Lyapunov approach to synchronization of chaotic systems with vanishing nonlinear perturbations: From static to dynamic couplings

Synchronization of chaotic dynamics can be pursued by means of different coupling strategies. Definitely, master-slave coupling represents one of the most adopted solutions, even if it presents some limitations due to the coupling term's selection strategy. In this paper, we investigate the role of different structures of coupling terms on the synchronization properties of master-slave chaotic system configurations. Here, Lyapunov theory for linear systems with nonlinear vanishing perturbations is exploited. The obtained results allow to determine the capability of a static, dynamic, or mixed coupling connection in stabilizing the synchronization manifold, using linear techniques based on the root locus. This knowledge allows to design the coupling structure considering also the synchronization error transient features, which are, here, shown to improve in the presence of higher-order dynamic couplings. A number of cases of study, involving classical chaotic nonlinear systems, show the efficacy and simplicity of the application of the strategy proposed.