Features of Bifurcations of Periodic Solutions of the Ikeda Equation
2018, Vol. 14, no. 3, pp. 301-324
Author(s): Kubyshkin E. P., Moriakova A. R.
This work is licensed under a Creative Commons Attribution-NoDerivs 3.0 Unported License
Author(s): Kubyshkin E. P., Moriakova A. R.
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.
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