Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics (Applied Mathematical Sciences)

Book Description

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.

Customer Reviews:

Excellent Book!.......2000-12-20

This is one of my favorite maths books. My interest in it is not so much for the discussion of chaotic systems, but more for that on stochastically perturbed systems. As far as this topic goes, this is the best book I've found in the field by far. The treatment is careful and self-contained, and the proofs are clear throughout. Lots of intuition is given with each result.

Book Description

The main focus of this book is on the development of the theory of Graph Directed Markov Systems. This far-reaching generalization of the theory of conformal iterated systems can be applied in many situations, including the theory of dynamical systems. Dan Mauldin and Mariusz Urbanski include much of the necessary background material to increase the appeal of this book to graduate students as well as researchers. They also include an extensive list of references for further reading.

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The main focus of this book is the exploration of the geometric and dynamic properties of a far reaching generalization of a conformal iterated function system - a Graph Directed Markov System. These systems are very robust in that they apply to many settings that do not fit into the scheme of conformal iterated systems. The basic theory is laid out here and the authors have touched on many natural questions arising in its context. However, they also emphasise the many issues and current research topics which can be found in original papers. For example the detailed analysis of the structure of harmonic measures of limit sets, the examination of the doubling property of conformal measures, the extensive study of generalized polynomial like mapping or multifractal analysis of geometrically finite Kleinian groups. This book leads readers onto frontier research in the field, making it ideal for both established researchers and graduate students.

Customer Reviews:

A nice entry into a new trend in dynamics and limit sets........2005-03-23

The geometry and dynamics of limit sets is one of the more exciting new trends in mathematics; a subject at the interface of analysis and discrete mathematics. The book is addressed to the beginning graduate level, but it will likely be useful to anyone who wishes to get a feeling for this active and new mathematical area. The examples include conformal measures, iteration of polynomial like mappings, Kleinian groups, multifractal analysis. The main topics are (by chapter) symbolic dynamics, families of functions and their conformal measures, graph and Markov systems, iterated function systems, and topics in dynamics, e.g., rigidity, parabolic systems, Hausdorff and packing measures.) Each subject is attractively presented and followed up to the level of current research.
The presentation is clear and concise, and the book can be used in the class room, or for self study. The topics progress logically, and the book represents a coherent unity of ideas, but it can also be used by anyone who wishes to learn just one part of the material, for example conformal iterated function systems.
Reviewed by Palle Jorgensen, March 2005.

Topological Dynamics of Random Dynamical Systems (Oxford Mathematical Monographs)

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Topological Dynamics of Random Dynamical Systems (Oxford Mathematical Monographs)
Nguyen Dinh Cong
Manufacturer: Oxford University Press, USA
ProductGroup: Book
Binding: Hardcover

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Book Description

This book is devoted to the theory of topological dynamics of random dynamical systems. The theory of random dynamical systems is a relatively new and fast expanding field of research which attracts the attention of researchers from various fields of science. It unites and develops the classical deterministic theory of dynamical systems and probability theory, hence finding many applications in a very wide range of disciplines from physics to biology to engineering, finance, and economics. Recent developments call for a systematic presentation of the theory. Topological dynamics of random dynamical systems is the first book to deal with the theory of topological dynamics of random dynamical systems. It presents in detail the solutions to the most fundamental problems of topological dynamics: linearization of nonlinear smooth systems, classification, and structural stability of linear hyperbolic systems. Employing the tools and methods of algebraic ergodic theory, the theory presented in the book has surprisingly beautiful results showing the richness of random dynamical systems as well as giving a gentle generalization of the classical deterministic theory. Mathematicians working in the theory of dynamical systems, stochastic dynamics, as well as those interested in applications of mathematical systems with random noise, will find this timely book a valuable reference and rich source of modern mathematical results and methods.

H-Infinity Optimal Control and Related Minimax Design Problems: A Dynamic Game Approach (Systems & Control: Foundations & Applications)

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H-Infinity Optimal Control and Related Minimax Design Problems: A Dynamic Game Approach (Systems & Control: Foundations & Applications)
Tamer Basar , and Pierre Bernhard
Manufacturer: Birkhäuser Boston
ProductGroup: Book
Binding: Hardcover

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ASIN: 0817638148

Book Description

One of the major concentrated activities of the past decade in control theory has been the development of the so-called 'H-infinity-optimal control theory', which addresses the issue of worst-case controller design for linear plants subject to unknown disturbances and plant uncertainties.

