Dual combination-combination synchronization of time delayed dynamical systems via adaptive sliding mode control under uncertainties and external disturbances

In this paper, the problem of dual combination-combination synchronization of time delay systems is studied for unknown parameters under uncertainties and external disturbances. By using adaptive sliding mode control method and Lyapunov stability criteria, nonlinear control functions and parameters updating equations are designed. This problem is first of its kind where synchronization of eight chaotic time delay systems is accomplished for unknown parameters. By performing rigorous calculations, sufficient condition is obtained for the asymptotically stable synchronization state. To implement the methodology two examples are presented. In first example, pairs of time delay Shimizu-Morioka and time delay Bhalekar-Gejji systems are considered while pairs of time delay advanced Lorenz and time delay Lorenz systems are considered in second example. Runge-Kutta method for delay-differential equations is used to perform numerical simulations. Theoretical and graphical treatments are presented to show the efficiency of the designed controllers and parameters updating equations.