Abstract: Based on the fractional calculus theory, three synchronous control schemes were proposed to make the error system of the fractional Brussel system converge the error system states to the equilibrium point. An appropriate controller was designed in the first control scheme and the convergence of the error system was obtained by using Mittag-Leffler function. In the second control scheme, the fractional sliding mode surface was introduced, and the chaos synchronization of the fractional order Brussel master-slave systems was achieved based on the fractional version of the Lyapunov stability and the sliding mode control method. The effects of model uncertainties and external disturbances were fully taken into account in the third control scheme. A new sliding mode reaching law was designed and the fast convergence of the error system to the equilibrium point was obtained based on fractional order terminal sliding mode control. It was proved that master-slave systems were chaos synchronization under proper controllers. Numerical simulations were presented to illustrate the effectiveness and applicability of the proposed schemes and to validate the theoretical results of the paper.

Key words: Brussel system, fractional-order, chaos synchronization, Mittag-Leffler function, terminal slding mode