Practical cluster synchronization of heterogeneous sytems on graphs with acyclic topology

We study the problem of practical cluster synchronization in networks of heterogeneous dynamical systems. The considered framework involves groups of identical dynamical systems, interacting with each other through linear couplings. The control objective studied in this paper is to achieve synchronization up to a possibly small error of all identical systems. Based on the two assumptions that all systems satisfy the QUAD condition and that the global coupling structure is acyclic, a constructive procedure for a coupling design ensuring practical cluster synchronization is proposed. For establishing the desired result, first, the synchronization of identical systems under external disturbances is studied. The main contribution is the extension of this result to the complete heterogeneous network. The theoretical results are illustrated and tested numerically on an exemplary network composed of several Van der Pol oscillators and Chua´s circuits.