Dynamical modes and nonlinear phenomena in modified autooscillatory system with frequency-phase control

In the proposed paper, we investigate the dynamical behavior of the modified system with frequency-phase control, which uses two-channel discriminator in the circuit of phase control and multi-frequency discriminator with periodic nonlinearity in the circuit of frequency control. We consider the case of identical low-pass filters of the third order in the both control circuits. Mathematical model of analyzed frequency-phase system is presented by a nonlinear dynamical system in the four-dimensional cylindrical phase space. The model is characterized by a great number of equilibrium states. The aim of this work is to reveal new dynamical modes and nonlinear phenomena due to the specified modification.
The study of the system under consideration dynamics consists in analysis of the phase synchronous mode, which is the main operating state in traditional applications of the systems with phase control, and non-synchronous modes of the frequency-phase system that serve as the basic working process in a new, non-traditional applications associated with the generation of chaotic oscillations. For solving these problems, we use computer modeling that based on qualitative and numerical methods of nonlinear dynamics. The conditions of the synchronous mode realization are determined. The existences of a great number various periodic and chaotic non-synchronous modes are established. Numerical analysis showed the existence in the system of qualitatively new quasi-synchronous and asynchronous modes, which are interesting for applied problems of generation of oscillations with chaotic modulation of their frequency and phase.
In the process of numerical simulation, we have analyzed the bifurcations leading to the emergence and disappearance of non-synchronous modes. High level of the non-synchronous modes multistability of frequency-phase system is discovered. The peculiarities of the dynamics of the system caused by the parameters that characterize the inertia of the control circuits and the degree of influence of the circuit frequency control are studied. The results are presented in the form of one- and two-parameter bifurcation diagrams, phase portraits, Poincare sections and waveforms of oscillations.
The revealed new non-synchronous modes of the modified system with frequency-phase control are of interest in the solution of applied problems of constructing generators of chaotic signals on the base of the systems with phase control.