#### For citation:

Kuznetsov A. P., Stankevich N. V. Dynamics of coupled generators of quasi-periodic oscillations with equilibrium state. *Izvestiya VUZ. Applied Nonlinear Dynamics*, 2018, vol. 26, iss. 2, pp. 41-58. DOI: 10.18500/0869-6632-2018-26-2-41-58

# Dynamics of coupled generators of quasi-periodic oscillations with equilibrium state

Subject of the study. Recently, the problems of synchronization of systems demonstrating quasi-periodic oscillations arouse interest. In particular, it can be generators of quasi-periodic oscillations that allow a radiophysical realization. In this paper we consider the dynamics of two coupled oscillators of quasi-periodic oscillations with a single equilibrium state. Novelty. The difference from the already studied case of coupled modified Anishchenko– Astakhov generators consists in engaging of two-parameter analysis and analysis in a much wider range of parameter changes, as well as a more dimensionless equation for an individual generator. Methods. The method of charts of Lyapunov exponents is used, which reveals areas of various types of dynamics, up to four-frequency oscillations. The bifurcation mechanisms of complete synchronization are investigated. Results. The possibility of synchronous quasiperiodicity is demonstrated, when the phases of the generators are locked, but the dynamics of the system is generally quasi-periodic. The possibility of the effect of «death of oscillations» arising due to the dissipative character of coupling is revealed. The possibility of the effect of broadband quasi-periodicity is demonstrated. Its peculiarity consists in the fact that twofrequency oscillations arise in a certain range of variation of the coupling parameter and a wide range of frequency mismatch. The bifurcation mechanisms of this effect are presented. It is shown that a certain degeneracy is characteristic for it, which is removed when nonidentity is introduced along the control parameters of individual generators. A bifurcation analysis is presented for this case. Two-parameter analysis allowed us to identify points of quasiperiodic bifurcations of codimension two QSNF (Quasi-periodic saddle-node fan) on the parameter plane, associated with the synchronization of multi-frequency tori. These points are the tips of the tongues of the two-frequency regimes, which have a threshold for the coupling coefficient. In their vicinity, three- and four-frequency quasi-periodic regimes are also observed. Discussion. Synchronization of quasi-periodic generators has a number of new moments that are established in two-parameter analysis in a wide range of parametric changes.

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