This second edition features a more streamlined presentation of the results included in the first edition and also contains substantial new material, reflecting new developments in the field since 1991. Among these are the nonlinear theory; connections between H-infinity-optimal control and risk sensitive stochastic control problems; H-infinity filtering for linear and nonlinear systems; and robustness considerations in the presence of regular and singular perturbations. Also included are a rather detailed description of the relationship between frequency-and time-domain approaches to robust controller design, and a complete set of results on the existence of value and characterization of optimal policies in finite- and infinite-horizon LQ differential games.

Book Description

This book treats nonlinear dynamics in both Hamiltonian and dissipative systems. The emphasis is on the mechanics for generating chaotic motion, methods of calculating the transitions from regular to chaotic motion, and the dynamical and statistical properties of the dynamics when it is chaotic. The book is intended as a self consistent treatment of the subject at the graduate level and as a reference for scientists already working in the field. It emphasizes both methods of calculation and results. It is accessible to physicists and engineers without training in modern mathematics. The new edition brings the subject matter in a rapidly expanding field up to date, and has greatly expanded the treatment of dissipative dynamics to include most important subjects. It can be used as a graduate text for a two semester course covering both Hamiltonian and dissipative dynamics.

Statistical Thermodynamics and Stochastic Theory of Nonlinear Systems Far from Equilibrium (Advanced Series in Statistical Mechanics)

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Statistical Thermodynamics and Stochastic Theory of Nonlinear Systems Far from Equilibrium (Advanced Series in Statistical Mechanics)
Werner Ebeling , and Igor M. Sokolov
Manufacturer: World Scientific Publishing Company
ProductGroup: Book
Binding: Hardcover

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The Trouble With Physics: The Rise of String Theory, The Fall of a Science, and What Comes Next

ASIN: 9810213824

Product Description

This book presents both the fundamentals and the major research topics in statistical physics of systems out of equilibrium. It summarizes different approaches to describe such systems on the thermodynamic and stochastic levels, and discusses a variety of areas including reactions, anomalous kinetics, and the behavior of self-propelling particles.

The Art of Modeling Dynamic Systems: Forecasting for Chaos, Randomness, and Determinism (Scientific and Technical Computation Series)

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Mathematical Depth and Descriptive Simplicity
A masterpiece
A broad overview
Best modeling book on the market !
Single best reference for "modern" applied modeling

The Art of Modeling Dynamic Systems: Forecasting for Chaos, Randomness, and Determinism (Scientific and Technical Computation Series)
Foster Morrison
Manufacturer: *Ronald Press
ProductGroup: Book
Binding: Hardcover

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ASIN: 0471520047

Book Description

In the coverage of dynamics, there is a definite gap between ``picture-book'' popularizations and the technical literature. This work fills that gap. Shows engineers and scientists how, by the application of statistical methods, coordinate transformations and mathematical analysis, any complex, unpredictable dynamical system can be mapped--transformed into a simpler, predictable system. The various modeling tools available, their benefits and their limitations are described. Examples and analogies are used in place of theorems and proofs, making this an immediately practical book. By showing how to make models more meaningful and useful, it will be particularly helpful in clearing up the impasse between economics and system dynamics. Features a number of carefully selected references to more mathematical treatments, examples of some of the more specialized techniques and case histories of some models.

Customer Reviews:

Mathematical Depth and Descriptive Simplicity.......2005-05-21

I find myself agreeing with all the comments made so far.

It's not too common to find a book that is able to describe in simple terms, such a large and diverse range of mathematical tools.

The author uses a framework - to tie together tools used in describing and handling deterministic, semi deterministic, and stochastic systems. For an example of Deterministic, try ODE's (ordinary differential equations), for semi deterministic - try Periodic but noisy wave-forms (some stock prices), and finally Stochastic - Random looking waveforms that have underlying patterns that can be described using either using Chaotic indicators (Hurst, Liapunov ) or probability type descriptors.

This book is the kind of thing you needed to help steer you through those dry mathematical books that are divorced from reality - A sort of classification system for deciphering what kind of gunpowder was used in those display's of intellectual fireworks from the tops of ivory towers. Kinda "So thats what all that maths means, but in plain english".

A depth of understanding, for practical application, without intellectual egotism and opaqueness. (But then maybe I'm just a bit thick ... :)

I'd tend to call this book as an equivalent to the Rosetta Stone for the maths of dynamical systems.

You may not use it directly - but you will benefit and grow in understanding from its' plain and simple sign posts along your journey.

It has its place on my book shelf.

