Mutual synchronization of dissipatively coupled memristive self-oscillators

Background and Objectives: Dynamical systems containing memristive elements, i.e. elements with “memory”, represent a special class of dynamical systems that can be named memristive systems. In memristive systems, the synchronization of oscillations has some features. However, a complete description of these features is still lacking. For the most part, this concerns the synchronization of memristive self-oscillators, both identical and with frequency detuning. In this paper, we study two resistively coupled self-oscillators containing memristive conductivities. We consider both the complete synchronization of identical selfoscillators and the frequency synchronization of self-oscillators with detuning. The features of synchronization effects associated with the memristive nature of partial systems are revealed. The influence of “non-ideality” of memristive elements and the duration of the establishment processes in this case are analyzed. Materials and Methods: Using numerical integration methods for various parameter values, approximate solutions of a system of ordinary differential equations that describe the dynamics of two coupled memristive self-oscillators are obtained. Projections of phase trajectories are plotted on various planes, as well as regions of synchronization of oscillations of the system under study. Results: It is shown that the memristive systems under consideration demonstrate both the effect of complete synchronization (with full identity of partial systems) and the effect of frequency synchronization (in the presence of frequency detuning). The complete synchronization of oscillations is characterized by the presence of a threshold for the coupling, the value of which continuously depends on the initial conditions, in particular, on the initial values of variables that specify the states of memristive elements. With a constant value of the coupling coefficient, there is a continuous dependence of the boundaries of the frequency synchronization region on the initial conditions. The introduction of a parameter characterizing the rate of memristors “forgetting” their initial state (“non-ideality”) leads to the disappearance of the dependence of the type of the steady state on the initial conditions. Conclusion: The fundamental phenomenon of synchronization is inherent in memristive self-oscillators, which allows them to be attributed to the class of self-oscillating systems. The influence of the initial conditions on the effects of synchronization can be considered as a general property of “ideal” memristive systems.