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Dynamical Systems

Dynamical Systems PDF

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Author: George David Birkhoff
Publisher: American Mathematical Soc.
ISBN: 082181009X
Size: 14.88 MB
Format: PDF, Mobi
Category : Mathematics
Languages : en
Pages : 305
View: 3041

Book Description: His research in dynamics constitutes the middle period of Birkhoff's scientific career, that of maturity and greatest power. --Yearbook of the American Philosophical Society The author's great book ... is well known to all, and the diverse active modern developments in mathematics which have been inspired by this volume bear the most eloquent testimony to its quality and influence. --Zentralblatt MATH In 1927, G. D. Birkhoff wrote a remarkable treatise on the theory of dynamical systems that would inspire many later mathematicians to do great work. To a large extent, Birkhoff was writing about his own work on the subject, which was itself strongly influenced by Poincare's approach to dynamical systems. With this book, Birkhoff also demonstrated that the subject was a beautiful theory, much more than a compendium of individual results. The influence of this work can be found in many fields, including differential equations, mathematical physics, and even what is now known as Morse theory. The present volume is the revised 1966 reprinting of the book, including a new addendum, some footnotes, references added by Jurgen Moser, and a special preface by Marston Morse. Although dynamical systems has thrived in the decades since Birkhoff's book was published, this treatise continues to offer insight and inspiration for still more generations of mathematicians.


Dynamical Systems In Neuroscience

Dynamical Systems in Neuroscience PDF

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Author: Eugene M. Izhikevich
Publisher: MIT Press
ISBN: 0262090430
Size: 57.55 MB
Format: PDF, Kindle
Category : Medical
Languages : en
Pages : 441
View: 865

Book Description: In order to model neuronal behavior or to interpret the results of modeling studies, neuroscientists must call upon methods of nonlinear dynamics. This book offers an introduction to nonlinear dynamical systems theory for researchers and graduate students in neuroscience. It also provides an overview of neuroscience for mathematicians who want to learn the basic facts of electrophysiology. Dynamical Systems in Neuroscience presents a systematic study of the relationship of electrophysiology, nonlinear dynamics, and computational properties of neurons. It emphasizes that information processing in the brain depends not only on the electrophysiological properties of neurons but also on their dynamical properties. The book introduces dynamical systems, starting with one- and two-dimensional Hodgkin-Huxley-type models and continuing to a description of bursting systems. Each chapter proceeds from the simple to the complex, and provides sample problems at the end. The book explains all necessary mathematical concepts using geometrical intuition; it includes many figures and few equations, making it especially suitable for non-mathematicians. Each concept is presented in terms of both neuroscience and mathematics, providing a link between the two disciplines. Nonlinear dynamical systems theory is at the core of computational neuroscience research, but it is not a standard part of the graduate neuroscience curriculum—or taught by math or physics department in a way that is suitable for students of biology. This book offers neuroscience students and researchers a comprehensive account of concepts and methods increasingly used in computational neuroscience. An additional chapter on synchronization, with more advanced material, can be found at the author's website, www.izhikevich.com.


Dynamical Systems

Dynamical Systems PDF

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Author: Clark Robinson
Publisher: AMACOM Div American Mgmt Assn
ISBN: 9780849384950
Size: 66.49 MB
Format: PDF, ePub, Docs
Category : Mathematics
Languages : en
Pages : 506
View: 5380

Book Description: Several distinctive aspects make Dynamical Systems unique, including: treating the subject from a mathematical perspective with the proofs of most of the results included; providing a careful review of background materials; introducing ideas through examples and at a level accessible to a beginning graduate student; focusing on multidimensional systems of real variables. The book treats the dynamics of both iteration of functions and solutions of ordinary differential equations. Many concepts are first introduced for iteration of functions where the geometry is simpler, but results are interpreted for differential equations. The dynamical systems approach of the book concentrates on properties of the whole system or subsets of the system rather than individual solutions. The more local theory discussed deals with characterizing types of solutions under various hypothesis, and later chapters address more global aspects. What's New in the Second Edition?: A revised discussion of the saddle node bifurcation; a new section on the horseshoe for a flow with a transverse homoclinic point; material on horseshoes for nontransverse homoclinic points, indicating recent extensions to the understanding of how horseshoes arise; information proving the ergodicity of a hyperbolic toral automorphism; a new chapter on Hamiltonian systems.


Dynamical Systems And Cosmology

Dynamical Systems and Cosmology PDF

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Author: A.A. Coley
Publisher: Springer Science & Business Media
ISBN: 9401703272
Size: 13.64 MB
Format: PDF, ePub
Category : Science
Languages : en
Pages : 195
View: 6137

Book Description: Dynamical systems theory is especially well-suited for determining the possible asymptotic states (at both early and late times) of cosmological models, particularly when the governing equations are a finite system of autonomous ordinary differential equations. In this book we discuss cosmological models as dynamical systems, with particular emphasis on applications in the early Universe. We point out the important role of self-similar models. We review the asymptotic properties of spatially homogeneous perfect fluid models in general relativity. We then discuss results concerning scalar field models with an exponential potential (both with and without barotropic matter). Finally, we discuss the dynamical properties of cosmological models derived from the string effective action. This book is a valuable source for all graduate students and professional astronomers who are interested in modern developments in cosmology.


