplicity of (3.1) conceals a complicated dynamics, which has not been clarified completely till now. However, the results concerning local stability (instability) of a stationary solution are well-known (Pesin (1974); Chow (1974); Kaplan and Yorke (1977)). A series of papers has been devoted to the equation (3.1) with f being a step (or close to it) function (Aliev (1984); Aliev et al. (1984); Heiden and Mackey (1982); Heiden (1983); Heiden and Walther (1983, 1984); Peters (1983)). As was shown, the behavior of solutions can be rather complicated and, in particular, chaotic.