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This book applied the techniques of dynamical systems and bifurcation theories to the study of nonlinear oscillations. Taking the cue from Poincare, the authors stress the geometrical and topological properties of solutions of differential equations and iterated maps. Numerous exercises, some of which require nontrivial algebraic manipulations and computer work, convey the important analytical underpinnings of problems in dynamical systems and help the reader develop an intuitive feel for the properties involved.In this fifth printing the authors have corrected further errors, oversights and updates. ... Read more

Customer Reviews (4)

Will never collect dust....
This book has been a continuing source of information and guidance for 18 years now. Students and researchers in many different fields have used this book due to its breadth and detail of coverage. The book does require a fairly advanced mathematical background, but the authors do include a glossary for the reader lacking this.

Chapter one is an overview of differential equations and dynamical systems. All the concepts needed for a study of such systems are discussed in great detail and also very informally, stressing instead the understanding of the concepts, and not merely their definition. Some of the proofs of the main results, such as the Hartman-Grobman and the stable manifold theorems, are omitted however.

This is followed in Chapter 2 by a very intuitive discussion of the van der Pols equation, Duffings equation, the Lorenz equations, and the bouncing ball. Numerical calculations are effectively employed to illustrate some of the main properties of the systems modeled by these equations.

A taste of bifurcation theory follows in Chapter 3. Center manifolds are defined and many examples are given, but the proof of the center manifold theorem is omitted unfortunately. Normal forms and Hopf bifurcations are treated in detail.

Averaging methods are discussed in Chapter 4, with part of the averaging theorem proved using a version of Gronwall's lemma. Several interesting examples of averaging are given, along with a discussion of to what extent the bifurcation properties of the averaged equations carry over to the original equations. Most importantly, this chapter discusses the Melnikov function, so very important in the study of small perturbations of dynamical systems with a hyperbolic fixed point. A full proof that simple zeros of the Melnikov function imply the transversal intersection of the stable and unstable manifolds is given.

Chapter 5 moves on to results of a more purely mathematical nature, where symbolic dynamics and the Smale horseshoe map are discussed. The proofs of the stable manifold theorem and the Palis lambda lemma are, however, omitted. Markov partitions and the shadowing lemma are discussed also but the latter is not proven. The authors do however give a proof of the Smale-Birkhoff homoclinic theorem. A purely mathematical overview of attractors is given along with measure-theoretic (ergodic) properties of dynamical systems.

The (local) bifurcation theory of Chapter 3 is extended to global bifurcations in the next chapter. A very detailed discussion of rotation numbers is given but the KAM theory is only briefly mentioned. The main emphasis is on 1-dimensional maps, the Lorentz system, and Silnikov theory. The authors give a very detailed treatment of wild hyperbolic sets.

The book ends with a discussion of bifurcations from equilibrium points that have multiple degeneracies. The discussion is more motivated from a physical standpont than the last few chapters. But some interesting mathematical constructions are employed, namely the role of k-jets, which havefascinating connections with algebraic goemetry, via the "blowing-up" techniques.

The concepts in the book have proven to have enduring value in the study of dynamical systems, and this book will no doubt continue to serve students and researchers in the years to come.

Background
Guckenheimer is one of my favourite book in nonlinear science. Another absolute reference. This books deserved to be milestone in nonlinear dynamics.

Changed the Nature of Science As We Know It.
This book has clearly withstood the test of time in over 15 years of continuous publication.On my bookcase, it stands among my most treasured and well-worn classics of fluid mechanics and differentialequations--Hirsch and Smale, Birkhoff and Rota, Chandrasekhar, Bachelor,Lamb, Landau and Lifschitz...It changed many of the unquestionedassumptions of many fields besides my own.It redefined the terms of manyscientific debates.And, it changed my life.

