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DOI: 10.1177/0954406212465236
1535 originally published online 29 October 2012
2013 227:
Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science
Chun-Yen Ho, Hsien-Keng Chen and Zheng-Ming Ge
Chaos synchronization of Yin and Yang T-S fuzzy models of Hénon map system
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Chaos synchronization of Yin and Yang T-S
fuzzy models of He´non map system
Chun-Yen Ho
1, Hsien-Keng Chen
2and Zheng-Ming Ge
1Abstract
Based on the Chinese philosophy, Yin is the decreasing, negative, historical or feminine principle in nature, while Yang is the increasing, positive, contemporary or masculine principle in nature. Yin and Yang are two fundamental opposites in Chinese philosophy. Since the discrete He´non map system is an invertible map, Yin–Yang T-S fuzzy model of chaotic He´non map systems with increasing and decreasing argument can be studied. The Yang T-S fuzzy model of chaotic He´non map system is presented. The Yin T-S fuzzy model of chaotic He´non map system and Yin T-S fuzzy rules are obtained by invertible matrix in linear system theory. Chaos synchronization of Yang T-S fuzzy model of He´non map systems is achieved by parallel distributed compensation technique, and the fuzzy controller for chaos synchronization of Yin T-S fuzzy model of He´non map systems is also obtained by invertible matrix in linear system theory. The design of the Yin fuzzy controller is fleetly obtained by the inverse of Yang fuzzy controller, the concern is the Yin chaotic system must be an inverse of the Yang chaotic system. T-S fuzzy model scheme is used in the nonlinear discrete chaotic map system, so the nonlinear discrete chaotic map system can be analyzed by linear system theory.
Keywords
Chinese philosophy, Yin chaos, Yang chaos, Yin–Yang chaotic He´non map system, inversed He´non map system, Yin–Yang T-S fuzzy model, chaos synchronization, inverse matrix theory
Date received: 25 February 2012; accepted: 19 September 2012
Introduction
Chaos is an interesting nonlinear phenomenon, has been intensively investigated in the last three dec-ades;1–7it has many characteristics, such as sensitive dependence on initial conditions and parameters, mixing randomness in the time domain, broadband power spectrum, ergodicity, etc.8 Chaos is a common phenomenon in discrete time nonlinear sys-tems9,10 such as He´non map,11–13 logistic map14–16 and generalized He´non map, where the generalized He´non map system has hyperchaotic behavior when it has one positive Lyapunov exponent more than He´non map system.17–20 In 1990, Pecora and Carroll21showed the possibility of chaotic synchron-ization and started a new research interest. The appli-cations of chaos synchronization have found in the fields of engineering and science such as in secure communications, chemical reactions, power con-verters, biological systems, and information process-ing, etc.22 In secure communication field, the main purpose for chaos and hyperchaos phenomenon in discrete time dynamical systems is to send a secret message from a transmitter to the receiver through a public channel and the secret message is sent safely when they are Synchronized.23,24 In many papers,
chaos synchronization25–29 of two-dimensional He´non map system are studied, of which the Yang chaos is also studied.30 In 1985, Takagi–Sugeno (T-S) fuzzy model was firstly proposed.31Recently, it has become quite popular to adopt T-S fuzzy models to represent a nonlinear system. T-S fuzzy model can be used for nonlinear system, the state spaces are repre-sented by linear models. So the conventional linear system theory can be applied to the analysis and syn-thesis of chaos synchronization.32–35
In this article, Yin chaos and the Yin T-S fuzzy model of the inverse He´non map system are firstly proposed. Since the He´non map system is an invert-ible map, the Yin T-S fuzzy model and Yin chaos synchronization of He´non map system are acquired by invertible matrix in linear system theory.
1
Department of Mechanical Engineering, National Chiao Tung University, Hsinchu, Taiwan
2
Department of Mechanical Engineering, Hsiuping University of Science and Technology, Dali City, Taichung, Taiwan
Corresponding author:
Zheng-Ming Ge, Department of Mechanical Engineering, National Chiao Tung University, Hsinchu, Taiwan.
