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


Introduction
Chaotic behavior in nonlinear system is an interesting phenomenon, which is very sensitive in initial conditions. Chaos plays an important role in nonlinear science field 1-4 . In 5 , the Yin chaos in continuous time nonlinear system is firstly proposed. In 1990, Pecora and Carroll proposed chaotic synchronization scheme and started a new research interest 6 . In many papers, chaos synchronization of two-dimensional Hênon map system is studied 7-11 . In 1985, Takagi and Sugeno 12 proposed T-S fuzzy model and has been used in complex nonlinear systems successfully 13 . The T-S fuzzy model of the nonlinear systems can exactly be represented as a fuzzy aggregation of some local linear systems. Generally, the implementation of synchronization 14 and control 15,16 in T-S fuzzy model of chaotic systems uses the parallel distributed compensation PDC scheme.

Abstract and Applied Analysis
The features of this paper are as follows. 1 We develop the chaotic Henon map with decreasing argument, which is called Yin chaotic Henon map.
2 Yin and Yang T-S fuzzy models of chaotic Henon maps are proposed.
3 The nonlinear discrete time Yin Henon map is the inverse of Yang Henon map. Besides, the design of PDC controllers uses linear invertible matrix theory to achieve chaos synchronization of T-S fuzzy Yin and Yang Henon maps.
This paper is organized as follows. In Section 2, the two-dimensional invertible maps, Yin and Yang chaos are introduced. In Section 3, the Yang and Yin chaos of Henon maps are presented. Section 4, Takagi-Sugeno fuzzy model scheme is given. In Section 5, Yang and Yin T-S fuzzy models of the chaotic Henon maps are given. In Section 6, the design of PDC controllers for Yin and Yang T-S fuzzy synchronization of chaotic Henon maps is given by the linear invertible matrix theory. In Section 7, simulation results are given. In Section 8, conclusions are drawn.

Two-Dimensional Invertible Maps
Consider the general two-dimensional map form as follows: where x 1 n 1 1 , x 2 n 1 1 are the functions of x 1 n 1 , x 2 n 1 , n 1 is the positive argument, that is, n 1 0, 1, 2, 3, . . . , n. The map is invertible 17 if 2.1 can be solved uniquely for x 1 n 1 and x 2 n 1 as functions of x 1 n 1 1 and x 2 n 1 1 . The inversed map system can be written from 2.1 : where n 2 is the negative argument, that is, n 2 0, −1, −2, −3, . . . , −n. The chaos obtained for negative argument is called Yin chaos, while that obtained for positive argument is called Yang chaos.

The Yang Chaos of Henon Map System and Yin Chaos of Inversed Henon Map System
Consider the Henon map system 18 : x 1 n 1 1 −a 1 x 2 1 n 1 x 2 n 1 1,  Figure 1. The Henon map system is invertible if b 1 / 0. The inversed Henon map system can be described as follows:
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.

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: . . , and p q n is M iq , where i 1, 2, . . . , r r is the number of fuzzy rules , x n ∈ R j represents the state vector, A i ∈ R j×j are known matrix, p 1 n , p 2 n , . . . , p q n are premise variables, M ih is a fuzzy set h 1, 2, . . . , q , and c ∈ R j is a constant vector. The final discrete time T-S fuzzy system is inferred as follows: x n 1 Yang T-S fuzzy model of chaotic Henon map system:

5.1
Yin T-S fuzzy model of chaotic Henon map system:

5.7
The Yang T-S fuzzy model of chaotic Henon map system is given in Figure 3. It is clear to see that the derived Yang T-S fuzzy model is equivalent to the original chaotic map in Figure 1. Otherwise, the Yin T-S fuzzy model of chaotic Henon map system is given in Figure 4 and is equivalent to the original chaotic map in Figure 2.

Controller Design for Chaos Synchronization of Yin and Yang T-S Fuzzy Model of Henon Maps
Let Yang T-S fuzzy chaotic map 5.5 be as the drive system and the response system as follows: According to stability theory for discrete linear system, it is known that a matrix H if and only if the eigenvalues of matrix H are less than 1 in absolute value 20 , the error dynamics 6.3 is asymptotically stable, in other words, the chaos synchronization is achieved. Let a Schur stable matrix H 1 for Yang controller design satisfy the following equation: 6.5

Abstract and Applied Analysis
Since the system is the T-S fuzzy model of invertible map and the determinants of K 1 , K 2 are not equal to zero, the Yin fuzzy controllers meet the following equation: So the Schur matrix H 2 for Yin controller design satisfy the following equation: We need to find the K 1 , K 2 , K 1 , K 2 to meet 6.5 and 6.7 , that is, if all the eigenvalues of H 1 , H 2 are less than 1 in absolute value, the Yang and Yin T-S fuzzy chaotic Henon maps are synchronized individually. Because 6.7 is not the invertible of 6.5 the invertible of 6.5 is

Simulation Results
In order to apply the proposed Yin-Yang chaos synchronization scheme by PDC technique, the T-S fuzzy model is used. Yang T-S fuzzy model of Henon map system as drive system is described in 5.1 , 5.3 , 5.5 , and 5.7 . Yang T-S fuzzy model of Henon map system as response system is described by 6.1 and 6.2 :

7.3
Yang T-S fuzzy model of chaotic Henon maps as drive and response system is simulated with initial conditions Simulation results show that the error dynamics approach to be asymptotically stable in Figures 5 and 6. The Yin fuzzy controllers can be obtained by 6.6 : −0.857. Simulation results show that the error dynamics approach to be asymptotically stable in Figures 7 and 8. In other words, chaos synchronization is achieved.

Conclusions
Yin chaos of inversed Henon map system and the Yin T-S fuzzy model of chaotic Henon map system are proposed. The T-S fuzzy model of a chaotic system can exactly be represented as a fuzzy aggregation of some local linear systems. As a result, the conventional linear system theory can be applied. Since nonlinear discrete time Henon map is an invertible map, we develop the Yang and Yin T-S fuzzy model of Henon map system. The design of the Yin fuzzy controller is fleetly obtained by the inverse of Yang fuzzy controller, the concern is that the Yin chaotic system must be an inverse of the Yang chaotic system and meet the stability theory for discrete linear system. In secure communication, the chaotic system as transmitter and the inverse chaotic system as receiver are hard to achieve due to the nonlinearity; however the Yin and Yang T-S fuzzy models of Henon map system can overcome these difficulties.