Dynamics of the generator with three circuits in the feedback loop. Multistability formation and transition to chaos

Background and Objectives: Studying the dynamical mechanisms of the emergence of nonlinear phenomena that are characteristic for multimode self-oscillating systems consisting of interacting oscillators and an ensemble of passive oscillators or representing active nonlinear systems with complex feedback channels is an important urgent task. The simplest example of a self-oscillating system with a complex feedback is the well-known classical van der Pol oscillator with an additional linear oscillatory circuit included in the feedback channel. We investigate the behavior of the multimode system increasing the number of oscillatory circuits in the oscillator's feedback loop. The research in this paper can help to better understand the mechanisms of multistability formation in infinite-dimensional self-oscillating systems such as a generator with delayed feedback and a generator with distributed feedback. Materials and Methods: The system equations were derived for the electronic scheme of the self-oscillating system. To describe the existing dynamic modes by numerical simulation methods, the projections of the phase portraits and the Poincare sections were obtained. To study the mechanisms of formation of multistable states, the bifurcation analysis methods were used. Results: It was found that the mechanism underlying the multistability formation is based on a sequence of two supercritical Andronov - Hopf bifurcations and a subcritical Neymark - Saker bifurcation. Therefore, the multistability emerges as a result of gaining stability by the unstable limit set that existed before the multistability appears. Conclusion: The discovered mechanism of multistability formation opens up wide possibilities for managing the multistability, which are inaccessible for systems in which the multistability is realized through tangential bifurcations. In contrast to the tangential bifurcation, the subcritical Neymark - Sacker bifurcation assumes the existence of a limit cycle both before and after the bifurcation. Thus, it is possible to use a wide range of methods and tools to stabilize saddle limit cycles in order to control the boundaries of the multistability region in the space of control parameters of the system.