The Relevant Theory Of (Intuitionistic) Fuzzy Sets And Its Application In Dynamical Systems

The concept of fuzzy topology may have very important applications in quan-tumparticle physics particularly in connection with both string and∈~∞theory. One of the most important problems in fuzzy topology is to obtain an appropriate concept of fuzzy metric space. As a generalization of fuzzy sets, the concept of intuitionistic fuzzy set has been studied by many authors, especially the structures of the intuitionistic fuzzy metric spaces and the (commonly) fixed point theorems, etc. Fuzzy systems were first discussed in 1965 by Zadeh. Since then, the potential application to systems theory has significantly motivated and influenced the direction of development of the theory of fuzzy sets. Fuzzy controllers have been used successfully in a variety of practical situations. However, they have been synthesized without recourse to an underlying theory. The introduction of fuzzy set theory into the field of dynamical systems theory has been motivated mainly by the needs for a theory of systems whose structure and/ or behavior involves uncertainties. The present papre is organized as follows:In Chapter One, the background and history of fuzzy topology and dynamical systems are briefly addressed.In Chapter Two, we establish a necessary and sufficient condition for metrization of the fuzzy topological spaces.In Chapter Three, we prove two common fixed point theorems for commuting mappings and compatible mappings of a fuzzy metric space, respectively. In addition, we obtain a common coincidence point theorem for nonfuzzy mapping and a sequence of fuzzy mappings in a linear metric space under the .R-weakly commuting condition.In Chapter Four, we first introduce the notion of fuzzy chaos. We also discuss topological transitivity, periodic orbits and sensitive dependence on initial conditions for fuzzy compact metric space. Secondly, we obtain a theorem for the existence and uniqueness of the solution of a fuzzy relational equation.In Chapter Five, we investigate the dynamical systems in the context of topological semigroups actions on intuitionistic fuzzy metric spaces. What is more,we offer a common fixed point theorem under the linear contractive condition in the setting of an intuitionistic fuzzy metric space. Moreover, we also give some common fixed point theorems under nonlinear contractive conditions for mappings in intuitionistic fuzzy symmetric spaces.In Chapter Six, we introduce the concepts of L-intuitionistic fuzzy point and I-remote neighborhood, respectively. And then, we turn to the investigation of Moore- Smith convergence in the intuitionistic fuzzy topological spaces.In Chapter Seven, we study the control of Hopf bifurcation for a plankton population model with a non-integer exponent of closure. Numerical simulations are provided to show the effectiveness and feasibility of the developed methods.In Chapter Eight, we present a systematic design procedure to synchronize the Chua's circuit without uncertain parameters based on back-stepping procedure. Furthermore, we propose an approach of adaptive synchronizing uncertain Chua's circuit. This approach can realize synchronization and parameters identification simultaneously. Some numerical simulation results are included to visualize the effectiveness and the feasibility of the developed methods.