Synchronization problems for unidirectional feedback coupled nonlinear systems

TL;DR

This paper develops three nonlinear feedback control methods to synchronize unidirectionally coupled nonlinear systems, including chaotic ones, by driving the slave system to a reference trajectory, using theories like perturbation, nonsmooth analysis, and singular perturbation.

Contribution

It introduces three novel nonlinear feedback synchronization methods tailored for different synchronization problems in coupled nonlinear systems, including chaotic dynamics.

Findings

Methods successfully synchronize chaotic systems to reference trajectories.

Simulations demonstrate effectiveness of the proposed control strategies.

Approaches leverage advanced theories like perturbation and nonsmooth analysis.

Abstract

In this paper we consider three different synchronization problems consisting in designing a nonlinear feedback unidirectional coupling term for two (possibly chaotic) dynamical systems in order to drive the trajectories of one of them, the slave system, to a reference trajectory or to a prescribed neighborhood of the reference trajectory of the second dynamical system: the master system. If the slave system is chaotic then synchronization can be viewed as the control of chaos; namely the coupling term allows to suppress the chaotic motion by driving the chaotic system to a prescribed reference trajectory. Assuming that the entire vector field representing the velocity of the state can be modified, three different methods to define the nonlinear feedback synchronizing controller are proposed: one for each of the treated problems. These methods are based on results from the small parameter perturbation theory of autonomous systems having a limit cycle, from nonsmooth analysis and from the singular perturbation theory respectively. Simulations to illustrate the effectiveness of the obtained results are also presented.