Evgeny Kubyshkin

This paper deals with bifurcations from the equilibrium states of periodic solutions of the Ikeda equation, which is well known in nonlinear optics as an equation with a delayed argument, in two special cases that have not been considered previously. Written in a characteristic time scale, the equation contains a small parameter with a derivative, which makes it singular. Both cases share a single mechanism of the loss of stability of equilibrium states under changes of the parameters of the equation associated with the passage of a countable number of roots of the characteristic equation through the imaginary axis of the complex plane, which are in this case in certain resonant relations. It is shown that the behavior of solutions of the equation with initial conditions from fixed neighborhoods of the studied equilibrium states in the phase space of the equation is described by countable systems of nonlinear ordinary differential equations that have a minimal structure and are called the normal form of the equation in the vicinity of the studied equilibrium state. An algorithm for constructing such systems of equations is developed. These systems of equations allow us to single out one “fast” variable and a countable number of “slow” variables, which makes it possible to apply the averaging method to the systems of equations obtained. Equilibrium states of the averaged system of equations of “slow” variables in the original equation correspond to periodic solutions of the same nature of sustainability. In the special cases under consideration, the possibility of simultaneous bifurcation from equilibrium states of a large number of stable periodic solutions (multistability bifurcation) and evolution of these periodic solutions to chaotic attractors with changing bifurcation parameters is shown. One of the special cases is associated with the formation of paired equilibrium states (a stable and an unstable one). An analysis of bifurcations in this case provides an explanation of the formation of the “boiling points of trajectories”, when a periodic solution arises “out of nothing” at some point in the phase space under changes of the parameters of the equation and quickly becomes chaotic.

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.