A ring generator of two- and three-frequency quasiperiodic self-oscillations based on the van der Pol oscillator.

Abstract

In this work, we present a model of an autonomous three-mode ring generator based on the van der Pol oscillator, where periodic, two-frequency quasiperiodic, three-frequency quasiperiodic, and chaotic self-oscillations are observed. The transitions to chaos occur as a result of a sequence of torus doubling bifurcations. When the control parameters are varied, the resonant limit cycles appear on a two-dimensional torus, and two-dimensional tori appear on a three-dimensional torus as a result of synchronization. We used a time series of dynamic variables, projections of phase portraits, Poincaré sections, and spectra of Lyapunov characteristic exponents to study the dynamics of the ring generator.