Local dynamics of singular pertubed equations with two delays

Ilia Kashchenko (Yaroslavl State University)

Yaroslavl State University, Russia

Consider the equation with two delays $$\dot x + x = ax(t-T_1)+bx(t-T_2) + f(x. x(t-T_1), x(t-T_2)), \quad T_1 > T_2 > 0.$$ Here $f(x)$ is nonlinear function. Main assumption is that $T_1$ is asymptotically large. There are three main situations: (1) $T_1$ is large, $T_2$ is fixed; (2) $T_1$ and $T_2$ are both large and proportional; (3) $T_1$ and $T_2$ are both large and $T_1T_2^{-1}$ is large too. The problem to research is to determine the behavior of solutions in some small (but independed of $\varepsilon$) neighbourhood of zero equilibrium state. The method of investigations is so-called method of quasinormal forms. Depending on the parameters critical cases in the problem of the stability of the equilibrium state are identified. In all critical cases special evolutionary equations (quasinormal forms) are built. Their non-local dynamics determines the local behavior of solutions of the original equations.