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The research of finding hidden attractors in nonlinear dynamical systems has attracted much consideration because of its practical and theoretical importance. A new fractional order four-dimensional system, which can exhibit some hidden hyperchaotic attractors, is proposed in this paper. The predictor–corrector method of the Adams–Bashforth–Moulton algorithm and the parameter switching algorithm are used to numerically study this system. It is interesting that three different kinds of hidden hyperchaotic attractors with two positive Lyapunov exponents are found, and the fractional order system can have a line of equilibria, no equilibrium point, or only one stable equilibrium point. Moreover, a self-excited attractor is also recognized with the change of its parameters. Finally, the synchronization behavior is studied by using a linear feedback control method.

In this paper, for synchronizing two actual nonidentical fractional-order hyperchaotic systems disturbed by model uncertainty and external disturbance, the fractional matrix and inverse matrix projective synchronization methods are presented and the methods' correctness and effectiveness are proved. Especially, under certain degenerative conditions, the methods are reduced to study the complete synchronization, antisynchronization, projective (or inverse projective) synchronization, modified (or modified inverse) projective synchronization, and stabilization problem for the disturbed (or undisturbed) fractional-order hyperchaotic systems. In addition, as the fractional matrix and inverse matrix projective synchronization methods' applications, the fractional-order hyperchaotic Chen and Rabinovich systems disturbed by model uncertainty and external disturbance are constructed, and the matrix and inverse matrix projective synchronizations between the two disturbed systems are achieved, respectively. This work constructs a basic theoretical framework of fractional matrix and inverse matrix projective synchronization methods and provides a general method for synchronizing the actual disturbed fractional-order hyperchaotic systems that are related to science and engineering. KEYWORDS disturbed hyperchaotic system, fractional inverse matrix projective synchronization, fractional matrix projective synchronization, fractional-order derivative Math Meth Appl Sci. 2018;41:6907-6920.wileyonlinelibrary.com/journal/mma

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