Coupled van der Pol – Duffing oscillators: phase dynamics and structure of synchronization tongues

Synchronization in the system of coupled non-identical nonisochronous Van fer Pol – Duffing oscillators with inertial and dissipative coupling is discussed. Generalized Adler’s equation is obtained and investigated in the presence of all relevant factors affecting the synchronization (nonisochronism of the oscillators, their nonidentity, coupling of the dissipative and inertial type). Characteristic symmetries are revealed for the Adler’s equation responsible for equivalence of some of the factors. Numerical study of the parameters space of the initial differential equations is carried out with a use of the method of charts of dynamic regimes in the parameter planes. Results obtained by both these approaches are compared and discussed. Introduction Systems of coupled self-oscillators are regarded as basic models in the theory of oscillators and nonlinear dynamics dealing with synchronization phenomenon and attendant effects. In general, this problem is characterized by a large number of parameters and by a complex bifurcation picture in the parameter space [1-13]. For the coupled systems one can observe many interesting oscillatory effects. Obviously, this problem is more complex than the problem concerning dynamics of one oscillator driven by an external periodic force [1,17]. Dynamics of coupled van der Pol oscillators and coupled van der Pol – Duffing oscillators is a subject of wide studies applicable in various areas of science and technology, relating, for example, to physical, chemical, biological and other systems. In this context, we have to mention, first of all, a fundamental monograph [1]; see also [2-16] and references therein. However, because of a relatively large number of parameters affecting dynamical phenomena in these systems, it is rather difficult to describe the general picture. Therefore, the studies usually are concentrated on certain fragments of the entire picture. For example, a case of a system of identical oscillators with dissipative coupling was analyzed in terms of the shortened, or amplitude equations [2]. A certain case of nonlinear coupling was studied in [10]. In paper [3] authors studied a system of identical and isochronous oscillators with combined (inertial and dissipative) coupling in terms of the generalized Adler’s equation. As seen from the literature, identity of the oscillators is one of the commonly used traditional assumptions. A certain exception is the recent paper [13]. However, in this work nonisochronism of the oscillators was not taken into account. This lack seems essential; indeed, the nonisochronism has to be taking into account to write down a well-known normal form of Andronov – Hopf bifurcation for a single oscillator; so the isochronous system does not represent the generic case [1, 18]. Moreover, in accordance with [1], the nonisochronism provides one of mechanisms responsible for appearance of in-phase or out-phase synchronization. Thus, it seems necessary to generalize, systematize and interpret all the available results. In the present work we intend to account in complex the following factors relevant for the dynamics of the coupled self-oscillators: • dissipative coupling,