Email: [email protected]
Proc IMechE Part C:
J Mechanical Engineering Science 227(7) 1535–1543
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The design of the Yin fuzzy controller can be fleetly obtained by the inverse of Yang fuzzy controller, the concern is the Yin chaotic system must be an inverse of the Yang chaotic system. The controller design by parallel distributed compensation (PDC) technology to achieve chaos synchronization of T-S fuzzy model of Yin He´non map system that can be acquired by the inverse of the controller for chaos synchronization of Yang He´non map system. It is more efficient than other approaches.
This article is organized as follows. In the following section, the two-dimensional invertible maps are introduced. In the next section, the Yin chaos of inversed He´non map system is presented and the Yang chaos of He´non map system is also shown for comparison. In the later section, T-S fuzzy model scheme is given. Further, Yin–Yang T-S fuzzy models of the chaotic He´non map systems and controller design by PDC technique for chaos syn-chronization of Yang T-S fuzzy model of He´non map system are presented, and a simple method for Yin fuzzy controller is given. In the final sections, numerical analysis is given and conclusions are drawn.
Two-dimensional invertible maps
Consider the general two-dimensional map form as follows
x1½n1þ1 ¼ f1 x1½n1, x2½n1
x2½n1þ1 ¼ f2 x1½n1, x21½n1
ð1Þ
where x1½n1þ1, x2½n1þ1 are the functions of x1½n1,
x2½n1, n1 is the positive argument, i.e. n1¼0, 1, 2,
3, . . . , n. The map is invertible36 if equation (1) can be solved uniquely for x1½n1, and x2½n1as functions
of x1½n1þ1 and x2½n1þ1. The inversed map system
can be written from equation (1) ^ x1½n21 ¼ g1 x^1½n2, ^x2½n2 ^ x2½n21 ¼ g2 x^1½n2, ^x2½n2 ð2Þ
where n2 is the negative argument, i.e. n2¼0, 1,
2, 3, . . . , n. The chaos obtained for negative argument is called Yin chaos, while that obtained for positive argument is called Yang chaos. Yin chaos for map system is studied in this article firstly. In Ge and Li,37 the chaos obtained for negative time for differential equations is also called Yin chaos, while that obtained for positive time for differential equations is called Yang chaos.
Consider the He´non map system30 x1½n1þ1 ¼ a1x12½n1 þx2½n1 þ1
x2½n1þ1 ¼ b1x1½n1
ð3Þ The He´non map system is invertible if b1 6¼0.
The inversed He´non map system can be written by
equation (3) ^ x1½n21 ¼ 1 b1 ^ x2½n2 ^ x2½n21 ¼ ^x1½n2 þ a1 b2 1 ^ x22½n2 1 ð4Þ
The Yang chaos of He´non map system
and Yin chaos of inversed He´non
map system
The system parameters for equation (3) are: a1¼1:4,
b1 ¼0:3. The Yang chaotic behaviors of He´non map
system with a1¼1:4, b1¼0:3 are quoted38by phase
portrait in Figure 1.
The system parameters for equation (4) are replaced as b2¼b11, a2 ¼ab12
1
. The Yin chaotic behaviors of inversed He´non map system with b2¼0:3, a2¼1:4
are shown by the phase portrait in Figure 2.
Comparing Figures 1 and 2, it is surprisingly noted that Figure 2 gives many new informations for famous He´non system. Since 1976,30 traditional stu-dies of He´non system have only devoted to its behav-iors with positive argument (Yang chaos). Now it is discovered that with negative argument or negative time (Yin chaos), a new continent is waiting for us in the future study of either nonlinear map systems or nonlinear continuous systems.
T-S fuzzy model
T-S fuzzy model was given by Takagi and Sugeno.31 The T-S fuzzy model can represent a general non-linear system and the T-S fuzzy rules are obtained by IF-THEN fuzzy rules.
