Synopses & Reviews
"In recent years there has been an explosive growth in the study of physical, biological, and economic systems that can be profitably studies using densities. Because of the general inaccessibility of the mathematical literature to the nonspecialist, little diffusion of the applicable mathematics into the study of these ""chaotic"" systems has taken place. This book will help bridge that gap. To show how densities arise in simple deterministic systems, the authors give a unified treatment of a variety of mathematical system generating densities, ranging from one-dimensional discrete time transformations through continuous time systems described by integro-partial-differential equations. Examples have been drawn from many fields to illustrate the utility of the concepts and techniques presented, and the ideas in this book should thus prove useful in the study of a number of applied sciences. The authors assume that the reader has a knowledge of advanced calculus and differential equations. Basic concepts from measure theory, ergodic theory, the geometry of manifolds, partial differential equations, probability theory and Markov processes, and stochastic integrals and differential equations are introduced as needed. Physicists, chemists, and biomathematicians studying chaotic behavior will find this book of value. It will also be a useful reference or text for mathematicians and graduate students working in ergodic theory and dynamical systems."
"In the second edition of their book, the authors have added one section on discrete time systems with multiplicative perturbations and six sections dealing with the property of sweeping and the corresponding Foguel alternative in various contexts (Markov operators, stochastic semigroups in continuous time, solutions of the Fokker-Planck equation). A completely new chapter includes sections on Markov and Foias operators, several notions of asymptotic stability for dynamical systems as well as a final section devoted to iterated function systems and fractals. In addition, the book now contains almost 100 exercises. ZENTRALBLATT MATH"
"This is a truly outstanding book. It is so well and carefully written that, despite the complexity of the subject matte and the difficulty of some of the mathematics, virtually everything is completely clear on a first reading. Indeed, the book is much more absorbing than most novels." Probability in the Engineering + Informational Sciences
The first edition of this book was originally published in 1985 under the ti- tle "Probabilistic Properties of Deterministic Systems. " In the intervening years, interest in so-called "chaotic" systems has continued unabated but with a more thoughtful and sober eye toward applications, as befits a ma- turing field. This interest in the serious usage of the concepts and techniques of nonlinear dynamics by applied scientists has probably been spurred more by the availability of inexpensive computers than by any other factor. Thus, computer experiments have been prominent, suggesting the wealth of phe- nomena that may be resident in nonlinear systems. In particular, they allow one to observe the interdependence between the deterministic and probabilistic properties of these systems such as the existence of invariant measures and densities, statistical stability and periodicity, the influence of stochastic perturbations, the formation of attractors, and many others. The aim of the book, and especially of this second edition, is to present recent theoretical methods which allow one to study these effects. We have taken the opportunity in this second edition to not only correct the errors of the first edition, but also to add substantially new material in five sections and a new chapter.
This book gives a unified treatment of a variety of mathematical systems generating densities, ranging from one-dimensional discrete time transformations through continuous time systems described by integro-partial-differential equations. Examples have been drawn from a variety of the sciences to illustrate the utility of the techniques presented. This material was organized and written to be accessible to scientists with knowledge of advanced calculus and differential equations. In various concepts from measure theory, ergodic theory, the geometry of manifolds, partial differential equations, probability theory and Markov processes, and chastic integrals and differential equations are introduced. The past few years have witnessed an explosive growth in interest in physical, biological, and economic systems that could be profitably studied using densities. Due to the general inaccessibility of the mathematical literature to the non-mathematician, there has been little diffusion of the concepts and techniques from ergodic theory into the study of these "chaotic" systems. This book intends to bridge that gap.
This book offers a unified treatment of a variety of mathematical systems generating densities, from one-dimensional discrete time transformations through continuous time systems described by integro-partial-differential equations. Includes numerous examples.
Table of Contents
Preface to the 2nd edition
x Preface to the 1st edition
x 1 Introduction
x 2 The Toolbox
x 3 Markov and Frobenius-Perron Operators
x 4 Studying Chaos with Densities
x 5 The Asymptotic Properties of Densities
x 6 The Behavior of Transformations on Intervals and Manifolds
x 7 Continuous Time Systems: An Introduction
x 8 Discrete Time Processes Embedded in Continuous Time Systems
x 9 Entropy
x 10 Stochastic Perturbation of Discrete Time Systems
x 11 Stochastic Perturbation of Continuous Time Systems
x 12 Markov and Foias Operators
x Notation and Symbols