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Zhong Li Wolfgang A Halang Guanrong Chen (Eds.) Integration of Fuzzy Logic and Chaos Theory ABC Dr Zhong Li Professor Wolfgang A Halang FernUniversität in Hagen FB Elektrotechnik Postfach 940, 55084 Hagen Germany E-mail: zhong.li@fer_n_uni-hagen.de wolfgang.halang@fernuni-hagen.de Professor Guanrong Chen Department of Electronic Engineering City University of Hong Kong Tat Chee Avenue, Kowloon Hong Kong/PR China E-mail: gchen@ee.cityu.edu.hk Library of Congress Control Number: 2005930453 ISSN print edition: 1434-9922 ISSN electronic edition: 1860-0808 ISBN-10 3-540-26899-5 Springer Berlin Heidelberg New York ISBN-13 978-3-540-26899-4 Springer Berlin Heidelberg New York This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlm or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer Violations are liable for prosecution under the German Copyright Law Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2006 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, etc in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use Typesetting: by the authors and TechBooks using a Springer LATEX macro package Printed on acid-free paper SPIN: 11353379 89/TechBooks 543210 Preface The 1960s were perhaps a decade of confusion, when scientists faced diﬃculties in dealing with imprecise information and complex dynamics A new set theory and then an inﬁnite-valued logic of Lotﬁ A Zadeh were so confusing that they were called fuzzy set theory and fuzzy logic; a deterministic system found by E N Lorenz to have random behaviours was so unusual that it was lately named a chaotic system Just like irrational and imaginary numbers, negative energy, anti-matter, etc., fuzzy logic and chaos were gradually and eventually accepted by many, if not all, scientists and engineers as fundamental concepts, theories, as well as technologies In particular, fuzzy systems technology has achieved its maturity with widespread applications in many industrial, commercial, and technical ﬁelds, ranging from control, automation, and artiﬁcial intelligence to image/signal processing, pattern recognition, and electronic commerce Chaos, on the other hand, was considered one of the three monumental discoveries of the twentieth century together with the theory of relativity and quantum mechanics As a very special nonlinear dynamical phenomenon, chaos has reached its current outstanding status from being merely a scientiﬁc curiosity in the mid-1960s to an applicable technology in the late 1990s Finding the intrinsic relation between fuzzy logic and chaos theory is certainly of signiﬁcant interest and of potential importance The past 20 years have indeed witnessed some serious explorations of the interactions between fuzzy logic and chaos theory, leading to such research topics as fuzzy modeling of chaotic systems using Takagi–Sugeno models, linguistic descriptions of chaotic systems, fuzzy control of chaos, and a combination of fuzzy control technology and chaos theory for various engineering practices A deep-seated reason to study the interactions between fuzzy logic and chaos theory is that they are related at least within the context of human reasoning and information processing In fact, fuzzy logic resembles human approximate reasoning using imprecise and incomplete information with inaccurate and even self-conﬂicting data to generate reasonable decisions under such uncertain environments, while chaotic dynamics play a key role in human brains for processing massive amounts of information instantly It is believed that the capability of humans in controlling chaotic dynamics in their brains is more than just an accidental by-product of the brain’s complexity, but VI Preface rather, it could be the chief property that makes the human brain diﬀerent from any artiﬁcial-intelligence machines It is also believed that to understand the complex information processing within the human brain, fuzzy data and fuzzy logical inference are essential, since precise mathematical descriptions of such models and processes are clearly out of question with today’s limited scientiﬁc knowledge With this book we attempt to present some current research progress and results on the interplay of fuzzy logic and chaos theory More speciﬁcally, in this book we collect some state-of-the-art surveys, tutorials, and application examples written by some experts working in the