Dynamic Systems on Measure Chains

Author: V. Lakshmikantham,S. Sivasundaram,B. Kaymakcalan
Publisher: Springer Science & Business Media
ISBN: 1475724497
Category: Mathematics
Page: 294
View: 4806

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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: The readership of this book consists of applied mathematicians, engineering scientists, research workers in dynamic systems, chaotic theory and neural nets.

Dynamic Equations on Time Scales

An Introduction With Applications
Author: Martin Bohner,Allan Peterson
Publisher: Springer Science & Business Media
ISBN: 9780817642259
Category: Computers
Page: 358
View: 3172

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The study of dynamic equations on a measure chain (time scale) goes back to its founder S. Hilger (1988), and is a new area of still fairly theoretical exploration in mathematics. Motivating the subject is the notion that dynamic equations on measure chains can build bridges between continuous and discrete mathematics. Further, the study of measure chain theory has led to several important applications, e.g., in the study of insect population models, neural networks, heat transfer, and epidemic models. Key features of the book: * Introduction to measure chain theory; discussion of its usefulness in allowing for the simultaneous development of differential equations and difference equations without having to repeat analogous proofs * Many classical formulas or procedures for differential and difference equations cast in a new light * New analogues of many of the "special functions" studied * Examination of the properties of the "exponential function" on time scales, which can be defined and investigated using a particularly simple linear equation * Additional topics covered: self-adjoint equations, linear systems, higher order equations, dynamic inequalities, and symplectic dynamic systems * Clear, motivated exposition, beginning with preliminaries and progressing to more sophisticated text * Ample examples and exercises throughout the book * Solutions to selected problems Requiring only a first semester of calculus and linear algebra, Dynamic Equations on Time Scales may be considered as an interesting approach to differential equations via exposure to continuous and discrete analysis. This approach provides an early encounter with many applications in such areas as biology, physics, and engineering. Parts of the book may be used in a special topics seminar at the senior undergraduate or beginning graduate levels. Finally, the work may serve as a reference to stimulate the development of new kinds of equations with potentially new applications.

Advances in Dynamic Equations on Time Scales

Author: Martin Bohner,Allan C. Peterson
Publisher: Springer Science & Business Media
ISBN: 0817682309
Category: Mathematics
Page: 348
View: 3595

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Excellent introductory material on the calculus of time scales and dynamic equations.; Numerous examples and exercises illustrate the diverse application of dynamic equations on time scales.; Unified and systematic exposition of the topics allows good transitions from chapter to chapter.; Contributors include Anderson, M. Bohner, Davis, Dosly, Eloe, Erbe, Guseinov, Henderson, Hilger, Hilscher, Kaymakcalan, Lakshmikantham, Mathsen, and A. Peterson, founders and leaders of this field of study.; Useful as a comprehensive resource of time scales and dynamic equations for pure and applied mathematicians.; Comprehensive bibliography and index complete this text.

Nonoscillation and Oscillation Theory for Functional Differential Equations

Author: Ravi P. Agarwal,Martin Bohner,Wan-Tong Li
Publisher: CRC Press
ISBN: 0203025741
Category: Mathematics
Page: 400
View: 5340

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This book summarizes the qualitative theory of differential equations with or without delays, collecting recent oscillation studies important to applications and further developments in mathematics, physics, engineering, and biology. The authors address oscillatory and nonoscillatory properties of first-order delay and neutral delay differential equations, second-order delay and ordinary differential equations, higher-order delay differential equations, and systems of nonlinear differential equations. The final chapter explores key aspects of the oscillation of dynamic equations on time scales-a new and innovative theory that accomodates differential and difference equations simultaneously.

The Stability of Dynamical Systems

Author: J. P. LaSalle
Publisher: SIAM
ISBN: 9781611970432
Category: Difference equations
Page: 73
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An introduction to aspects of the theory of dynamial systems based on extensions of Liapunov's direct method. The main ideas and structure for the theory are presented for difference equations and for the analogous theory for ordinary differential equations and retarded functional differential equations. The latest results on invariance properties for non-autonomous time-varying systems processes are presented for difference and differential equations.

Symbolic Dynamics and Its Applications

Author: Roy L. Adler,Peter Walters
Publisher: American Mathematical Soc.
ISBN: 0821851462
Category: Mathematics
Page: 451
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This volume contains the proceedings of the conference, Symbolic Dynamics and its Applications, held at Yale University in the summer of 1991 in honor of Roy L. Adler on his sixtieth birthday. The conference focused on symbolic dynamics and its applications to other fields, including ergodic theory, smooth dynamical systems, information theory, automata theory, and statistical mechanics. One hundred thirty-nine participants attended from thirteen countries, representing mathematics, applied mathematics, electrical engineering, and physics departments in universities and in industry. Featuring a range of contributions from some of the leaders in the field, this volume presents an excellent overview of the subject.

Shadowing in Dynamical Systems

Theory and Applications
Author: K.J. Palmer
Publisher: Springer Science & Business Media
ISBN: 9780792361794
Category: Mathematics
Page: 300
View: 8249

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In this book the theory of hyperbolic sets is developed, both for diffeomorphisms and flows, with an emphasis on shadowing. We show that hyperbolic sets are expansive and have the shadowing property. Then we use shadowing to prove that hyperbolic sets are robust under perturbation, that they have an asymptotic phase property and also that the dynamics near a transversal homoclinic orbit is chaotic. It turns out that chaotic dynamical systems arising in practice are not quite hyperbolic. However, they possess enough hyperbolicity to enable us to use shadowing ideas to give computer-assisted proofs that computed orbits of such systems can be shadowed by true orbits for long periods of time, that they possess periodic orbits of long periods and that it is really true that they are chaotic. Audience: This book is intended primarily for research workers in dynamical systems but could also be used in an advanced graduate course taken by students familiar with calculus in Banach spaces and with the basic existence theory for ordinary differential equations.