Consider the following discrete time T-S fuzzy rules
Ri: IF p1ðnÞ is Mi1, . . . , and pqðnÞ is Miq;
THEN xðn þ 1Þ ¼ AixðnÞ þ c,
ð5Þ where i ¼ 1, 2, . . . , r (r is the number of fuzzy rules), xðnÞ 2 Rj represents the state vector, A
i2Rjj are
known matrix, p1ðnÞ, p2ðnÞ, . . . , pqðnÞare premise
vari-ables, Mih is a fuzzy set (h ¼ 1, 2, . . . , q), c 2 Rj is a
constant vector. Equation (5) represents local linear models by T-S fuzzy rules. The final discrete time T-S fuzzy system is inferred by the fuzzy strategies of singleton fuzzier, product fuzzy inference and weighted average defuzzier as follows
xðn þ1Þ ¼ Pr i¼1!iðpðnÞÞðAixðnÞ þ cÞ Pr i¼1!iðpðnÞÞ ð6Þ where !iðpðnÞÞ ¼ Yq h¼1 MihðphðnÞÞ ð7Þ
in which MihðphðnÞÞ is the degree of membership of
phðnÞin Mih. The following conditions must be satisfied
Pr i¼1 !iðpðnÞÞ4 0, !iðpðnÞÞ50, 8 < : i ¼1, 2, . . . , r: ð8Þ Let iðpðnÞÞ ¼ !iðpðnÞÞ=Pri¼1!iðpðnÞÞ, equation (6)
can be rewritten as xðn þ1Þ ¼X r i¼1 iðpðnÞÞðAixðnÞ þ cÞ ð9Þ Note that Pr i¼1 iðpðnÞÞ ¼1, iðpðnÞÞ50, 8 < : i ¼1, 2, . . . , r: ð10Þ
Yin and Yang T-S fuzzy models of the
chaotic He´non map system
From the phase portraits of Yang He´non map system in Figure 1, we can get the range of the state x1ðnÞ is
Figure 2. Phase portrait of the chaotic Yin He´non map system with a2¼1:4, b2¼0:3. Figure 1. Phase portrait of the chaotic Yang He´non map system with a1¼1:4, b1¼0:3.
from 1.4 to 1.4 (x1ðnÞ 2 ½1:4, 1:4). The Yang T-S
fuzzy model of chaotic He´non map system can be obtained as follows R1: IF x1ðn1Þis M11, THEN xðn1þ1Þ ¼ A1xðn1Þ þc R2: IF x1ðn1Þis M21, THEN xðn1þ1Þ ¼ A2xðn1Þ þc ð11Þ where xðn1Þ ¼ ðx1ðn1Þ, x2ðn1ÞÞT, n1¼0, 1, 2, 3, . . . , n, A1¼ ha 1ð1:4Þ 1 b1 0 i , A2¼t ha 1ð1:4Þ 1 b1 0 i , c ¼ ð1, 0ÞT.
The membership functions are chosen as M11ðx1ðn1ÞÞ ¼ 1 2 1 þ x1ðn1Þ 1:4 ð12Þ M21ðx1ðn1ÞÞ ¼ 1 2 1 x1ðn1Þ 1:4 ð13Þ Then, applying the product-inference rule, single-ton fuzzifier, and the center of gravity defuzzifier to the above fuzzy rule base, the overall Yang fuzzy cha-otic map can be inferred as
xðn1þ1Þ ¼ X2 i¼1 iðx1ðn1ÞÞðAixðn1Þ þcÞ ð14Þ where iðx1ðn1ÞÞ ¼ Mi1ðx1ðn1ÞÞ P2 i¼1 ðMi1ðx1ðn1ÞÞÞ ð15Þ
The Yang T-S fuzzy model of chaotic He´non map system is given in Figure 3. From Figure 3, it is clearly to see that the derived Yang T-S fuzzy model is equivalent to the original chaotic map in Figure 1.
