Ludmila Turukina

Ensembles of several chaotic R¨ossler oscillators are considered. It is shown that a typical phenomenon for such systems is the emergence of invariant tori of different and sufficiently high dimension. The possibility of a quasi-periodic Hopf bifurcation and of the cascade of such bifurcations based on tori of increasing dimension is demonstrated. The domains of resonant tori are revealed whose boundaries correspond to a saddle-node bifurcation. Within areas of resonant modes the torus-doubling bifurcations and tori destruction are observed.

A phenomenon of interaction between three reactively coupled Van der Pol oscillators is considered in the paper. Phase equations are being obtained in second order approximation. Bifurcation analysis and Lyapunov’s exponent maps are used for illustrating system behavior. The paper consider significant features of reactive coupling. Complicated original system behavior with steering parameter growth is being considered too.

The conditions are discussed for which the ensemble of interacting oscillators may demonstrate Landau–Hopf scenario of successive birth of multi-frequency regimes. A model is proposed in the form of a network of five globally coupled oscillators, characterized by varying degree of excitement of individual oscillators. Illustrations are given for the birth of the tori of increasing dimension by successive quasi-periodic Hopf bifurcation.

The problem of external driving by the harmonic signal of two coupled self-oscillators is investigated. Comparison with the synchronization picture for phase oscillators is given. We discuss the configuration of periodic, two- and three-frequency regimes in the parameter space of external signal. The illustrations of three-frequency tori and resonance two-frequency tori are given. A number of significant differences from the bifurcation mechanisms for the destruction of synchronization are found compared with the case of phase oscillators.

The problem of the dynamics of phase oscillators is discussed with an increasing their numbers. We discuss the organization of the parameters plane responsible for the frequency detunings of the oscillators and amplitude of the dissipative coupled. The region of complete synchronization, quasi-periodic oscillations of different dimension and chaos are are observed. We discuss the changing of the synchronization picture with an increasing of the number of oscillators in the chain. We use the method of charts of Lyapunov exponents and modification of the method of charts of dynamical regimes visualized two-frequency resonant tori of different types.

The pulse driven Rossler system before the saddle-node bifurcation in the regime of divergence is considered. It is shown that external pulses initiate stable periodic and quasi-periodic regimes in non-autonomous system. The effect of synchronous response due interaction between external signal and own rhythm of autonomous system concerned with the «rotation» of the representation point in the three-dimensional phase space is observed. It is revealed that the torus doublings in the stroboscopic section exist in the certain area on the parameter plane of external force in this system.