interdisciplinary ﬁelds overlapping fuzzy logic and chaos theory The content of the book covers fuzzy deﬁnition of chaos, fuzzy modeling and control of chaotic systems using both Mamdani and Takagi–Sugeno models, fuzzy model identiﬁcation using genetic algorithms and neural network schemes, bifurcation phenomena and self-referencing in fuzzy systems, complex fuzzy systems and their collective behaviors, as well as some applications of combining fuzzy logic and chaotic dynamics, such as fuzzy–chaos hybrid controllers for nonlinear dynamic systems, and fuzzy model based chaotic cryptosystems It is our hope that this book can serve as a handy reference for researchers working in the interdisciplines related, among others, to both fuzzy logic and chaos theory We would like to thank all authors for their signiﬁcant contributions, without which the publication of this book would have not been possible We are very grateful to Prof Janusz Kacprzyk for recommending this book to the Springer series, Studies in Fuzziness and Soft Computing, with appreciation going to the editorial and production staﬀ of Springer-Verlag in Heidelberg for their ﬁne work and kind cooperation May 2005 Zhong Li Wolfgang A Halang Guanrong Chen Contents Beyond the Li–Yorke Deﬁnition of Chaos Peter Kloeden and Zhong Li Chaotic Dynamics with Fuzzy Systems Domenico M Porto 25 Fuzzy Modeling and Control of Chaotic Systems Hua O Wang and Kazuo Tanaka 45 Fuzzy Model Identiﬁcation Using a Hybrid mGA Scheme with Application to Chaotic System Modeling Ho Jae Lee, Jin Bae Park, and Young Hoon Joo 81 Fuzzy Control of Chaos Oscar Calvo 99 Chaos Control Using Fuzzy Controllers (Mamdani Model) Ahmad M Harb and Issam Al-Smadi 127 Digital Fuzzy Set-Point Regulating Chaotic Systems: Intelligent Digital Redesign Approach Ho Jae Lee, Jin Bae Park, and Young Hoon Joo 157 Anticontrol of Chaos for Takagi–Sugeno Fuzzy Systems Zhong Li, Guanrong Chen, and Wolfgang A Halang 185 Chaotiﬁcation of the Fuzzy Hyperbolic Model Huaguang Zhang, Zhiliang Wang, and Derong Liu 229 Fuzzy Chaos Synchronization via Sampled Driving Signals Juan Gonzalo Barajas-Ram´ırez 259 Bifurcation Phenomena in Elementary Takagi–Sugeno Fuzzy Systems Federico Cuesta, Enrique Ponce, and Javier Aracil 285 VIII Contents Self-Reference, Chaos, and Fuzzy Logic Patrick Grim 317 Chaotic Behavior in Recurrent Takagi–Sugeno Models Alexander Sokolov and Michael Wagenknecht 361 Theory of Fuzzy Chaos for the Simulation and Control of Nonlinear Dynamical Systems Oscar Castillo and Patricia Melin 391 Complex Fuzzy Systems and Their Collective Behavior Maide Bucolo, Luigi Fortuna, and Manuela La Rosa 415 Real-Time Identiﬁcation and Forecasting of Chaotic Time Series Using Hybrid Systems of Computational Intelligence Yevgeniy Bodyanskiy and Vitaliy Kolodyazhniy 439 Fuzzy–Chaos Hybrid Controllers for Nonlinear Dynamic Systems Keigo Watanabe, Lanka Udawatta, and Kiyotaka Izumi 481 Fuzzy Model Based Chaotic Cryptosystems Chian-Song Chiu and Kuang-Yow Lian 507 Evolution of Complexity Pavel Oˇsmera 527 Problem Solving via Fuzziness-Based Coding of Continuous Constraints Yielding Synergetic and Chaos-Dependent Origination Structures Osamu Katai, Tadashi Horiuchi, and Toshihiro Hiraoka 579 Some Applications of Fuzzy Dynamic Models with Chaotic Properties Alexander Sokolov 603 Beyond the Li–Yorke Deﬁnition of Chaos Peter Kloeden and Zhong Li Abstract Extensions of the well-known deﬁnition of chaos due to Li and Yorke for diﬀerence equations in R1 are reviewed for diﬀerence equations in Rn with either a snap-back repeller or saddle point as well as for mappings in Banach spaces and complete metric spaces A further extension applicable to mappings in a space of fuzzy sets, namely the metric space (ξ n , D) of fuzzy sets on the base space Rn , is then discussed and some illustrative examples are presented The aim is to provide a theoretical foundation for further studies on the interaction between fuzzy logic and chaos theory Introduction Chaos may well be considered together with relativity and quantum mechanics as one of the three monumental discoveries of the twentieth century Over the past four decades chaos has matured as a science (though is still evolving) and has given us deep insights into previously intractable and inherently nonlinear natural phenomena The term chaos associated with an interval map was ﬁrst formally introduced into mathematics by Li and Yorke