Since the He´non map system is a nonlinear invert-ible map and the Yang T-S fuzzy model of the chaotic map system in different state space regions are repre-sented by linear model, the Yin T-S fuzzy model of chaotic He´non map system can be inferred by inverse matrix in linear system theory as follows
^ A1¼A11 ¼ 1 detðA1Þ 0 1 b1 a1ð1:4Þ ¼ 0 1 b1 1 a1 b1 ð1:4Þ 2 6 6 4 3 7 7 5¼ 0 b2 1 a2ð1:4Þ ð16Þ ^ A2¼A12 ¼ 1 detðA2Þ 0 1 b1 a1ð1:4Þ ¼ 0 1 b1 1 a1 b1 ð1:4Þ 2 6 6 4 3 7 7 5¼ 0 b2 1 a2ð1:4Þ ð17Þ where b2¼b11, a2¼ab11 .
From the matrix ^A1, ^A2, and phase portraits of Yin
He´non map system (x2ðnÞ 2 ½1:4, 1:4) in Figure 2,
the Yin T-S fuzzy rules are as follows R1: IF ^x2ðn2Þis ^M11,
THEN ^xðn21Þ ¼ ^A1xðn^ 2Þ þ^c,
R2: IF ^x2ðn2Þis ^M21,
THEN ^xðn21Þ ¼ ^A2xðn^ 2Þ þ^c,
ð18Þ
Figure 3. Phase portrait of Yang T-S fuzzy model of He´non map system with a1¼1:4, b1¼0:3.
where ^
xðn2Þ ¼ ðx^1ðn2Þ, ^x2ðn2ÞÞT,
n2¼0, 1, 2, 3, . . . , n:
The membership functions are chosen as ^ M11ðx^2ðn2ÞÞ ¼ 1 2 1 þ ^ x2ðn2Þ 1:4 ð19Þ ^ M11ðx^2ðn2ÞÞ ¼ 1 2 1 ^ x2ðn2Þ 1:4 ð20Þ
The Yin T-S fuzzy model of chaotic He´non map system is given in Figure 4 and is equivalent to the original chaotic map in Figure 2.
Controller design for chaos
synchroniza-tion of Yin and Yang T-S fuzzy model of
He´non map system
Let equation (14) as the drive system and the response system as follows
yðn þ1Þ ¼X
2
i¼1
iðy1ðnÞÞðAiyðnÞ þ cÞ þ uðnÞ ð21Þ
The target for the design of a fuzzy controller uðnÞ is synchronized the two discrete time chaotic map sys-tems by the PDC technique as follows
uðnÞ¼X 2 i¼1 iðx1ðnÞÞKixðnÞ X2 i¼1 iðy1ðnÞÞKiyðnÞ ð22Þ
Define the error dynamics
eðnÞ ¼ xðnÞ yðnÞ ð23Þ From equations (21) and (22), the closed-loop syn-chronization error dynamics is arranged as
eðn þ1Þ ¼X 2 i¼1 iðx1ðnÞÞðAiþKiÞxðnÞ X 2 i¼1 iðy1ðnÞÞðAiþKiÞyðnÞ ð24Þ
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,39the error dynamics (equation (23)) is asymp-totically stable, in other words, the chaos synchron-ization is achieved. Let a Schur stable matrix H satisfied the following equation
A1þK1¼A2K2¼H ð25Þ
From equation (25), equation (24) can be rewritten as follows eðn þ1Þ ¼X 2 i¼1 iðx1ðnÞÞHxðnÞ X2 i¼1 iðy1ðnÞÞHyðnÞ ð26Þ Since the system is an invertible map, the Yin fuzzy controller can be obtained by invertible matrix the-orem as follows
^
K1¼K11 , ^K2¼K12 ð27Þ
Simulation results
In order to apply the proposed Yin–Yang chaos syn-chronization scheme by PDC technique, the T-S fuzzy model is used. Yang T-S fuzzy model of He´non map system as drive system is described in equations (11) to (15). Yang T-S fuzzy model of He´non map sys-tem as response syssys-tem is described by equations (21) and (22)
yðn1þ1Þ ¼
X2 i¼1
iðy1ðn1ÞÞðAiyðn1Þ þcÞ þ uðn1Þ
where A1¼ a1ð1:4Þ 1 b1 0 " # , A2¼ a1ð1:4Þ 1 b1 0 , c ¼ ½1, 0T, and uðnÞ ¼X 2 i¼1 iðx1ðn1ÞÞKixðn1Þ X2 i¼1 iðy1ðn1ÞÞKiyðn1Þ
According to the proposed Yin–Yang chaos syn-chronization by PDC technique, the stable matrix H are chosen as
H ¼ 0:27 0 0 0:1 " #
