Dynamic Systems on Measure Chains - B. Kaymakcalan

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Résumé :
From a modelling point of view, it is more realistic to model a phenomenon by a dynamic system which incorporates both continuous and discrete times, namely, time as an arbitrary closed set of reals called time-scale or measure chain. It is therefore natural to ask whether it is possible to provide a framework which permits us to handle both dynamic systems simultaneously so that one can get some insight and a better understanding of the subtle differences of these two different systems. The answer is affirmative, and recently developed theory of dynamic systems on time scales offers the desired unified approach. In this monograph, we present the current state of development of the theory of dynamic systems on time scales from a qualitative point of view. It consists of four chapters. Chapter one develops systematically the necessary calculus of functions on time scales. In chapter two, we introduce dynamic systems on time scales and prove the basic properties of solutions of such dynamic systems. The theory of Lyapunov stability is discussed in chapter three in an appropriate setup. Chapter four is devoted to describing several different areas of investigations of dynamic systems on time scales which will provide an exciting prospect and impetus for further advances in this important area which is very new. Some important features of the monograph are as follows: It is the first book that is dedicated to a systematic development of the theory of dynamic systems on time scales which is of recent origin. It demonstrates the interplay of the two different theories, namely, the theory of continuous and discrete dynamic systems, when imbedded in one unified framework. It provides an impetus to investigate in the setup of time scales other important problems which might offer a better understanding of the intricacies of a unified study.?/LIST? Audience: Thereadership of this book consists of applied mathematicians, engineering scientists, research workers in dynamic systems, chaotic theory and neural nets.

Sommaire:
Preface. 1: 1.0. Introduction. 1.1. Measure Chains and Time Scales. 1.2. Differentiation. 1.3 Mean Value Theorem and Consequences. 1.4. Integral and Antiderivative. 1.5. Notes. 2: 2.0. Introduction. 2.1. Local Existence and Uniqueness. 2.2. Dynamic Inequalities. 2.3. Existence of Extremal Solutions. 2.4. Comparison Results. 2.5. Linear Variation of Parameters. 2.6. Continuous Dependence. 2.7. Nonlinear Variation of Parameters. 2.8. Global Existence and Stability. 2.9. Notes. 3: 3.0. Introduction. 3.1. Comparison Theorems. 3.2. Stability Criteria. 3.3. A Technique in Stability Theory. 3.4. Stability of Conditionally Invariant Sets. 3.5. Stability in Terms of Two Measures. 3.6. Vector Lyapunov Functions and Practical Stability. 3.7. Notes. 4: 4.0. Introduction. 4.1. Monotone Iterative Technique. 4.2. Method of Quasilinearization. 4.3. Monotone Flows and Stationary Points. 4.4. Invariant Manifolds. 4.5. Practical Stability of Large-Scale Uncertain Dynamic Systems. 4.6. Boundary Value Problems. 4.7. Sturmian Theory. 4.8. Convexity of Solutions Relative to the Initial Data. 4.9. Invariance Principle. 4.10. Notes. References. Subject Index.