A study of synchronization processes of nonlinear systems in the difference space of phase variables - Politansky, Leonid; Politanskyi, Ruslan; Lesynsky, Valentin (Eastern-European Journal of Enterprise Technologies, 2017)
The analysis of trajectories in the phase space of the systems of ordinary differential equations has been made. Classification of phase trajectories has been developed. Synchronization in Rössler systems, coupled by the scheme “main–controlled” system, has been studied. In the controlled system, variables in the right –hand side are replaced by functions of time, which are solutions to the main system. The analysis of processes in nonlinear systems was made by means of replacement with the help of synchronization matrix and transfer to the linearized system of variables equal to the difference of phase variables of the main and controlled systems. As a result of this analysis, there have been set the values of the synchronization matrix elements in which there occur different types of synchronization: complete, phase and topological. It is shown that even in the absence of communication between Rössler systems in the difference space of phase variables of the main and controlled systems with nonlinear dynamics, there occurs topological synchronization and there is formed an attractor with low spatial complexity that is an open trajectory of limited values. The criterion for the absence of synchronization of nonlinear systems is the unlimited growth of the difference of phase variables