Application of calculation of local Lyapunov exponents to analyze characteristics of intermittent generalized synchronization

In the work we investigated the main characteristics of intermittent generalized synchronization, such as the distributions of the durations of laminar phases at a fixed value of the coupling parameter and the dependence of the mean duration of laminar phases on the supercriticality parameter, by calculating the local Lyapunov exponents. This method should make it possible to carry out research not only in the case of unidirectional coupling of interacting systems, but also in the case of mutual one. The analyzed systems were Rossler oscillators with a relatively simple topology of the attractor (hyperbolic type) and Lorenz oscillators with a relatively complex (two-sheeted) topology of the attractor. It was found that in the first case there is an on-off intermittency described by functions of a power-law type, and in the second case there is a hop-intermittency subject to exponential laws. Initially, generalized synchronization in the context of continuous dynamic systems means establishing a connection between the state vectors of systems in the form of a functional relationship. Later it was proved that, in the general case, there is a connection in the form of a functional, i.e. there is a dependence on the history of the systems. The method for calculating local Lyapunov exponents is the most universal and allows one to correctly analyze the behavior of systems in both cases. The behavior of systems is controlled not only by their own control parameters, but also bv a coupling parameter that characterizes a kind of degree of synchronization. With an increase in the coupling parameter, at a certain critical value, a continuous (strong) generalized synchronization is established, characterized by a smooth functional relationship. Intermittent generalized synchronization occurs at values of the coupling parameter slightly less than the critical one and is characterized by a fractal functional relationship. The regime is called weak generalized synchronization and it is this regime that is considered in the work. Assessment of the characteristics of intermittency would be impossible without the use of the method of identifying characteristic phases of the interacting systems’ behavior. Intermittent behavior is characterized by the fact that time intervals of synchronous, in the sense of generalized synchronization, oscillations (laminar phases) alternate with time intervals corresponding to asynchronous bursts (turbulent phases). At the same time, with an increase in the coupling parameter between systems, an increase in the mean duration of the laminar phases of behavior and a simultaneous decrease in the average duration of turbulent phases is observed. The nature of the dependence of the average duration on the coupling parameter depends on the type of intermittency.