One Dimensional Dynamical Systems

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One-Dimensional Dynamics

Author : Welington de Melo
Publisher : Springer Science & Business Media
ISBN 13 : 3642780431
Total Pages : 616 pages
Book Rating : 4.6/5 (427 download)

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Book Synopsis One-Dimensional Dynamics by : Welington de Melo

Download or read book One-Dimensional Dynamics written by Welington de Melo and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 616 pages. Available in PDF, EPUB and Kindle. Book excerpt: One-dimensional dynamics has developed in the last decades into a subject in its own right. Yet, many recent results are inaccessible and have never been brought together. For this reason, we have tried to give a unified ac count of the subject and complete proofs of many results. To show what results one might expect, the first chapter deals with the theory of circle diffeomorphisms. The remainder of the book is an attempt to develop the analogous theory in the non-invertible case, despite the intrinsic additional difficulties. In this way, we have tried to show that there is a unified theory in one-dimensional dynamics. By reading one or more of the chapters, the reader can quickly reach the frontier of research. Let us quickly summarize the book. The first chapter deals with circle diffeomorphisms and contains a complete proof of the theorem on the smooth linearizability of circle diffeomorphisms due to M. Herman, J.-C. Yoccoz and others. Chapter II treats the kneading theory of Milnor and Thurstonj also included are an exposition on Hofbauer's tower construction and a result on fuB multimodal families (this last result solves a question posed by J. Milnor).

One-Dimensional Dynamical Systems

Author : Ana Rodrigues
Publisher : CRC Press
ISBN 13 : 1000427978
Total Pages : 119 pages
Book Rating : 4.0/5 (4 download)

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Book Synopsis One-Dimensional Dynamical Systems by : Ana Rodrigues

Download or read book One-Dimensional Dynamical Systems written by Ana Rodrigues and published by CRC Press. This book was released on 2021-08-10 with total page 119 pages. Available in PDF, EPUB and Kindle. Book excerpt: • Example-driven approach • Suitable as supplementary reading for a graduate or advanced undergraduate course in dynamical systems

Dynamics of One-Dimensional Maps

Author : A.N. Sharkovsky
Publisher : Springer Science & Business Media
ISBN 13 : 940158897X
Total Pages : 268 pages
Book Rating : 4.4/5 (15 download)

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Book Synopsis Dynamics of One-Dimensional Maps by : A.N. Sharkovsky

Download or read book Dynamics of One-Dimensional Maps written by A.N. Sharkovsky and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 268 pages. Available in PDF, EPUB and Kindle. Book excerpt: maps whose topological entropy is equal to zero (i.e., maps that have only cyeles of pe 2 riods 1,2,2 , ... ) are studied in detail and elassified. Various topological aspects of the dynamics of unimodal maps are studied in Chap ter 5. We analyze the distinctive features of the limiting behavior of trajectories of smooth maps. In particular, for some elasses of smooth maps, we establish theorems on the number of sinks and study the problem of existence of wandering intervals. In Chapter 6, for a broad elass of maps, we prove that almost all points (with respect to the Lebesgue measure) are attracted by the same sink. Our attention is mainly focused on the problem of existence of an invariant measure absolutely continuous with respect to the Lebesgue measure. We also study the problem of Lyapunov stability of dynamical systems and determine the measures of repelling and attracting invariant sets. The problem of stability of separate trajectories under perturbations of maps and the problem of structural stability of dynamical systems as a whole are discussed in Chap ter 7. In Chapter 8, we study one-parameter families of maps. We analyze bifurcations of periodic trajectories and properties of the set of bifurcation values of the parameter, in eluding universal properties such as Feigenbaum universality.

Topics from One-Dimensional Dynamics

Author : Karen M. Brucks
Publisher : Cambridge University Press
ISBN 13 : 9780521838962
Total Pages : 312 pages
Book Rating : 4.8/5 (389 download)

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Book Synopsis Topics from One-Dimensional Dynamics by : Karen M. Brucks

Download or read book Topics from One-Dimensional Dynamics written by Karen M. Brucks and published by Cambridge University Press. This book was released on 2004-07-12 with total page 312 pages. Available in PDF, EPUB and Kindle. Book excerpt: One-dimensional dynamics has generated many results, and avenues of active mathematical research with numerous inroads to this research remain to be pursued by the advanced undergraduate or beginning graduate student. While much of the material in this book is not covered elsewhere, some aspects present new research topics whose connections are drawn to other research areas from the text. Although the material presented is not meant to be approached in a linear fashion, anybody with an interest in dynamics will find many topics of interest.

