Combination-combination synchronisation of time-delay chaotic systems for unknown parameters with uncertainties and external disturbances

In this article, an adaptive control method is proposed to study the combination-combination synchronisation phenomenon of four non-identical time-delayed chaotic systems for fully unknown parameters with parametric uncertainties and external disturbances. Based on the Lyapunov-Krasovskii functional theory, an appropriate adaptive controller is constructed so that a globally and asymptotically stable synchronisation state can be established between the master and the slave systems. Unknown parameters are identified by designing suitable parameter update laws. To elaborate the presented scheme, double-delay Rossler and time-delay Chen systems are considered as the master systems and time-delay Shimizu-Morioka and time-delay modified Lorenz systems are considered as the slave systems. Numerical simulations are presented to justify the theoretical analysis.