in 1975 [1], where they established a simple criterion for chaos in one-dimensional diﬀerence equations, i.e., the well-known “period three implies chaos.” There is, however, still no uniﬁed, universally accepted, and rigorous mathematical deﬁnition of chaos in the scientiﬁc literature to provide a fundamental basis for studying such exotic phenomena Various alternative, but closely related deﬁnitions of chaos have been proposed, among which those of Li–Yorke and Devaney seem to be the most popular Consider a one-dimensional discrete dynamical system [1, 2]: xk+1 = f (xk ), k = 0, 1, 2, , (1) where xk ∈ J (an interval) and f : J → J is a continuous mapping For x ∈ J, f (x) denotes x, while f n+1 (x) denotes f (f n (x)) for n = 0, 1, 2, A point x∗ is called a period point with period n (or an n-period point) if x∗ ∈ J and x∗ = f n (x∗ ) but x∗ = f k (x∗ ) for ≤ k < n and if n = 1, then x∗ = f (x∗ ) is called a ﬁxed point A point x∗ is said to be periodic or P Kloeden and Z Li: Beyond the Li–Yorke Deﬁnition of Chaos, StudFuzz 187, 1–23 (2006) c Springer-Verlag Berlin Heidelberg 2006 www.springerlink.com P Kloeden and Z Li is called a periodic point if it is an n-periodic point for some n ≥ With this terminology, Li and Yorke introduced the ﬁrst mathematical deﬁnition of chaos and established a very simple criterion, i.e., “period three implies chaos” for its existence This criterion, which plays an key role in predicting and analyzing one-dimensional chaotic dynamic systems, was described by Li and Yorke as follows: Theorem (Li–Yorke Theorem) Let J be an interval and f : J → J be continuous Assume that there is one point a ∈ J, for which the points b = f (a), c = f (a), and d = f (a) satisfy d ≤ a < b < c (or d ≥ a > b > c) Then (i) for every k = 1, 2, , there is a k-periodic point in J (ii) there is an uncountable set S ⊂ J, containing no periodic points, which satisﬁes the following conditions: (a) For every ps , qs ∈ S with ps = qs , lim sup |f n (ps ) − f n (qs )| > n→∞ and lim inf |f n (ps ) − f n (qs )| = n→∞ (b) For every ps ∈ S and periodic points qper ∈ J, with ps = qper , lim sup |f n (ps ) − f n (qper )| > n→∞ The set S in part (a) of conclusion (ii) was called a a scrambled set by Li and Yorke The ﬁrst part of the Li–Yorke theorem is, in fact, a special case of Sharkovsky’s theorem [3], which was proved by the Ukrainian mathematician A.N Sharkovsky in 1964 It is, however, the second part of the Li–Yorke theorem that thoroughly unveils the nature and characteristics of chaos, speciﬁcally, the sensitive dependence on initial conditions and the resulting unpredictable nature of the long-term behavior of the dynamics In 1978 F.R Marotto generalized the Li–Yorke theorem to higher dimensional discrete dynamical systems [4] He proved that if a diﬀerence equation in Rn has a snap-back repeller, then it has a scrambled set similar to that deﬁned in the Li–Yorke theorem and thus exhibits chaotic behavior Consider the following n-dimensional system: xk+1 = f (xk ), k = 0, 1, 2, , (2) where xk ∈ Rn and f : Rn → Rn is a continuous mapping, which is usually nonlinear Denote by Br (x) the closed ball in Rn of radius r centered at point x, and by Br0 (x) its interior Also, let x be the usual Euclidean norm of x in Rn Then, assuming f to be diﬀerentiable in Br (x), Marotto claimed that the logical relationship A ⇒ B (⇒ means “implying”) holds, where Beyond the Li–Yorke Deﬁnition of Chaos (a) all eigenvalues of the Jacobian Df (z) of system (2) at the ﬁxed point z = f (z) are greater than in norm (b) there exist some s > and r > such that f (x) − f (y) > s x − y for all x, y ∈ Br (z) In other words, if (a) is satisﬁed, then (b) also holds, i.e., f is expanding in Br (z) Then, Marotto introduced the following concepts Deﬁnition (Marotto Deﬁnitions) (1) Expanding ﬁxed point: Let f be diﬀerentiable in Br (z) The point z ∈ Rn is an expanding ﬁxed point of f in Br (z) if f (z) = z and all eigenvalues of Df (x) exceed in norm for all x ∈ Br (z) (2) Snap-back repeller: Assume that z is an expanding ﬁxed point of f in Br (z) for some r > Then z is said to be a snap-back repeller of f if there exists a point x0 ∈ Br (z) with x0 = z, f M (x0 ) = z and the determinant |Df M (x0 )| = for some positive integer M Marotto showed that the presence of a snap-back repeller is a suﬃcient criterion for the existence of chaos [4] Theorem (Marotto Theorem) If f possesses a snap-back