Fractional dynamical controllers for generalized multi-synchronization of commensurate fractional order Liouvillian chaotic systems

In a master slave configuration of strictly different commensurate fractional order Liouvillian systems, the generalized synchronization problem of multiple decoupled families of Liouvillian systems is addressed. The main key ingredient is to find canonical forms for the original systems, from a family of fractional differential primitive elements based on output of each system, taking into account the Liouvillian feature. Fractional order dynamical controllers are designed to solve the generalized multi-synchronization problem. Moreover, it is shown that adding diffusive coupling terms in the dynamical controllers solves the synchronization problem with complex interaction between slave systems, with any type of interplay. Finally, some numerical examples show the effectiveness of the proposed approach.