We propose a new autonomous dynamical system of dimension N = 4 that demonstrates the regime of stable two-frequency motions and period-doubling bifurcations of a two-dimensional torus. It is shown that the period-doubling bifurcation of the two-dimensional torus is not followed by the resonance phenomenon, and the two-dimensional ergodic torus undergoes a period-doubling bifurcation. The interaction of two generators is also analyzed. The phenomenon of external and mutual synchronization of two-frequency oscillations is observed, for which winding number locking on a two-dimensional torus takes place.
The peculiarities of external synchronization of a resonant limit cycle on a torus are studied in an autonomous oscillator of quasiperiodic oscillations with two basic frequencies. We show numerically and experimentally that in the resonance conditions the synchronization effect takes place only at one of the two basic frequencies of the system, while the oscillations at the second basic frequency remain unsynchronized. Our results convincingly indicate a principal difference between synchronization of the resonant limit cycle on the torus and of a typical limit cycle. This is in contrast to the well-established theory of synchronization of a limit cycle. This finding opens new strategies for controlling systems with multiple time scales.
We study synchronization of a resonant limit cycle on a two-dimensional torus with an external harmonic signal. The regime of the resonant limit cycle is realized in a system of two coupled Van der Pol oscillators; we consider the resonances 1:1 and 1:3. We analyze the influence of coupling strength between the oscillators. We show that the resonant limit cycle can be generally synchronized on the torus through the resonance destruction followed by the locking of one and then another one of the basic frequencies. We consider the bifurcational mechanism of the synchronization effect.
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