Quasi-Stationary Distributions

Markov Chains, Diffusions and Dynamical Systems
Author: Pierre Collet,Servet Martínez,Jaime San Martín
Publisher: Springer Science & Business Media
ISBN: 3642331319
Category: Mathematics
Page: 280
View: 5657

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Main concepts of quasi-stationary distributions (QSDs) for killed processes are the focus of the present volume. For diffusions, the killing is at the boundary and for dynamical systems there is a trap. The authors present the QSDs as the ones that allow describing the long-term behavior conditioned to not being killed. Studies in this research area started with Kolmogorov and Yaglom and in the last few decades have received a great deal of attention. The authors provide the exponential distribution property of the killing time for QSDs, present the more general result on their existence and study the process of trajectories that survive forever. For birth-and-death chains and diffusions, the existence of a single or a continuum of QSDs is described. They study the convergence to the extremal QSD and give the classification of the survival process. In this monograph, the authors discuss Gibbs QSDs for symbolic systems and absolutely continuous QSDs for repellers. The findings described are relevant to researchers in the fields of Markov chains, diffusions, potential theory, dynamical systems, and in areas where extinction is a central concept. The theory is illustrated with numerous examples. The volume uniquely presents the distribution behavior of individuals who survive in a decaying population for a very long time. It also provides the background for applications in mathematical ecology, statistical physics, computer sciences, and economics.

Dynamical Systems and Linear Algebra

Author: Fritz Colonius,Wolfgang Kliemann
Publisher: American Mathematical Society
ISBN: 0821883194
Category: Mathematics
Page: 284
View: 8101

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This book provides an introduction to the interplay between linear algebra and dynamical systems in continuous time and in discrete time. It first reviews the autonomous case for one matrix A via induced dynamical systems in ℝd and on Grassmannian manifolds. Then the main nonautonomous approaches are presented for which the time dependency of A(t) is given via skew-product flows using periodicity, or topological (chain recurrence) or ergodic properties (invariant measures). The authors develop generalizations of (real parts of) eigenvalues and eigenspaces as a starting point for a linear algebra for classes of time-varying linear systems, namely periodic, random, and perturbed (or controlled) systems. The book presents for the first time in one volume a unified approach via Lyapunov exponents to detailed proofs of Floquet theory, of the properties of the Morse spectrum, and of the multiplicative ergodic theorem for products of random matrices. The main tools, chain recurrence and Morse decompositions, as well as classical ergodic theory are introduced in a way that makes the entire material accessible for beginning graduate students.

Mathematical Foundations of Quantum Information and Computation and Its Applications to Nano- and Bio-systems

Author: Masanori Ohya,I. Volovich
Publisher: Springer Science & Business Media
ISBN: 9789400701717
Category: Science
Page: 760
View: 979

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This monograph provides a mathematical foundation to the theory of quantum information and computation, with applications to various open systems including nano and bio systems. It includes introductory material on algorithm, functional analysis, probability theory, information theory, quantum mechanics and quantum field theory. Apart from standard material on quantum information like quantum algorithm and teleportation, the authors discuss findings on the theory of entropy in C*-dynamical systems, space-time dependence of quantum entangled states, entangling operators, adaptive dynamics, relativistic quantum information, and a new paradigm for quantum computation beyond the usual quantum Turing machine. Also, some important applications of information theory to genetics and life sciences, as well as recent experimental and theoretical discoveries in quantum photosynthesis are described.

Seminar on Stochastic Analysis, Random Fields, and Applications IV

Centro Stefano Franscini, Ascona, May 2002
Author: Robert C. Dalang
Publisher: Springer Science & Business Media
ISBN: 9783764371319
Category: Mathematics
Page: 328
View: 4534

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"This volume contains twenty refered research and review papers presented at the 4th Seminar on Stochastic Processes, Random Fields and Applications which took place at the Centro Stefano Franscini (Monte Verita) in Ascona, Switzerland, from May 19 to 24, 2002. The seminar focused mainly on stochastic partial differential equations, stochastic models in mathematical physics, and financial engineering. Mechanical Engineering was the subject of a minisymposium on stochastic methods in financial models."

Extremal Fuzzy Dynamic Systems

Theory and Applications
Author: Gia Sirbiladze
Publisher: Springer Science & Business Media
ISBN: 1461442494
Category: Business & Economics
Page: 400
View: 1973

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Presenting a fresh approach to the fuzzy modeling of dynamic processes in systems science research, this book’s coverage of both theory and key applications incorporates time as a source of fuzziness and treats applied EFD systems such as software libraries.

Advances in Discrete Dynamical Systems

Author: Saber Elaydi
Publisher: Advanced Studies in Pure Mathe
Category: Mathematics
Page: 398
View: 1375

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This volume contains the proceedings of talks presented at the 11th International Conference on Difference Equations and Applications (ICDEA 2006). ICDEA 2006 was held on July 2006 in Kyoto at the 15th MSJ International Research Institute. These proceedings comprise new results at the leading edge of many areas in difference equations and discrete dynamical systems and their various applications to the sciences, engineering, physics, and economics.


Author: W. A. Coppel
Publisher: Springer
ISBN: 3540369058
Category: Mathematics
Page: 156
View: 5915

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