Synchronization phenomena in coupled dynamical systems with hidden attractors

Abstract

Recently, Leonov and Kuznetsov introduced a new class of nonlinear dynamical systems, which is called systems with hidden attractors, in contrary to the well-known class of systems with self-excited attractors. In this class, dynamical systems with infinite number of equilibrium points, with stable equilibria, or without equilibrium are classified. Since then, the study of chaotic systems with hidden attractors has become an attractive research topic because this new class of dynamical systems could play an important role not only in theoretical problems but also in engineering applications. In this direction, the proposed chapter presents the bidirectional and unidirectional coupling schemes between two identical dynamical chaotic systems with no-equilibrium points. As it is observed, when the value of the coupling coefficient is increased in both coupling schemes, the coupled systems undergo a transition from desynchronization mode to complete synchronization. Also, the simulation results reveal the richness of the coupled system’s dynamical behavior, especially in the bidirectional case, showing interesting nonlinear dynamics, with a transition between periodic, quasiperiodic and chaotic behavior as the coupling coefficient increases, as well as synchronization phenomena, such as complete and anti-phase synchronization. Various tools of nonlinear theory for the study of the proposed coupling method, such as bifurcation diagrams, phase portraits and Lyapunov exponents have been used.