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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.

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.

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,

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.

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 , the , the , and the . 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.

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.

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.

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.

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.

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.