Boosting synchronization in chaotic systems: Combining past and present interactions

A classical interconnection for inducing synchronization in dynamical systems is static diffusive coupling, i.e., a coupling without dynamics that considers the output or state differences of neighboring systems. However, this type of coupling has some well-known limitations: 1) for some chaotic systems and certain input-output combinations, it is impossible to synchronize a pair or a network of identical systems, 2) the interval of coupling strength values for which synchronization is achieved is limited. Here, we show that these limitations may be removed, or at least considerably relaxed, by using a coupling with memory, i.e., a coupling composed by a standard diffusive-like term plus a delayed version of itself. As a particular example, we consider a pair of Rössler systems in a master-slave configuration, where we analyze the local stability of the synchronous solution using the linearized error dynamics in combination with the D-decomposition method. Also, analytic conditions for tuning the coupling strength values corresponding to the non-delayed and delayed terms of the overall coupling are provided. Furthermore, the applicability of the proposed coupling for boosting synchronization in other chaotic systems is also discussed. In particular, we show how the proposed coupling with memory is applicable for synchronizing a pair of uni-directionally coupled van der Pol oscillators with sinusoidal excitation and a pair of Hindmarsh-Rose neurons. Ultimately, the results presented in this paper suggest that the synchronizability of the coupled systems is enhanced when the present and past interactions are taken into account.