I obtained Guckenheimerand Holmes' classic when it first came out in 1983. It was so clear,concise and intellectually engaging that it inspired me to wonder whetherthe system of equations I was studying for my Ph.D. research at thetime--the governing equations of thermal convection at infinite Prandtlnumber (which govern plate tectonics in the earth's mantle)--might have achaotic solution.Guckenheimer and Holmes outlined a clear methodology tofind out the answer.

My advisor at the University of Chicago thoughtnot.Only steady solutions could be admitted in the absence of externalforcing due to the lack of momentum transfer--this belief was widely heldat the time, despite certain oscillatory solutions found by Fritz Busse(then at UCLA) and chaotic solutions found in certain limiting cases byAndrew Fowler at Oxford.

In despair, I left my studies at Chicago towork as a Unix sysadmin at my undergraduate alma mater --Cornell, where(unbeknownst to me when I took the job) John Guckenheimer had justrelocated from UCSC.Delighted to find him there, I sat in on his courses. Later, with his help, I wrote a proposal to NASA to support the completionof my thesis--with him and Donald Turcotte serving as my advisors.

The3-year fellowship was approved, and during this time I demonstrated andpublished that thermal convection at infinite Prandtl number--a conditionthat pervades many planetary interiors including our own--is indeed chaoticin the absence of external forcing.

Prior to this, planetary convectioncodes primarily looked for steady state solutions.Since, numericalanalysts in the field have upgraded to time-dependent models. The source ofchaos at infinite Prandtle number I identified--the heat advection term--isnow widely accepted as the source of what is now called "ThermalTurbulence" in planetary interiors.

The defense at Chicago wasquite an event. Since my new advisors were flown in from Ithaca, you mightsay my thesis--The Nonlinear Dynamics of Thermal Convection at InfinitePrandtl Number--passed with flying colors.Someone at Chicago mightdisagree, but his opinion is irrelevant.

Demonstrating the many possiblesolutions to a single set of equations and showing how the choice ofsolution depends very sensitively on the rather poorly-constrained initialconditions of the earth--does render mantle modeling itself rathersuperfluous and indeed, scientifically suspect.However, many importantprofessors who stayed in the field nonetheless continue to run theirtime-dependent mantle convection codes, and never cease to wonder at thefact that they all get different results.It's rather amusing,really.

When all that too has passed away, the truths so beautifully putforth in Guckenheimer and Holmes will remain.Like I said, it's a classic.Furthermore, being number 42 in its series, it's got to be the answer tothe ultimate question of life, the universe and everything. Was for me,anyway.

Basic and clasic
For the moment it is "the" book on Dynamical Systems, through the world. Its first chapter is a good introduction on the mathematics needed to aboard the subject. The second introduces chaos, and the rest isfor a good understanding of the newest and prolific science. ... Read more

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This book breaks new ground by integrating dynamical systems concepts with a contextual behavioral view of development. The two make a dramatically good fit and something new emerges from the combination: a "behavioral systems approach" to development. With its consistent, natural science, non-linear, contextual orientation, Novak has fashioned an undergraduate text that will have broad appeal for courses in developmental psychology.

Topics include:

-Modern Developmental Theory

-Behavioral Genetics

-Prenatal Development

-The Remarkable Newborn

-Habituation & Respondent Learning

-Operant Learning

-Cognitive Development

-The Development of Communication

-Personality and the Self

-Prosocial Development

-Development of Antisocial Behaviors

-The Family System

-Schools ... Read more

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Presents dynamical systems in the infinite dimension, especially those generated by dissipative partial differential equations. New revised edition. DLC: Differentiable dynamical systems. ... Read more

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Time-delay occurs in many dynamical systems such as biological systems, chemical systems, metallurgical processing systems, nuclear reactor, long transmission lines in pneumatic, hydraulic systems and electrical networks. Especially, in recent years, time-delay which exists in networked control systems has brought more complex problem into a new research area. Frequently, it is a source of the generation of oscillation, instability and poor performance. Considerable effort has been applied to different aspects of linear time-delay systems during recent years. Because the introduction of the delay factor renders the system analysis more complicated, in addition to the difficulties caused by the perturbation or uncertainties, in the control of time-delay systems, the problems of robust stability and robust stabilization are of great importance.

