Parametric Resonance in the Logistic Equation with Delay under a Two-Frequency Perturbation

Abstract

The logistic equation with delay feedback circuit and with periodic perturbation parameters is considered. Parameters of the problem (coe cient of linear growth and delay) are chosen close to the critical values at which cycle is bifurcated from equilibrium point. We assume that these values have double-frequency relation to the time and the frequency of action and doubled frequency of the natural vibration are close. Asymptotic analysis is performed under these assumptions and leads to a two-dimensional system of ordinary di˙erential equations. Linear part of this system is periodic. If the parameter that defines frequency detuning of an external action is large or small we can apply standard asymptotic methods to the resulting system. Otherwise numerical analysis is performed. Using results of numerical analysis, we clarify the main scenarios of phase transformations and find the region of chaotic oscillations. It is main conclusion that in the case of parametric resonance the dynamics of the problem with double-frequency perturbation is more complicated than dynamics of the problem with single-frequency perturbation.