Synchronization of coupled generators of quasi-periodic oscillations upon destruction of invariant curve

The purpose of this study is to describe the complete picture of synchronization of two coupled generators of quasi-periodic oscillations, to classify various types of synchronization, to study features of occurrence and destruction of multi-frequency quasi-periodic oscillations. Methods. The object of the research is systems of ordinary differential equations of various dimensions. The work uses the fourth-order Runge-Kutta method to solve a system of differential equations. Main analysis of the dynamics is carried out on the basis of calculated spectrum of Lyapunov exponents depending on parameters of systems, so-called charts of Lyapunov exponents. Bifurcation trees, winding numbers, phase portraits and Poincare maps were also visualized. Results. Study of the dynamics of two coupled quasi-periodic generators for two sets of operating parameters of the subsystems is carried out. Two cases were studied when a two-frequency torus or chaotic oscillations (destroyed torus) are observed in the first oscillator. The dynamics of the second oscillator demonstrates different types of dynamics with a variation of the frequency detuning: periodic, quasi-periodic and chaotic. It is shown that for all parameters, phase synchronization of generators, broadband synchronization, and the phenomenon of oscillator death are observed. Dynamical regimes picture of the parameter plane frequency detuning the coupling strength has a universal structure. Exit from the broadband synchronization region with a decrease in the coupling strength is accompanied by a quasiperiodic Hopf bifurcation and the birth of a three-frequency torus. Conclusion. The interaction of the simplest generators of quasi-periodic oscillations gives a rich picture of dynamic regimes: multi-frequency quasi-periodic oscillations with a different numbers of frequencies, chaotic behavior characterized by a different spectrum of Lyapunov exponents. Despite the variety of dynamic regimes, the synchronization picture of two dissipatively coupled quasi-periodic generators has a universal structure. Quasi-periodic phase and broadband synchronization are observed. The destruction of a torus in a subsystem leads to the destruction of multi-frequency tori in the system of two coupled oscillators, as well as a decrease in the variety of types of chaotic attractors.