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Extensions of the well-known definition of chaos due to Li and Yorke for difference equations in ℝ are reviewed for difference equations in ℝ 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 (, ) of fuzzy sets on the base space ℝ, 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.

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.

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.

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.

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.

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.

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.

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.

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.

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 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.