Method for characteristic phase detection in systems with complex topology of attractor being near the boundary of generalized synchronization

The aim of the paper consists in the development of universal method for the detection of characteristic phases of the behavior in systems with complex topology of attractor being in the regime of intermittent generalized synchronization. The method is based on an analysis of the location of representation points on the attractors of interacting systems coupled unidirectionally or mutually. The result of this work is the verification of the performance of the proposed method on systems with unidirectional coupling (two unidirectionally coupled Lorenz oscillators being in chaotic regime) that allow the analysis of intermittency using the auxiliary system method. It was found that the jump of the representation points to different sheets of attractors of interacting systems precedes the appearance of the turbulent phase of the behavior detected using the auxiliary system method. Using both methods, the statistical characteristics of intermittency, i.e. the distributions of the laminar phase lengths for several fixed values of the coupling parameter, were calculated and they were compared with each other. It was found that in all considered cases the results of both methods almost exactly coincide with each other, while the distributions of the laminar phase lengths obey the exponential laws, which is not typical for systems with a simple enough topology of attractor. It was assumed that in systems with a complex topology of attractor a new type of intermittency called by jump intermittency is observed.