Endless Process of Bifurcations in Delay Differential Equations

Abstract

The paper is devoted to studying the singularly perturbed equations and systems with one or multiple delays. In focus is a bifurcation effect that, as the small parameter present in the system tends to zero, an endless process of forward and backward bifurcations repetition occur. To describe this effect and give its analytical background, we use the analog of the normal form, the quasinormal form, a nonlinear partial differential equation that regularly depends on a small parameter. The solutions of the quasinormal form give the main part of the asymptotic approximation of the solution of the differential equation with delay. Also effects of endless process of bifurcations is demonstrated in detail in two examples. In the first example, we consider second order equation with large delay, and in the second example — first order equation with two delays.