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The 1960s were perhaps a decade of confusion, when scientists faced d- culties in dealing with imprecise information and complex dynamics. A new set theory and then an in?nite-valued logic of Lot? A. Zadeh were so c- fusing 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 gr- ually 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 ?elds, ranging from control, automation, and arti?cial intelligence to image/signal processing,patternrecognition,andelectroniccommerce.Chaos,ontheother hand,wasconsideredoneofthethreemonumentaldiscoveriesofthetwentieth 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 scienti?c curiosity in the mid-1960s to an applicable technology in the late 1990s. Finding the intrinsic relation between fuzzy logic and chaos theory is certainlyofsigni?cantinterestandofpotentialimportance.Thepast20years have indeed witnessed some serious explorations of the interactions between fuzzylogicandchaostheory,leadingtosuchresearchtopicsasfuzzymodeling 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.



Beyond the Li-Yorke Definition of Chaos

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.
Peter Kloeden, Zhong Li

Chaotic Dynamics with Fuzzy Systems

In this chapter a new approach for modeling chaotic dynamics is proposed. It is based on a linguistic description of chaotic phenomena, which can be easily related to a fuzzy system design. This approach allows building chaotic generators by means of few fuzzy sets and using a small number of fuzzy rules. It is also possible to create chaotic signals with assigned characteristics (e.g., Lyapunov exponents). Fuzzy descriptions of well-known discrete chaotic maps are therefore introduced, denoting an improved robustness to parameter changes.
Domenico M. Porto

Fuzzy Modeling and Control of Chaotic Systems

In this chapter, fuzzy modeling techniques based on Takagi-Sugeno (TS) fuzzy model are first proposed to model chaotic systems; then, a unified approach is presented for stabilization, synchronization, and chaotic model following control for the chaotic TS fuzzy systems using linear matrix inequality (LMI) technique; finally, illustrative examples are presented.
Hua O. Wang, Kazuo Tanaka

Fuzzy Model Identification Using a Hybrid mGA Scheme with Application to Chaotic System Modeling

In constructing a successful fuzzy model for a complex chaotic system, identification of its constituent parameters is an important yet difficult problem, which is traditionally tackled by a time-consuming trial-and-error process. In this chapter, we develop an automatic fuzzy-rule-based learning method for approximating the concerned system from a set of input—output data. The approach consists of two stages: (1) Using the hybrid messy genetic algorithm (mGA) together with a new coding technique, both structure and parameters of the zero-order Takagi—Sugeno fuzzy model are coarsely optimized. The mGA is well suited to this task because of its flexible representability of fuzzy inference systems: (2) The identified fuzzy inference system is then fine-tuned by the gradient descent method. In order to demonstrate the usefulness of the proposed scheme, we finally apply the method to approximating the chaotic Mackey—Glass equation.
Ho Jae Lee, Jin Bae Park, Young Hoon Joo

Fuzzy Control of Chaos

In this chapter a Mamdani fuzzy model based fuzzy control technique is proposed to control chaotic systems, whose dynamics is complex and unknown, to the unstable periodic orbits (UPO). Some empirical tricks are introduced for building up a proper fuzzy rule base and designing a fuzzy controller. Finally, an example of fuzzy control of the Chua’s circuit is presented to illustrate the effectiveness of the proposed approach.
Oscar Calvo

Chaos Control Using Fuzzy Controllers (Mamdani Model)

Controlling a strange attractor, or say, a chaotic attractor, is introduced in this chapter. Because of the importance to control the undesirable behavior in systems, researchers are investigating the use of linear and nonlinear controllers either to get rid of such oscillations (in power systems) or to match two chaotic systems (in secure communications). The idea of using the fuzzy logic concept for controlling chaotic behavior is presented. There are two good reasons for using the fuzzy control: first, mathematical model is not required for the process, and second, the nonlinear controller can be developed empirically, without complicated mathematics. The two systems are well-known models, so the first reason is not a big deal, but we can take advantage from the second reason.
Ahmad M. Harb, Issam Al-Smadi

Digital Fuzzy Set-Point Regulating Chaotic Systems: Intelligent Digital Redesign Approach

