A study of synchronization processes of nonlinear systems in the difference space of phase variables

Leonid Politansky,Valentin Lesynsky,Ruslan Politanskyi

doi:10.15587/1729-4061.2017.103955

Abstract

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 Rossler 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 Rossler 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

Highlights

Due to the prospects of using systems with nonlinear dynamics in information and communication networks, the study of synchronous work of systems with nonlinear dynamics is important and promising in various fields of modern science [1], despite the fact that the methods of solving differential equations with nonlinear functions are already known
It should be noted that the known methods for solving differential equations by linearization technique do not exclude the possibility of having incorrect solutions in the process of system buckling analysis
The case when the solutions to the systems of differential equations with nonlinear right-hand side form an attractor in the phase space is feasible for systems with nonlinear dynamics

Summary

Introduction

Due to the prospects of using systems with nonlinear dynamics in information and communication networks, the study of synchronous work of systems with nonlinear dynamics is important and promising in various fields of modern science [1], despite the fact that the methods of solving differential equations with nonlinear functions are already known. It should be noted that the known methods for solving differential equations by linearization technique do not exclude the possibility of having incorrect solutions in the process of system buckling analysis. The use of linearization techniques provides the study of only complete synchronization of the two systems, the essence of which is that over time solutions of the main and controlled systems are identical. 3/4 ( 87 ) 2017 main and controlled system and form closed trajectories in the phase space. The case when the solutions to the systems of differential equations with nonlinear right-hand side form an attractor in the phase space is feasible for systems with nonlinear dynamics. Synchronization is not possible in case of an unlimited increase of the distance between phase trajectories

Literature review and problem statement

Conclusions