Grammatical Complexity and One-Dimensional Dynamical Systems

Author : H-M Xie
Publisher : World Scientific
ISBN 13 : 9814499897
Total Pages : 288 pages
Book Rating : 4.8/5 (144 download)

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Book Synopsis Grammatical Complexity and One-Dimensional Dynamical Systems by : H-M Xie

Download or read book Grammatical Complexity and One-Dimensional Dynamical Systems written by H-M Xie and published by World Scientific. This book was released on 1996-11-23 with total page 288 pages. Available in PDF, EPUB and Kindle. Book excerpt: A combinatorial method is developed in this book to explore the mysteries of chaos, which has became a topic of science since 1975. Using tools from theoretical computer science, formal languages and automata, the complexity of symbolic behaviors of dynamical systems is classified and analysed thoroughly. This book is mainly devoted to explanation of this method and apply it to one-dimensional dynamical systems, including the circle and interval maps, which are typical in exhibiting complex behavior through simple iterated calculations. The knowledge for reading it is self-contained in the book. Contents:Strings and Languages:Free MonoidsDynamical LanguagesGrammatical Complexity of Unimodal Maps:Languages of Unimodal MapsRegular Languages of Unimodal MapsA General Discussion of Kneading SequencesNon-Regular Languages of Unimodal MapsDEB of Unimodal MapsTopological Entropy of Unimodal MapsGrammatical Complexity of Circle Homeomorphisms:Languages of Circle HomeomorphismsComplexity Levels of Circle HomeomorphismsAutomata of Circle HomeomorphismsAppendices:Finite Automata and Regular LanguagesNon-Regular LanguagesL Systems and Languages Readership:Scientists interested in chaos and nonlinear science. keywords:Grammatical Complexity;Dynamical Systems;Symbolic Dynamics;Unimodal Maps;Circle Homeomorphisms;Kneading Sequences;Formal Languages;Chomsky Hierarchy;L Systems;Distinct Excluded Blocks;Topological Entropy

Dynamical Systems in Neuroscience

Author : Eugene M. Izhikevich
Publisher : MIT Press
ISBN 13 : 0262514206
Total Pages : 459 pages
Book Rating : 4.2/5 (625 download)

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Book Synopsis Dynamical Systems in Neuroscience by : Eugene M. Izhikevich

Download or read book Dynamical Systems in Neuroscience written by Eugene M. Izhikevich and published by MIT Press. This book was released on 2010-01-22 with total page 459 pages. Available in PDF, EPUB and Kindle. Book excerpt: Explains the relationship of electrophysiology, nonlinear dynamics, and the computational properties of neurons, with each concept presented in terms of both neuroscience and mathematics and illustrated using geometrical intuition. 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.

Lectures on One-dimensional Dynamics

Author : Welington de Melo
Publisher :
ISBN 13 : 9788524400414
Total Pages : 252 pages
Book Rating : 4.4/5 (4 download)

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Book Synopsis Lectures on One-dimensional Dynamics by : Welington de Melo

Download or read book Lectures on One-dimensional Dynamics written by Welington de Melo and published by . This book was released on 1990* with total page 252 pages. Available in PDF, EPUB and Kindle. Book excerpt:

One-Dimensional Dynamics

Author : Welington de Melo
Publisher : Springer
ISBN 13 : 9783642780455
Total Pages : 606 pages
Book Rating : 4.7/5 (84 download)

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Book Synopsis One-Dimensional Dynamics by : Welington de Melo

Download or read book One-Dimensional Dynamics written by Welington de Melo and published by Springer. This book was released on 2011-12-16 with total page 606 pages. Available in PDF, EPUB and Kindle. Book excerpt: One-dimensional dynamics has developed in the last decades into a subject in its own right. Yet, many recent results are inaccessible and have never been brought together. For this reason, we have tried to give a unified ac count of the subject and complete proofs of many results. To show what results one might expect, the first chapter deals with the theory of circle diffeomorphisms. The remainder of the book is an attempt to develop the analogous theory in the non-invertible case, despite the intrinsic additional difficulties. In this way, we have tried to show that there is a unified theory in one-dimensional dynamics. By reading one or more of the chapters, the reader can quickly reach the frontier of research. Let us quickly summarize the book. The first chapter deals with circle diffeomorphisms and contains a complete proof of the theorem on the smooth linearizability of circle diffeomorphisms due to M. Herman, J.-C. Yoccoz and others. Chapter II treats the kneading theory of Milnor and Thurstonj also included are an exposition on Hofbauer's tower construction and a result on fuB multimodal families (this last result solves a question posed by J. Milnor).

