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In this work, we study the multistability and chaos phenomena in a classical two-mode van der Pol generator which consists of a nonlinear element and two linear oscillators. We show that the configuration of the connections of the linear oscillators in the two-mode self-oscillating system significantly affects its oscillation regimes and bifurcational transitions. In the case of the feedback loop including one oscillator, the two-mode system demonstrates the well-known effect of frequency entrainment, including bistability and hysteresis phenomena. If the feedback loop involves both linear oscillators, the entrainment effect disappears; however, two new complex regimes of quasi-periodicity and chaotic self-oscillations emerge. We present here the results of the bifurcation analysis of the multistability formation and transition to chaos.
Multi-mode systems are of great interest in different fields of science. Such systems are used to describe various nonlinear processes governed by mathematical models in the form of ensembles of nonlinear oscillators with either limit cycles or more complex attractors. These ensembles may represent chains and networks of nonlinear dissipative oscillators with stable equilibria under external local or distributed forcing. Also, such ensembles may consist of excitable elements or include both types of oscillators with various characters and topologies of coupling. Self-oscillating systems of ring type and self-oscillating systems represented by ensembles of heterogeneous elements (e.g., when limit-cycle oscillators interact with dissipative oscillators characterized by stable equilibria) are also included in the diversity of multi mode self-oscillating systems. The two-mode van der Pol system serves the simplest model for either of these two types of systems: the two-mode van der Pol system can be formed by a classical van der Pol oscillator loaded by an additional linear oscillator or both linear oscillators can be included into the feedback loop of the self-oscillating system. These well-known paradigmatic models enable one to study in detail the bifurcational mechanisms and conditions for the emergence of multistability, frequency entrainment, quasiperiodicity, and chaos depending on the coupling configuration between the linear oscillators in a self-oscillating system. In this paper, we study the mechanisms of multistability formation and transition to chaos in a two-mode van der Pol system with two different types of connection of the linear oscillators. The results presented here and the results of our previous work1 show that in two interacting oscillators (one with a limit cycle and another with a stable focus in its phase space), the mechanism of multistability formation does not depend on the type of coupling between the linear oscillator and the non-linear van der Pol oscillator. In both cases of capacitive and inductive coupling, the coexisting attractors emerge through two super-critical Andronov-Hopf bifurcations and one sub-critical Neimark-Sacker bifurcation. In the case of a two-mode ring system, the frequency entrainment does not emerge, while more complex, quasiperiodic, and chaotic regimes are observed. This is due to the change in a Neimark-Sacker bifurcation from the sub-critical to the super-critical, leading to the emergence of a stable torus, its further destruction, and the appearance of a chaotic attractor.
