Globally exponential multi switching-combination synchronization control of chaotic systems for secure communications

Multi-Switching Combination Synchronization of Three Fractional-Order Delayed Systems

“…(ii) Supposing that f i = 0, g i = 0 and h i = 0 (i = 1, 2, 3), hybrid projective synchronization between the systems in Equations (24) and (30) can be achieved with the following controllers…”

“…The system parameters are given as a 1 = 10, b 1 = 8 3 , c 1 = 28, a 2 = 35, b 2 = 3, c 2 = 28, a 3 = 0.4, b 3 = 0.2, m 3 = 10, thus the systems in Equations (24), (27) and (30) exhibit chaotic behaviors, respectively. We assume f 1 = f 2 = f 3 = 1, g 1 = g 2 = g 3 = 1 and h 1 = h 2 = h 3 = 1, and the initial values are (x 1 (0), x 2 (0), x 3 (0)) = (−20, 2, 3), (y 1 (0), y 2 (0), y 3 (0)) = (7, 4.04, 20) and (z 1 (0), z 2 (0), z 3 (0)) = (1, 2, 40), respectively.…”

“…We assume f 1 = f 2 = f 3 = 1, g 1 = g 2 = g 3 = 1 and h 1 = h 2 = h 3 = 1, and the initial values are (x 1 (0), x 2 (0), x 3 (0)) = (−20, 2, 3), (y 1 (0), y 2 (0), y 3 (0)) = (7, 4.04, 20) and (z 1 (0), z 2 (0), z 3 (0)) = (1, 2, 40), respectively. Multi-switching combination synchronization between the systems in Equations (24), (27) and (30) can be realized with K = diag{−10, −10, −10}. Figure 4 illustrates synchronization errors e 123 , e 231 , e 312 .…”

“…Figure 4 illustrates synchronization errors e 123 , e 231 , e 312 . Figure 5 shows synchronization states x 2 + y 3 vs. z 1 , x 3 + y 1 vs. z 2 and x 1 + y 2 vs. z 3 of the drive systems in Equations (24) and (27) and the response system in Equation (30). Figure 6 displays synchronization errors e 112 , e 223 , and e 331 .…”

“…Figure 6 displays synchronization errors e 112 , e 223 , and e 331 . Figure 7 illustrates synchronization states x 1 + y 2 vs. z 1 , x 2 + y 3 vs. z 2 and x 3 + y 1 vs. z 3 of the drive systems in Equations (24) and (27) and the response system in Equation (30). In Figures 4-7, we can see that the multi-switching combination synchronization errors converge to zero, i.e., the multi-switching combination synchronizations for Cases 1 and 2 are achieved, respectively.…”