A masterpiece.......2002-01-25

This is the kind of book that should be available in every area of science and mathematics. A book that is not a research book, but yet is not a broad popularization. This book can benefit both persons who are just beginning a study of dynamic modeling and I dare say things will be learned by even those that are researchers in the area. Good work Foster Morrison whoever you are.

A broad overview.......2001-11-22

Very general overview for the subject of dynamic systems modeling, providing ample historical perspectives. Not suitable for folks looking for deep technical matters.

Best modeling book on the market !.......2001-01-20

This book is certainly the best book you'll find around for treating in depth the subject of modeling dynamic systems. The style is clear and very intuitive. It provides a well-structured identification of dynamic systems and explains the "what you can do and what you cannot" for modeling all types of dynamic systems. It's true that it can be hard to find but waiting some couple of weeks certainly is no reason for not trying to get this extraordinary overview of dynamic systems !

The only drawback you could find is that it is too short... reading it makes you want more...so using it as an introduction to dynamic systems can only motivate you at going further in the field of dynamic systems modeling !

Single best reference for "modern" applied modeling.......1998-09-30

This is a fantastic book and I'm sorry it's hard to get now. I found it by accident in a used book store in Madison, WI, in 1995 and found I learned a lot from it, even though by that time I had already taken multiple classes in Differential Equations, Linear Algebra, Statistics, etc. Dr. Morrison really sets out in lucid detail many of the important developments in applied modeling theory--dynamical systems, stochastic systems, the fudamentals that lie behind them--from a very useful perspective, and one that is often missing from the orthodox academic treatments. What he chooses not to cover in detail he references thoroughly. I wish that I had had this book when I was learning many of the topics. Anyone who is seriously considering learning dynamic modeling should read this book and I have little doubt that fairly experienced modelers will find something between the covers.

Complex Systems: Chaos and Beyond, A Constructive Approach with Applications in Life Sciences

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Beyond Chaos but not Beyond Comprehension

Complex Systems: Chaos and Beyond, A Constructive Approach with Applications in Life Sciences
Kunihiko Kaneko , and Ichiro Tsuda
Manufacturer: Springer
ProductGroup: Book
Binding: Hardcover

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ASIN: 3540672028

Book Description

Chaos in science has always been a fascinating realm since it challenges the usual scientific approach of reductionism. While carefully distinguishing between complexity, holism, randomness, incompleteness, nondeterminism and stochastic behaviour the authors show that, although many aspects of chaos have been phenomenologically understood, most of its defining principles are still difficult to grasp and formulate. Demonstrating that chaos escapes all traditional methods of description, the authors set out to find new methods to deal with this phenomenon and illustrate their constructive approach with many examples from physics, biology and information technology. While maintaining a high level of rigour, an overly complicated mathematical apparatus is avoided in order to make this book accessible, beyond the specialist level, to a wider interdisciplinary readership.

Customer Reviews:

Beyond Chaos but not Beyond Comprehension.......2000-12-22

This is a book I've long been waiting for. It gives an excellent overview of how the study of complex systems and chaos brings together various branches of science with a special focus on Physics and the Biological Sciences. The book is not too technical yet highly to the point with philosophical and fundamental discussions and thus of great value to both expert and novice.

Handbook of Dynamical Systems : Volume 1A

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Handbook of Dynamical Systems : Volume 1A

Manufacturer: North Holland
ProductGroup: Book
Binding: Hardcover

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ASIN: 0444826696

Book Description

Volumes 1A and 1B.

These volumes give a comprehensive survey of dynamics written by specialists in the various subfields of dynamical systems. The presentation attains coherence through a major introductory survey by the editors that organizes the entire subject, and by ample cross-references between individual surveys.

The volumes are a valuable resource for dynamicists seeking to acquaint themselves with other specialties in the field, and to mathematicians active in other branches of mathematics who wish to learn about contemporary ideas and results dynamics. Assuming only general mathematical knowledge the surveys lead the reader towards the current state of research in dynamics.

Book Description

This volume considers optimal stochastic decision processes from the viewpoint of stochastic programming. It focuses on theoretical properties and on approximate or numerical solution techniques for time-dependent optimization problems with random parameters (multistage stochastic programs, optimal stochastic decision processes). Methods for finding approximate solutions of probabilistic and expected cost based deterministic substitute problems are presented. Besides theoretical and numerical considerations, the proceedings volume contains selected refereed papers on many practical applications to economics and engineering: risk, risk management, portfolio management, finance, insurance-matters and control of robots.

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