Design of PDC Controllers by Matrix Reversibility for Synchronization of Yin and Yang Chaotic Takagi-Sugeno Fuzzy Henon Maps

and Applied Analysis 3 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 −1.5 −1 −0.5 0 0.5 1 1.5 x1 x 2 Figure 1: Phase portrait of the chaotic YangHenon map system with a1 1.4, b1 0.3. where a1 1.4, b1 0.3 and positive argument n1 0, 1, 2, 3, . . . , n. The Yang chaotic behaviors of Henon map system are quoted 19 by phase portraits in Figure 1. The Henon map system is invertible if b1 / 0. The inversed Henon map system can be described as follows: x̂1 n2 − 1 1 b1 x̂2 n2 , x̂2 n2 − 1 x̂1 n2 a1 b2 1 x̂2 2 n2 − 1. 3.2 The parameters 1/b1, a1/b 1 are replaced by b2, a2, respectively, where b2 0.3, a2 1.4 and negative argument n2 −1,−2,−3, . . . ,−n. The Yin chaotic behaviors of inversed Henon map system are shown in Figure 2. Comparing Figure 1 and Figure 2, it is surprisingly noted that Figure 2 gives new information for famous Henon map system. Since 1976 18 , traditional studies of Henon system have only been devoted to its behaviors with positive argument. Now it is discovered that with negative argument or negative time, a new continent is waiting for us in the future study of either nonlinear map systems or nonlinear continuous systems. 4. Takagi-Sugeno Fuzzy Model The T-S fuzzy model 12 can represent a nonlinear system by linear state space which is described by fuzzy IF-THEN rules. Consider the following discrete time T-S fuzzy rules: R : IF p1 n is Mi1, . . . , and pq n is Miq, THEN x n 1 Aix n c, 4.1 4 Abstract and Applied Analysis −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 −1.5 −1 −0.5 0 0.5 1 1.5