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

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