The Yang fuzzy controller can be obtained from equation (25) as follows K1¼ 2:23 1 0:3 0:1 , K2¼ 1:69 1 0:3 0:1 ,
Yang T-S fuzzy model of chaotic He´non map system as drive system and Yang T-S fuzzy model of chaotic He´non map system as response system is simulated with the parameters a1¼1:4, b1¼0:3,
and the initial conditions ½x1ð0Þ x2ð0Þ T¼
½0:63 0:19 T, ½ y1ð0Þ y2ð0Þ T¼ ½0:4 0:1 T.
Simulation results show that the error dynamics approach to asymptotically stable in Figures 5 and 6. The chaos synchronization of Yang T-S fuzzy model of He´non map system is achieved by PDC technique. The discrete time chaotic He´non map system is a nonlinear invertible map, the Yin– Yang T-S fuzzy model of chaotic He´non map system can be represented as a fuzzy aggregation of some local linear systems. So linear theory adopt to the T-S fuzzy model of chaotic He´non map system, the Yin–Yang chaotic He´non map systems are invertible systems for each other, the Yin fuzzy controller can be obtained by invertible matrix theorem by equation (27). ^ K1¼K11 ¼ 1:29 12:98 3:89 28:96 , ^ K2¼K12 ¼ 0:76 7:63 2:29 11:14
Figure 5. Error dynamics e1for chaos synchronization of Yang T-S fuzzy model of He´non map system.
Simulation results show that the error dynamics approach to asymptotically stable in Figures 7 and 8. The chaos synchronization of Yin T-S fuzzy model of He´non map system is achieved by PDC technique.
Conclusions
Yin chaos of inversed He´non map system and the Yin T-S fuzzy model of chaotic He´non map system are
firstly proposed. The T-S fuzzy model of a chaotic system can exactly be represented as a fuzzy aggrega-tion of some local linear systems. As a result, the conventional linear system theory can be applied. Since nonlinear discrete time He´non map is an invert-ible map, we develop the Yang T-S fuzzy model of He´non map system, then the invertible matrix the-orem in linear system is used to get the Yin T-S fuzzy model of He´non map system. Correspondingly, the Yin fuzzy controller is designed by invertible matrix
Figure 6. Error dynamics e2for chaos synchronization of Yang T-S fuzzy model of He´non map system.
theorem. The design of the Yin fuzzy controller is fleetly obtained by the inverse of Yang fuzzy control-ler, the concern is that the Yin chaotic system must be an inverse of the Yang chaotic system. According PDC technology, if the control parameters make Schur matrix stable, and the other set of system par-ameters have chaotic behavior, the chaos synchron-ization is achieved even if the Yin and Yang T-S fuzzy models of He´non map system include the small par-ameters. In secure communication, the chaotic system as transmitter and the inverse chaotic system as recei-ver are hard to achieve due to the nonlinearity, the Yin and Yang T-S fuzzy models of He´non map system can overcome these difficulties.
Funding
This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.
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