repeller, then system (2) is chaotic in the following generalized sense of Li–Yorke: (i) There is a positive integer N such that for each integer p ≥ N , f has a point of period p (ii) There is a “scrambled set” of f , i.e., an uncountable set S containing no periodic points of f , such that (a) f (S) ⊂ S (b) for every xs , ys ∈ S with xs = ys , lim sup f k (xs ) − f k (ys ) > k→∞ (c) for every xs ∈ S and any periodic point yper of f , lim sup f k (xs ) − f k (yper ) > k→∞ (iii) There is an uncountable subset S0 of S such that for every x0 , y0 ∈ S0 : lim inf f k (x0 ) − f k (y0 ) = k→∞ It is apparent that the existence of a snap-back repeller for the onedimensional mapping f is equivalent to the existence of a point of period-3 for the map f n for some positive integer n, see [4] Unfortunately, two counterexamples have been given in [2, 5] to show that A ⇒ B is not necessarily true Since the Marotto theorem is based on the concept of “snap-back repeller,” which was introduced from the assertion of A ⇒ B, there exists an error in the proof given by Marotto Recently, an P Kloeden and Z Li improved and corrected version of Marotto’s theorem was given by Li and Chen [2], where the essential meanings of the two concepts of an expanding ﬁxed point and a snap-back repeller of continuously diﬀerentiable maps in Rn are clearly explained For an earlier generalization of the Marroto theorem see [6] and for an extension to maps in metric spaces see [7] as well as below More generally, Devaney [8] calls a continuous map f : X → X in a metric space (X, d) chaotic on X, if (i) f is transitive on X: for any pair of nonempty open sets U, V ⊂ X, there exists an integer k > such that f k (U ) ∩ V is nonempty; (ii) the periodic points of f are dense in X; (iii) f has sensitive dependence on initial conditions: if there exists a δ > such that for any x ∈ X and for any neighborhood D of x, there exists a y ∈ D and an k ≥ such that d(f k (x), f k (y)) > δ It has been observed that conditions (i) and (ii) in this deﬁnition imply condition (iii) if X is not a ﬁnite set [9] and that condition (i) implies conditions (ii) and (iii) if X is an interval [10] Hence, condition (iii) is in fact redundant in the above deﬁnition For continuous time nonlinear autonomous systems it is much more diﬃcult to give a mathematically rigorous proof to the existence of chaos Even one of the classic icons of modern nonlinear dynamics, the Lorenz attractor, now known for 40 years, was not proved rigorously to be chaotic until 1999 Warwick Tucker of the University of Uppsala showed in his Ph.D dissertation [11, 12], using normal form theory and careful computer simulations, that Lorenz equations indeed possess a robust chaotic attractor A commonly agreed analytic criterion for proving the existence of chaos in continuous time systems is based on the fundamental work of Shil’nikov, known as the Shil’nikov method or Shil’nikov criterion [13], whose role is in some sense equivalent to that of the Li–Yorke deﬁnition in the discrete setting The Shil’nikov criterion guarantees that complex dynamics will occur near homoclinicity or heteroclinicity when an inequality (Shil’nikov inequality) is satisﬁed between the eigenvalues of the linearized ﬂow around the saddle point(s), i.e., if the real eigenvalue is larger in modulus than the real part of the complex eigenvalue Complex behavior always occurs when the saddle set is a limit cycle In this chapter we focus on discrete-time systems and discuss generalizations of Marotto’s work, which are applicable to ﬁnite dimensional diﬀerence equations with saddle points as well as to those with repellers and to mappings in Banach spaces and in complete metric spaces including mappings from a metric space of fuzzy sets into itself ... Anticontrol of Chaos for Takagi–Sugeno Fuzzy Systems Zhong Li, Guanrong Chen, and Wolfgang A Halang 185 Chaotiﬁcation of the Fuzzy Hyperbolic Model Huaguang Zhang, Zhiliang Wang, and. .. interdisciplinary ﬁelds overlapping fuzzy logic and chaos theory The content of the book covers fuzzy deﬁnition of chaos, fuzzy modeling and control of chaotic systems using both Mamdani and Takagi–Sugeno...Dr Zhong Li Professor Wolfgang A Halang FernUniversität in Hagen FB Elektrotechnik Postfach 940, 55084 Hagen Germany E-mail: zhong .li@ fer_n_uni-hagen.de wolfgang .halang@ fernuni-hagen.de Professor
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