An Introduction To Dynamical Systems

An Introduction to Dynamical Systems PDF

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Author: Rex Clark Robinson
Publisher: American Mathematical Soc.
ISBN: 0821891359
Size: 58.78 MB
Format: PDF, ePub, Docs
Category : Mathematics
Languages : en
Pages : 733
View: 3559

Book Description: This book gives a mathematical treatment of the introduction to qualitative differential equations and discrete dynamical systems. The treatment includes theoretical proofs, methods of calculation, and applications. The two parts of the book, continuous time of differential equations and discrete time of dynamical systems, can be covered independently in one semester each or combined together into a year long course. The material on differential equations introduces the qualitative or geometric approach through a treatment of linear systems in any dimension. There follows chapters where equilibria are the most important feature, where scalar (energy) functions is the principal tool, where periodic orbits appear, and finally, chaotic systems of differential equations. The many different approaches are systematically introduced through examples and theorems. The material on discrete dynamical systems starts with maps of one variable and proceeds to systems in higher dimensions. The treatment starts with examples where the periodic points can be found explicitly and then introduces symbolic dynamics to analyze where they can be shown to exist but not given in explicit form. Chaotic systems are presented both mathematically and more computationally using Lyapunov exponents. With the one-dimensional maps as models, the multidimensional maps cover the same material in higher dimensions. This higher dimensional material is less computational and more conceptual and theoretical. The final chapter on fractals introduces various dimensions which is another computational tool for measuring the complexity of a system. It also treats iterated function systems which give examples of complicated sets. In the second edition of the book, much of the material has been rewritten to clarify the presentation. Also, some new material has been included in both parts of the book. This book can be used as a textbook for an advanced undergraduate course on ordinary differential equations and/or dynamical systems. Prerequisites are standard courses in calculus (single variable and multivariable), linear algebra, and introductory differential equations.


Global Analysis Of Dynamical Systems

Global Analysis of Dynamical Systems PDF

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Author: H.W Broer
Publisher: CRC Press
ISBN: 9781420034288
Size: 15.69 MB
Format: PDF, Kindle
Category : Mathematics
Languages : en
Pages : 464
View: 7135

Book Description: Contributed by close colleagues, friends, and former students of Floris Takens, Global Analysis of Dynamical Systems is a liber amicorum dedicated to Takens for his 60th birthday. The first chapter is a reproduction of Takens's 1974 paper "Forced oscillators and bifurcations" that was previously available only as a preprint of the University of Utrecht. Among other important results, it contains the unfolding of what is now known as the Bogdanov-Takens bifurcation. The remaining chapters cover topics as diverse as bifurcation theory, Hamiltonian mechanics, homoclinic bifurcations, routes to chaos, ergodic theory, renormalization theory, and time series analysis. In its entirety, the book bears witness to the influence of Takens on the modern theory of dynamical systems and its applications. This book is a must-read for anyone interested in this active and exciting field.


The Stability Of Dynamical Systems

The Stability of Dynamical Systems PDF

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Author: J. P. LaSalle
Publisher: SIAM
ISBN: 9781611970432
Size: 24.45 MB
Format: PDF, Docs
Category : Difference equations
Languages : en
Pages : 73
View: 4594

Book Description: 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.


Differential Equations Dynamical Systems And Linear Algebra

Differential Equations  Dynamical Systems  and Linear Algebra PDF

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Author: Morris W. Hirsch
Publisher: Academic Press
ISBN: 0080873766
Size: 13.13 MB
Format: PDF
Category : Mathematics
Languages : en
Pages : 358
View: 4921

Book Description: This book is about dynamical aspects of ordinary differential equations and the relations between dynamical systems and certain fields outside pure mathematics. A prominent role is played by the structure theory of linear operators on finite-dimensional vector spaces; the authors have included a self-contained treatment of that subject.


Chaotic Transitions In Deterministic And Stochastic Dynamical Systems

Chaotic Transitions in Deterministic and Stochastic Dynamical Systems PDF

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Author: Emil Simiu
Publisher: Princeton University Press
ISBN: 0691144346
Size: 42.38 MB
Format: PDF, Docs
Category : Mathematics
Languages : en
Pages : 240
View: 4957

Book Description: The classical Melnikov method provides information on the behavior of deterministic planar systems that may exhibit transitions, i.e. escapes from and captures into preferred regions of phase space. This book develops a unified treatment of deterministic and stochastic systems that extends the applicability of the Melnikov method to physically realizable stochastic planar systems with additive, state-dependent, white, colored, or dichotomous noise. The extended Melnikov method yields the novel result that motions with transitions are chaotic regardless of whether the excitation is deterministic or stochastic. It explains the role in the occurrence of transitions of the characteristics of the system and its deterministic or stochastic excitation, and is a powerful modeling and identification tool. The book is designed primarily for readers interested in applications. The level of preparation required corresponds to the equivalent of a first-year graduate course in applied mathematics. No previous exposure to dynamical systems theory or the theory of stochastic processes is required. The theoretical prerequisites and developments are presented in the first part of the book. The second part of the book is devoted to applications, ranging from physics to mechanical engineering, naval architecture, oceanography, nonlinear control, stochastic resonance, and neurophysiology.


Random Dynamical Systems

Random Dynamical Systems PDF

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Author: Ludwig Arnold
Publisher: Springer Science & Business Media
ISBN: 3662128780
Size: 25.81 MB
Format: PDF, Docs
Category : Mathematics
Languages : en
Pages : 586
View: 792

Book Description: 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.