Analysis of Bifurcations of Periodic Solutions of Ikeda Equation

Abstract

We study the bifurcation of periodic solutions from the equilibrium state of the well-known in nonlinear optics Ikeda equation. The equation, written in a characteristic time scale, contains a small parameter at the derivative, which makes it singular. We apply uniform normalization method, which allows to reduce the study of the behavior of solutions in the neighborhood of the equilibrium state to the analysis of the countable system of ordinary differential equations. This system contains "fast" and "slow" variables. It is shown that the equilibrium states of "slow" variables equation determine periodic solutions. Analysis of equilibrium states allows us to study the bifurcation of the periodic solutions depending on the parameters and their stability. The possibility of simultaneous bifurcation of a large number of stable periodic solutions is shown.