Features of Bifurcations of Periodic Solutions of the Ikeda Equation

Abstract

We study equilibrium states and bifurcations of periodic solutions from the equilibrium state of the Ikeda delay-differential equation well known in nonlinear optics. This equation was proposed as a mathematical model of a passive optical resonator in a nonlinear environment. The equation, written in a characteristic time scale, contains a small parameter at the derivative, which makes it singular. It is shown that the behavior of solutions of the equation with initial conditions from the fixed neighborhood of the equilibrium state in the phase space of the equation is described by a countable system of nonlinear ordinary differential equations. This system of equations has a minimal structure and is called the normal form of the equation in the neighborhood of the equilibrium state. The system of equations allows us to pick out one "fast" and a countable number of "slow" variables and apply the averaging method to the system obtained. It is shown that the equilibrium states of a system of averaged equations with "slow" variables correspond to periodic solutions of the same type of stability. The possibility of simultaneous bifurcation of a large number of stable periodic solutions(multistability bifurcation) is shown. With further increase in the bifurcation parameter each of the periodic solutions becomes a chaotic attractor through a series of period-doubling bifurcations. Thus, the behavior of the solutions of the Ikeda equation is characterized by chaotic multistability.

Analysis of Special Cases in the Study of Bifurcations of Periodic Solutions of the Ikeda Equation

“…This work complements Ref. [2] and allows one to create a fairly complete picture of possible bifurcations of periodic solutions from equilibrium states of Eq. (1.1) and of the development of these solutions when changing the parameters of the equation to chaotic attractors, and to understand the structure of the phase space of the equation.…”

“…The first is the case of the loss of stability of the zero equilibrium state when c = 0 at the point μ = 1, the second is the case of formation of a pair of equilibrium equations -a simultaneous formation of stable and unstable equilibrium states "out of thin air". These two cases share a single mechanism of loss of stability, different from that considered in [2]. In both cases, the possibility of bifurcation of a large number of periodic solutions and their development into chaotic multistability will be shown.…”

“…(1.1) and their development depending on changes in the parameters of the equation. The authors of [2] investigated the equilibrium states of Eq. (1.1), their dynamics and stability depending on the parameters of the equation, as well as the nature of the loss of stability.…”

“…There is also a brief overview of the work of other authors. In [2], two special cases of loss of stability of equilibrium states, which are important for applications, have not been considered. The first is the case of the loss of stability of the zero equilibrium state when c = 0 at the point μ = 1, the second is the case of formation of a pair of equilibrium equations -a simultaneous formation of stable and unstable equilibrium states "out of thin air".…”

Analysis of the Conditions for the Emergence of Spatially Inhomogeneous Structures of Light Waves in Optical Information Transmission Systems

“…Для построения центрального многообразия и системы дифференциальных уравнений на нем воспользуемся подходом работы [13]…”