Modified Function Projective Synchronization between Different Dimension Fractional-Order Chaotic Systems

and Applied Analysis 3 where ‖ · ‖ is the Euclidean norm,M x is am×n real matrix, and matrix elementMij x i 1, 2, . . . m, j 1, 2, . . . , n are continuous bounded functions. ei yi − ∑n j 1 Mijxj i 1, 2, . . . m are called MFPS error. Remark 2.2. According to the view of tracking control, M x x can be chosen as a reference signal. The MFPS in our paper is transformed into the problem of tracking control, that is the output signal y in system 2.2 follows the reference signal M x x. In order to achieve the output signal y follows the reference signal M x x. Now, we define a compensation controller C1 x ∈ R for response system 2.2 via fractional-order derivative dr M x x /dtr . The compensation controller is shown as follows: C1 x dr M x x dtqr − Fr M x x , 2.4 and let controller C x, y as follows: C ( x, y ) C1 x C2 ( x, y ) , 2.5 where C2 x, y ∈ R is a vector function which will be designed later. By controller 2.5 and compensation controller 2.4 , the response system 2.2 can be changed as follows: dr e dtqr D1 ( x, y ) e C2 ( x, y ) , 2.6 where D1 x, y e Fr y − Fr M x x , and D1 x, y ∈ Rm×m. So, the MFPS between drive system 2.1 and response system 2.2 is transformed into the following problem: choose a suitable vector function C2 x, y such that system 2.6 is asymptotically converged to zero. In what followswe present the stability theorem for nonlinear fractional-order systems of commensurate order 22–25 . Consider the following nonlinear commensurate fractionalorder autonomous system