This book presents some basic theories of stability and stabilization of systems with time-delay, which are related to the main results in this book. More attention will be paid on synthesis of systems with time-delay. That is, sliding mode control of systems with time-delay, networked control systems with time-delay, networked data fusion with random delay.

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This volume consists of six articles covering different facets of the mathematical theory of dynamical systems. The topics range from topological foundations through invariant manifolds, decoupling, perturbations and computations to control theory. All contributions are based on a sound mathematical analysis. Some of them provide detailed proofs while others are of a survey character. In any case, emphasis is put on motivation and guiding ideas. Many examples are included.

The papers of this volume grew out of a tutorial workshop for graduate students in mathematics held at the University of Augsburg. Each of the contributions is self-contained and provides an in-depth insight into some topic of current interest in the mathematical theory of dynamical systems. The text is suitable for courses and seminars on a graduate student level. ... Read more

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The shared platform of the articles collected in this volume is used to advocate a dynamical systems approach to cognition. It is argued that recent developments in cognitive science towards an account of embodiment, together with the general approach of complexity theory and dynamics, have a major impact on behavioral and cognitive science. The book points out that there are two domains that follow naturally from the stance of embodiment: first, coordination dynamics is an established empirical paradigm that is best able to aid the approach; second, the obvious goal-directedness of intelligent action (i.e., intentionality) is nicely addressed in the framework of the dynamical synergetic approach. ... Read more

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This book is addressed to scientists, engineers and students of engineering departments, who make use of modelling and computer simulation. Since more and more physical experiments are being replaced by computer simulations the use of mathematical models of various engineering systems has become an especially important area of research. The book is devoted to selected problems of various engineering domains, such as control, electrical engineering or electrical metrology. They are based on different mathematical fields such as matrix theory, differential equations, function approximation with applications in dynamic modelling, methods of simplifying high-order models, determining mapping errors of simplified models, their optimization and the synthesis of suitable input signals. The book is easy to read and understand because all the needed mathematical transformations and formula are derived and explained by means of the examples enclosed. ... Read more

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The first systematic presentation of the theory of dynamical systems under the influence of randomness, this book includes products of random mappings as well as random and stochastic differential equations. The basic multiplicative ergodic theorem is presented, providing a random substitute for linear algebra. On its basis, many applications are detailed. Numerous instructive examples are treated analytically or numerically.

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Mathematical modelling of communicable diseases has in the past decades been the subject of intense research activity, on the part of both epidemiologists and biomathematicians; nonlinear forces of infection, spatial structure, age structure and other relevant features have been integrated to make the models more and more realistic and useful in prediction and control. The author's perspective in this book is that there is a concrete possibility of classifying most of the available models according to their mathematical structure, so to obtain a solid framework for analysing the behaviour of the modelled epidemic systems. This monograph suggests a possible classification of a large amount of models (bilinear, nonlinear, with or without structure), based on the Lyapunov stability theory and the theory of order preserving dynamical systems. The volume contains an original presentation of many worked out examples and case studies, mainly based on the author's experience, fully integrated with the exposition of the theory. It also contains a revisit of the most recent advances in the modelling of epidemics, including HIV/AIDS. Two appendices have been added for the ease of non-mathematicians. This monograph may be viewed as a research monograph for mathematically-oriented epidemiologists and for applied mathematicians. However, the detailed presentation of the methods make it a self-contained introduction to the mathematical modelling of infectious diseases so that it may also be used as a textbook in advanced courses of mathematical modelling in Biology and Medicine. The long and updated list of references makes this monograph a valuable survey of the subject.

In this second printing of the book the author has corrected all detected misprints, and updated the bibliography items.

