Chaotic systems demonstrate complex behaviour in their state variables and their parameters, which generate some challenges and consequences. This paper presents a new synchronisation scheme based on integral sliding mode control (ISMC) method on a class of complex chaotic systems with complex unknown parameters. Synchronisation between corresponding states of a class of complex chaotic systems and also convergence of the errors of the system parameters to zero point are studied. The designed feedback control vector and complex unknown parameter vector are analytically achieved based on the Lyapunov stability theory. Moreover, the effectiveness of the proposed methodology is verified by synchronisation of the Chen complex system and the Lorenz complex systems as the leader and the follower chaotic systems, respectively. In conclusion, some numerical simulations related to the synchronisation methodology is given to illustrate the effectiveness of the theoretical discussions.

This study concerns the control of uncertain fractional-order multi-inputs chaotic systems, using fractional dynamic sliding mode controller (FD-SMC). This controller removes the chattering of the system using a low-pass integrator fractal filter beforehand, which provides an augmented system with extra dimension. Hence, the dimension of the system state is more than the original system. These augmented state variables are known to implement FD-SMC. To solve this challenge, a novel fractional sliding observer (FSO) is constructed and usedin this study. The theory of Lyapunov is applied to construct the stability of both the proposed FSO and FDSMC. Comparison is also presented with conventional fractional sliding mode controller (CF-SMC). To have a reliable comparison, the proposed FSO is implemented in both FD-SMC and CF-SMC. This study demonstrates the advantage of the FD-SMC in removing the chattering, its simplicity in the concept and realisation, with CF-SMC where the chattering is usually present. Another advantage of the proposed FD-SMC is appeared in the coefficient of the sign function (switching gain), which is independent of the model uncertainties. Finally, an improved multiinputs fractional order Duffing–Holmes chaotic system (FO-DHCS) is proposed to validate this novel approach using simulation examples. The illustration of the simulation results shows a better performance of this proposed approach.