Topics from One-Dimensional Dynamics

Author : Karen M. Brucks
Publisher : Cambridge University Press
ISBN 13 : 9780521547666
Total Pages : 316 pages
Book Rating : 4.5/5 (476 download)

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Book Synopsis Topics from One-Dimensional Dynamics by : Karen M. Brucks

Download or read book Topics from One-Dimensional Dynamics written by Karen M. Brucks and published by Cambridge University Press. This book was released on 2004-06-28 with total page 316 pages. Available in PDF, EPUB and Kindle. Book excerpt: Publisher Description

Dynamics in One Dimension

Author : Louis S. Block
Publisher : Springer
ISBN 13 : 3540470239
Total Pages : 251 pages
Book Rating : 4.5/5 (44 download)

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Book Synopsis Dynamics in One Dimension by : Louis S. Block

Download or read book Dynamics in One Dimension written by Louis S. Block and published by Springer. This book was released on 2006-11-14 with total page 251 pages. Available in PDF, EPUB and Kindle. Book excerpt: The behaviour under iteration of unimodal maps of an interval, such as the logistic map, has recently attracted considerable attention. It is not so widely known that a substantial theory has by now been built up for arbitrary continuous maps of an interval. The purpose of the book is to give a clear account of this subject, with complete proofs of many strong, general properties. In a number of cases these have previously been difficult of access. The analogous theory for maps of a circle is also surveyed. Although most of the results were unknown thirty years ago, the book will be intelligible to anyone who has mastered a first course in real analysis. Thus the book will be of use not only to students and researchers, but will also provide mathematicians generally with an understanding of how simple systems can exhibit chaotic behaviour.

One-Dimensional Dynamical Systems

Author : Marek Rychlik
Publisher :
ISBN 13 : 9783540570554
Total Pages : 300 pages
Book Rating : 4.5/5 (75 download)

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Book Synopsis One-Dimensional Dynamical Systems by : Marek Rychlik

Download or read book One-Dimensional Dynamical Systems written by Marek Rychlik and published by . This book was released on 1997-12-30 with total page 300 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Combinatorial Dynamics And Entropy In Dimension One

Author : Alseda Luis
Publisher : World Scientific Publishing Company
ISBN 13 : 9814553220
Total Pages : 344 pages
Book Rating : 4.8/5 (145 download)

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Book Synopsis Combinatorial Dynamics And Entropy In Dimension One by : Alseda Luis

Download or read book Combinatorial Dynamics And Entropy In Dimension One written by Alseda Luis and published by World Scientific Publishing Company. This book was released on 1993-06-04 with total page 344 pages. Available in PDF, EPUB and Kindle. Book excerpt: In last thirty years an explosion of interest in the study of nonlinear dynamical systems occured. The theory of one-dimensional dynamical systems has grown out in many directions. One of them has its roots in the Sharkovski0 Theorem. This beautiful theorem describes the possible sets of periods of all cycles of maps of an interval into itself. Another direction has its main objective in measuring the complexity of a system, or the amount of chaos present in it. A good way of doing this is to compute topological entropy of the system. The aim of this book is to provide graduate students and researchers with a unified and detailed exposition of these developments for interval and circle maps. Many comments are added referring to related problems, and historical remarks are made. Request Inspection Copy

Mathematical Tools for One-Dimensional Dynamics

Author : Edson de Faria
Publisher : Cambridge University Press
ISBN 13 : 1139474847
Total Pages : 192 pages
Book Rating : 4.1/5 (394 download)

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Book Synopsis Mathematical Tools for One-Dimensional Dynamics by : Edson de Faria

Download or read book Mathematical Tools for One-Dimensional Dynamics written by Edson de Faria and published by Cambridge University Press. This book was released on 2008-10-02 with total page 192 pages. Available in PDF, EPUB and Kindle. Book excerpt: Originating with the pioneering works of P. Fatou and G. Julia, the subject of complex dynamics has seen great advances in recent years. Complex dynamical systems often exhibit rich, chaotic behavior, which yields attractive computer generated pictures, for example the Mandelbrot and Julia sets, which have done much to renew interest in the subject. This self-contained book discusses the major mathematical tools necessary for the study of complex dynamics at an advanced level. Complete proofs of some of the major tools are presented; some, such as the Bers-Royden theorem on holomorphic motions, appear for the very first time in book format. An appendix considers Riemann surfaces and Teichmüller theory. Detailing the very latest research, the book will appeal to graduate students and researchers working in dynamical systems and related fields. Carefully chosen exercises aid understanding and provide a glimpse of further developments in real and complex one-dimensional dynamics.