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This unique volume introduces the reader to the mathematical language for complex systems and is ideal for students who are starting out in the study of stochastical dynamical systems. Unlike other books in the field it covers a broad array of stochastic and statistical methods. ... Read more

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"The exposition is remarkably clear, definitions are separated explicitly, theorems are often provided together with the motivation for changing one or other hypothesis, as well as the relevance of certain generalisations... This study is an excellent review of the current situation for problems of stability of the solution of differential equations. It is addressed to all interested on non-linear differential problems, as much from the theoretical as from the applications angle."-Bulletin de la Société Mathématique de Belgique ... Read more

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Coherent Dynamics of Complex Quantum Systems is aimed at senior-level undergraduate students in the areas of atomic, molecular, and laser physics, physical chemistry, quantum optics and quantum informatics. It should help them put particular problems in these fields into a broader scientific context and thereby take advantage of the well-elaborated technique of the adjacent fields. ... Read more

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This study addresses both classical and previously unsolved optimization tasks in linear algebra, linear systems theory, control theory and signal processing. Exploiting developments in Computation such as Parallel Computing and Neural Networks, it gives a dynamical systems approach for tackling a wide class of constrained optimization tasks. Optimization and Dynamical Systems will be of interest to engineers and mathematicians. Engineers will learn the mathematics and the technical approach necessary to solve a wide class of constrained optimization tasks. Mathematicians will see how techniques from global analysis and differential geometry can be developed to achieve useful construction procedures for optimization on manifolds. ... Read more

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Discontinuous dynamical systems have played an important role in both theory and applications during the last several decades. This is still an area of active research and techniques to make the applications more effective are an ongoing topic of interest.Principles of Discontinuous Dynamical Systems is devoted to the theory ofdifferential equations with variable moments of impulses. It introduces a new strategy of implementing an equivalence to systems whose solutions have prescribed moments of impulses and utilizing special topologies in spaces of piecewise continuous functions. The achievements obtained on the basis of this approach are described in this book. The text progresses systematically, by covering preliminaries in the first four chapters. This is followed by more complex material and special topics such as Hopf bifurcation, Devaney's chaos, and the shadowing property are discussed in the last two chapters.This book is suitable for researchers and graduate students inmathematics and also in diverse areas such as biology, computer science, and engineering who deal with real world problems. ... Read more

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This book develops the theory of global attractors for a class of parabolic PDEs that includes reaction-diffusion equations and the Navier-Stokes equations, two examples that are treated in detail.A lengthy chapter on Sobolev spaces provides the framework that allows a rigorous treatment of existence and uniqueness of solutions for both linear time-independent problems (Poisson's equation) and the nonlinear evolution equations which generate the infinite-dimensional dynamical systemss of the title. Attention then switches to the global attractor, a finite-dimensional subset of the infinite-dimensional phase space which determines the asymptotic dynamics. In particular, the concluding chapters investigate in what sense the dynamics restricted to the attractor are themselves "finite-dimensional." The book is intended as a didactic text for first year graduates, and assumes only a basic knowledge of Banach and Hilbert spaces, and a working understanding of the Lebesgue integral. ... Read more

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This volume consists of a set of ten lectures conceived as both introduction and up-to-date survey on discrete integrable systems. It constitutes a companion book to "Integrability of Nonlinear Systems" (Springer-Verlag, 2004, LNP 638, ISBN 3-540-20630-2). Both volumes address primarily graduate students and nonspecialist researchers but will also benefit lecturers looking for suitable material for advanced courses and researchers interested in specific topics.

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This volume contains the invited papers presented at the 9th International Conference Dynamical Systems Theory and Applications held in Lódz, Poland, December 17-20, 2007, dealing with nonlinear dynamical systems. The conference brought together a large group of outstanding scientists and engineers, who deal with various problems of dynamics encountered both in engineering and in daily life.

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Award-winning monograph of the Ferran Sunyer i Balaguer Prize 1994. ... Read more