Phase synchronization in bi-directionally coupled chaotic ratchets

Multi-stability and basin crisis in synchronized parametrically driven oscillators

“…7 Trajectories of the attractors in the Poincaré sections illustrating the boundary crisis phenomenon during the transition to synchronized dynamics for the pendulum with parametric excitation for k = 0.43 ture of the distance between it and the basin boundary approaches zero so that the entire phase space is gradually being filled up with uncorrelated Poincaré points. This is an indicator of a more complex dynamics; and the phenomenon termed boundary or exterior crisis of the attractor has been reported earlier as a synchronization transition [13,19,38,39]. It is usually attributed to the collision of the attractor with an unstable periodic orbit on its basin boundary or, equivalently its stable manifold [18].…”

“…According to Lyapunov stability theory [28,29], the inequality in (19) represents a sufficient condition for global asymptotic stability of the system (7) at the equilibrium point. With A, K, P, M as defined earlier; (19) thus becomes…”

“…It is usually attributed to the collision of the attractor with an unstable periodic orbit on its basin boundary or, equivalently its stable manifold [18]. In the synchronized state of coupled oscillators, the chaotic attractor is rebuilt, although there could be residual Poincaré points that do not fall on the main body of the original attractor [38] leaving some spotted appearance in the open parts of the phase space.…”

“…They are usually related to sudden changes in chaotic attractors with parameter variation and such changes are caused by the collision of the chaotic attractor with an unstable periodic orbit or, equivalently its stable manifold [18]. Recent studies have also identified the crisis event in coupled oscillators when the oscillators transit from nonsynchronous to synchronous state [13,19,20]. Here, we report a different type of crisis-the basin crisis which is associated with the destruction of co-existing periodic attractors and the creation of different coexisting periodic attractors of the same or of different periodicity during the transition to synchronization; and has not been reported previously in the literature to the best of our knowledge.…”

Adaptive backstepping control and synchronization of a modified and chaotic Van der Pol-Duffing oscillator

“…Chaos synchronization is closely related to the observer problem in control theory and recent studies deal with the synchronization problem based on control theory approach. Chaos synchronization has been intensively investigated in the context of many specific problems arising from physical [3,4], chemical and ecological [5], and applications to secure communications [6∼8] to mention a few. Enormous research progress has been made in developing and understanding various types of synchronization schemes, such as adaptive control [9], active-backstepping design [10,11], active control [12∼14], backstepping [15], and sliding mode control [16].…”