Dynamics of One-Dimensional Quantum Systems

Author : Yoshio Kuramoto
Publisher : Cambridge University Press
ISBN 13 : 0521815983
Total Pages : 487 pages
Book Rating : 4.5/5 (218 download)

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Book Synopsis Dynamics of One-Dimensional Quantum Systems by : Yoshio Kuramoto

Download or read book Dynamics of One-Dimensional Quantum Systems written by Yoshio Kuramoto and published by Cambridge University Press. This book was released on 2009-08-06 with total page 487 pages. Available in PDF, EPUB and Kindle. Book excerpt: A concise and accessible account of the dynamical properties of one-dimensional quantum systems, for graduate students and new researchers.

An Introduction to Dynamical Systems

Author : Rex Clark Robinson
Publisher : American Mathematical Soc.
ISBN 13 : 0821891359
Total Pages : 763 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis An Introduction to Dynamical Systems by : Rex Clark Robinson

Download or read book An Introduction to Dynamical Systems written by Rex Clark Robinson and published by American Mathematical Soc.. This book was released on 2012 with total page 763 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Discrete Dynamical Systems

Author : Oded Galor
Publisher : Springer Science & Business Media
ISBN 13 : 3540367764
Total Pages : 159 pages
Book Rating : 4.5/5 (43 download)

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Book Synopsis Discrete Dynamical Systems by : Oded Galor

Download or read book Discrete Dynamical Systems written by Oded Galor and published by Springer Science & Business Media. This book was released on 2007-05-17 with total page 159 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides an introduction to discrete dynamical systems – a framework of analysis that is commonly used in the ?elds of biology, demography, ecology, economics, engineering, ?nance, and physics. The book characterizes the fundamental factors that govern the quantitative and qualitative trajectories of a variety of deterministic, discrete dynamical systems, providing solution methods for systems that can be solved analytically and methods of qualitative analysis for those systems that do not permit or necessitate an explicit solution. The analysis focuses initially on the characterization of the factors that govern the evolution of state variables in the elementary context of one-dimensional, ?rst-order, linear, autonomous systems. The f- damental insights about the forces that a?ect the evolution of these - ementary systems are subsequently generalized, and the determinants of the trajectories of multi-dimensional, nonlinear, higher-order, non- 1 autonomous dynamical systems are established. Chapter 1 focuses on the analysis of the evolution of state variables in one-dimensional, ?rst-order, autonomous systems. It introduces a method of solution for these systems, and it characterizes the traj- tory of a state variable, in relation to a steady-state equilibrium of the system, examining the local and global (asymptotic) stability of this steady-state equilibrium. The ?rst part of the chapter characterizes the factors that determine the existence, uniqueness and stability of a steady-state equilibrium in the elementary context of one-dimensional, ?rst-order, linear autonomous systems.

Dynamics of Very High Dimensional Systems

Author : Earl H Dowell
Publisher : World Scientific Publishing Company
ISBN 13 : 9813102276
Total Pages : 284 pages
Book Rating : 4.8/5 (131 download)

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Book Synopsis Dynamics of Very High Dimensional Systems by : Earl H Dowell

Download or read book Dynamics of Very High Dimensional Systems written by Earl H Dowell and published by World Scientific Publishing Company. This book was released on 2003-08-22 with total page 284 pages. Available in PDF, EPUB and Kindle. Book excerpt: Many books on dynamics start with a discussion of systems with one or two degrees of freedom and then turn to the generalization to the case of many degrees of freedom. For linear systems, the concept of eigenfunctions provides a compact and elegant method for decomposing the dynamics of a high dimensional system into a series of independent single-degree-of-freedom dynamical systems. Yet, when the system has a very high dimension, the determination of the eigenfunctions may be a distinct challenge, and when the dynamical system is nonconservative and/or nonlinear, the whole notion of uncoupled eigenmodes requires nontrivial extensions of classical methods. These issues constitute the subject of this book.