This chapter concerns digital control of chaotic systems represented by Takagi-Sugeno fuzzy systems, using intelligent digital redesign (IDR) technique. The term IDR involves converting an existing analog fuzzy set-point regulator into an equivalent digital counterpart in the sense of state-matching. The IDR problem is viewed as a minimization problem of norm distances between nonlinearly interpolated linear operators to be matched. The main features of the present method are that its constructive condition with global rather than local state-matching, for concerned chaotic systems, is formulated in terms of linear matrix inequalities; the stability property is preserved by the proposed IDR algorithm. A few set-point regulation examples of chaotic systems are demonstrated to visualize the feasibility of the developed methodology, which implies the safe digital implementation of chaos control systems.
Ho Jae Lee, Jin Bae Park, Young Hoon Joo

Anticontrol of Chaos for Takagi-Sugeno Fuzzy Systems

The current study on anticontrol of chaos for both discrete-time and continuous-time Takagi-Sugeno (TS) fuzzy systems is reviewed. To chaotifying discrete-time TS fuzzy systems, the parallel distributed compensation (PDC) method is employed to determine the structure of a fuzzy controller so as to make all the Lyapunov exponents of the controlled TS fuzzy system strictly positive. But for continuous-time ones, the chaotification approach is based on the fuzzy feedback linearization and a suitable approximate relationship between a time-delay differential equation and a discrete map. The time-delay feedback controller, chosen among several candidates, is a simple sinusoidal function of the delayed states of the system, which can have an arbitrarily small amplitude. These anticontrol approaches are all proved to be mathematically rigorous in the sense of Li and Yorke. Some examples are given to illustrate the effectiveness of the proposed anticontrol methodologies.
Zhong Li, Guanrong Chen, Wolfgang A. Halang

Chaotification of the Fuzzy Hyperbolic Model

In this chapter, the problem of chaotifying the continuous-time fuzzy hyperbolic model (FHM) is studied. We first use impulsive and nonlinear feedback control methods to chaotify the FHM and we show that the chaos produced by the present methods satisfy the three criteria of Devaney. We then design a controller based on inverse optimal control and adaptive parameter tuning methods to chaotify the FHM by tracking the dynamics of a chaotic system. Computer simulation results show that for any initial value the FHM can track a chaotic system asymptotically.
Huaguang Zhang, Zhiliang Wang, Derong Liu

Fuzzy Chaos Synchronization via Sampled Driving Signals

In this chapter the Tagaki-Sugeno fuzzy model representation of a chaotic system is used to find an alternative solution to the chaos synchronization problem. One of the advantages of the proposed approach is that it allows to express the synchronization problem as a fuzzy logic observer design in terms of linear matrix inequalities, which can be solved numerically using readily advailable software packages. Also, given the linear nature of this fuzzy representation, it is possible to use sophisticated methodologies to consider the more practical problem of digital implementation of a synchronization design. In particular, in this contribution the problem of a master-slave chaos synchronization design from sampled drive signals is considered and a solution is proposed as the state-matching digital redesign of the fuzzy logic observer designed to solve the continuous-time synchronization problem. The effectiveness of the proposed synchronization method is illustrated through numerical simulations of three well-known benchmark chaotic system, namely, Chua’s circuit, Chen’s equation, and the Duffing oscillator.
Juan Gonzalo Barajas-Ramírez

Bifurcation Phenomena in Elementary Takagi-Sugeno Fuzzy Systems

The relevance of bifurcation analysis in Takagi-Sugeno (T-S) fuzzy systems is emphasized mainly through examples. It is demonstrated that even the most simple cases can show a great variety of behaviors. Several local and global bifurcations (some of them, degenerate) are detected and summarized in the corresponding bifurcation diagrams. It is claimed that by carefully making this kind of analysis it is possible to overcome some criticism raised regarding the blind use of fuzzy systems.
Federico Cuesta, Enrique Ponce, Javier Aracil

Self-Reference, Chaos, and Fuzzy Logic

Self-reference and paradox introduce a spectrum of nonlinear phenomena in fuzzy logic. Working from the example of the Liar paradox, and using iterated functions to model self-reference, sentences can be constructed with the dynamical semantics of fixed-point attractors, fixed-point repellors, and full chaos on the [0,1] interval. The paper also extends the analysis to pairs and triples of mutually referential sentences, which generate strange attractors and semantic fractals in two and three dimensions.
Patrick Grim

