Synchronization of oscillators with hard excitation coupled with delay. Part 2. Amplitude-phase approximation

Aim of this work is to develop the theory of mutual synchronization of two oscillators with hard excitation associated with a delay. Taking into account the delay of a coupling signal is necessary, in particular, when analyzing synchronization at microwave frequencies, when the distance between the oscillators is large compared to the wavelength. Methods. A bifurcation analysis of the mutual synchronization of two generators with hard excitation in the amplitude-phase approximation is carried out. The results of the bifurcation analysis are compared with the results of numerical simulation of the system of differential equations with delay. Results. A complete bifurcation pattern of mutual synchronization on the plane "frequency mismatch - coupling parameter" is presented. In the case of small mismatch and weak coupling, the fixed points, which correspond to modes with dominance of one of the oscillators, merge with saddle fixed points and disappear when the coupling parameter increases. In the case of large mismatch, one of these points either vanishes or loses stability as a result of a subcritical Andronov-Hopf bifurcation. The other of these points remains stable at any values of the coupling parameter, and the oscillation amplitudes of both oscillators gradually equalize and the phase difference tends to zero, i.e., the oscillation mode with dominance of one of the oscillators gradually transforms into the in-phase synchronization mode. It has been found that with an increase in the coupling parameter, a transformation of the basin of attraction of a stable zero fixed point occurs. As a result of this transformation, if at the initial moment of time the oscillations of the generators are close to antiphase, the oscillations decay at any initial amplitudes. Conclusion. The synchronization pattern in the system of delay-coupled oscillators with hard excitation has been studied. It was discovered that in addition to mutual synchronization modes with approximately equal oscillation amplitudes, stationary modes with suppression of oscillations of one generator by another are also possible. The bifurcation mechanisms of the appearance and disappearance of multistability in the system have been examined.