Chaotic Behavior in Recurrent Takagi-Sugeno Models

We investigate dynamic systems which are modeled by recurrent fuzzy rule bases widely used in applications. The main question to be answered is “Under which conditions recurrent rule bases show chaotic behavior in the sense of Li-Yorke?” We determine the minimal number of rules of zero-order and first-order Takagi—Sugeno models with chaotic orbits. We also consider the case of an arbitrary number of rules in such models and high-order time delay case. This chapter is the first from a series of papers where we will consider arbitrary types of consequent functions, noncomplete or contradictory rule bases, vectors in the rule antecedents,
Alexander Sokolov, Michael Wagenknecht

Theory of Fuzzy Chaos for the Simulation and Control of Nonlinear Dynamical Systems

This chapter introduces the basic concepts of dynamical systems theory and several basic mathematical methods for controlling chaos. The main goal of this chapter is to provide an introduction to and a summary of the theory of dynamical systems, with particular emphasis on fractal theory, chaos theory, and chaos control. We first define what is meant by a dynamical system, then we define an attractor, and then the concept of the fractal dimension of a geometrical object. We also define the Lyapunov exponents as a measure of the chaotic behavior of a dynamical system. On the other hand, the fractal dimension can be used to classify geometrical objects because it measures the complexity of an object. The chapter also describes mathematical methods for controlling chaos in dynamic systems. These methods can be used to control a real dynamic system; however, due to efficiency and accuracy requirements we were forced to use fuzzy logic to model the uncertainty, which is present when numerical simulations are performed. We also describe in this chapter a new theory of chaos using fuzzy logic techniques. Chaotic behavior in nonlinear dynamical systems is very difficult to detect and control. Part of the problem is that mathematical results for chaos are difficult to use in many cases, and even if one could use them there is an underlying uncertainty in the accuracy of the numerical simulations of the dynamical systems. For this reason, we can model the uncertainty of detecting the range of values where chaos occurs, using fuzzy set theory. Using fuzzy sets, we can build a theory of fuzzy chaos, where we can use fuzzy sets to describe the behaviors of a system. We illustrate our approach with two cases: Chua’s circuit and Duffing’s oscillator.
Oscar Castillo, Patricia Melin

Complex Fuzzy Systems and Their Collective Behavior

This work aims at being a contribution for the characterization of a new class of complex systems built as arrays of coupled fuzzy logic based chaotic oscillators and an investigation on their collective dynamical features. Different experiments were carried out varying the parameters related to the single-unit dynamics, as Lyapunov exponent, and to the macrosystem structure, as the number of connections. Four types of global behaviors have been identified and characterized distinguishing their patterns as follows: the spatiotemporal chaos, the regular synchronized behavior, the transition phase, and the chaotic synchronized behavior. These collective behaviors and the synchronization capability have been highlighted by defining a mathematical indicator which weights the slight difference among a wide number of spatiotemporal patterns. To investigate the effects due to the network architecture on the synchronization characteristics, complex fuzzy systems have been reproduced using fuzzy chaotic cells connected through different topologies: regular, “small worlds,” and random.
Maide Bucolo, Luigi Fortuna, Manuela La Rosa

Real-Time Identification and Forecasting of Chaotic Time Series Using Hybrid Systems of Computational Intelligence

In this chapter, the problems of identification, modeling, and forecasting of chaotic signals are discussed. These problems are solved with the use of the conventional techniques of computational intelligence as radial basis neural networks and learning neuro-fuzzy architectures, as well as novel hybrid structures based on the Kolmogorov’s superposition theorem and using the neo-fuzzy neurons as elementary processing units. The need for the solution of the forecasting problem in real time poses higher requirements to the processing speed, so the considered hybrid structures can be trained with the proposed algorithms having high convergence rate and providing a compromise between the smoothing and tracking properties during the processing of nonstationary noisy signals.
Yevgeniy Bodyanskiy, Vitaliy Kolodyazhniy

Fuzzy-Chaos Hybrid Controllers for Nonlinear Dynamic Systems

Controlling of chaos is an interesting research topic while employing of deterministic chaos for controlling is more interesting. This chapter focuses on employing and utilizing of inherent chaotic features in a nonlinear dynamical system in a useful manner. When it comes to employing deterministic chaos, there are tremendous advantages such as low-energy consumption, robustness of the controller performance, information security, and simplicity of employing chaos whenever it has chaotic attractive features in the original systems itself. If the original system does not have chaotic properties, deterministic chaos will be introduced to the system. Keeping these objectives, the control algorithm is constructed in order to control nonlinear systems, which exhibit chaotic behavior. We introduce two phases of control: First phase uses open-loop control forming a chaotic attractor or using chaotic inherent features in a system itself. Fuzzy model based controller is employed under state feedback control in the second phase of control. The Henon map and the three-dimensional Lorenz attractor, which have chaotic attractive features in their original systems, are taken into consideration so as to utilize the benefits of chaos. Then, a two-link manipulator is considered to illustrate the design procedure with employing deterministic chaos. Simulation results show the effectiveness of the proposed controller.
Keigo Watanabe, Lanka Udawatta, Kiyotaka Izumi

Fuzzy Model Based Chaotic Cryptosystems

In this chapter, we address a fuzzy model based chaotic cryptosystem. For the crytosystem, the plaintext (message) is encrypted using the superincreasing sequence formed by chaotic signals at the drive system side. The resulting ciphertext is embedded to the output or state of the drive system and is sent to the response system end. The plaintext is retrieved via the synthesis approach for signal synchronization. We show that the chaotic synchronization problem can be solved using linear matrix inequalities. The advantages of this crytosystem are the systematic methodology of fuzzy model based design suitable for well-known Lure type discrete-time chaotic systems; flexibility in selection of chaotic signals for secure key generator; flexibility of masking the ciphertext using either the state or output; multiuser capabilities; and a time-varying superincreasing sequence. In light of the above advantages, the chaotic communication structure has a higher-level of security compared to traditional masking methods. In addition, numerical simulations and DSP-based experiments are carried out to verify the validity of theoretical results.
Chian-Song Chiu, Kuang-Yow Lian

Evolution of Complexity

The strength of physical science lies in its ability to explain phenomena as well as make prediction based on observable, repeatable phenomena according to known laws. Science is particularly weak in examining unique, nonrepeatable events. We try to piece together the knowledge of evolution with the help of biology, informatics, and physics to describe a complex evolutionary structure with unpredictable behavior. Evolution is a procedure where matter, energy, and information come together. Our research can be regarded as a natural extension of Darwin’s evolutionary view of the last century. We would like to find plausible uniformitarian mechanisms for evolution of complex systems. Workers with specialized training in overlapping disciplines can bring new insights to an area of study, enabling them to make original contributions. This chapter describes evolution of complexity as a basic principle of evolutionary computation.
Pavel Ošmera

Problem Solving via Fuzziness-Based Coding of Continuous Constraints Yielding Synergetic and Chaos-Dependent Origination Structures

Based on the comparison of artificial systems with natural systems to elucidate the differences of their characteristics, we will propose a framework of a double-layered architecture of a problem solving system for constraint satisfaction problems, where the upper layer has characteristics corresponding to the artificial systems and the lower layer has characteristics corresponding to the natural systems. These two layers are derived by “fuzzy coding” (coding by fuzziness) in order to “decompose” continuous constraints for problem reduction. Thereafter, two different approaches to problem solving by the layered system architecture are proposed. One way of problem solving is to make use of the synergetic and tacit of the known layered structure. The other way is to focus on the chaotic phenomena through the interaction between the two layers. Moreover, considerations are made on the cause and meaning of these chaos phenomena in order to suggest some directions to make good use of it.
Osamu Katai, Tadashi Horiuchi, Toshihiro Hiraoka

Some Applications of Fuzzy Dynamic Models with Chaotic Properties

In this paper an approach to recording and restoring of information is proposed. For this purpose the method for recording and information storing of nonlinear dynamic system trajectories is used. Such systems are based on Takagi—Sugeno recurrent models. Principles of recording, which allow us to reconstruct an arbitrary piece of the bit series, are proposed. Another application belongs to modeling of weak-formalized systems such as economic dynamic processes. We propose to consider the models of multiproject systems as a net of Mamdani rule-based recurrent models. The described approaches are implemented in a Matlab software environment. We investigate the chaotic properties of Mamdani recurrent rule bases as well. The result of investigation of the proposed algorithms are given and analyzed.
Alexander Sokolov
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