One Dimensional Dynamics

Author : Welington de Melo,Sebastian van Strien
Publisher : Springer Science & Business Media
Page : 616 pages
File Size : 46,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642780431

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One-Dimensional Dynamics by Welington de Melo,Sebastian van Strien Pdf

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).

Author : Welington de Melo,Sebastian van Strien
Publisher : Springer
Page : 606 pages
File Size : 52,5 Mb
Release : 2011-12-16
Category : Mathematics
ISBN : 3642780458

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One-Dimensional Dynamics by Welington de Melo,Sebastian van Strien Pdf

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).

Author : Welington de Melo
Publisher : Unknown
Page : 252 pages
File Size : 46,7 Mb
Release : 1990*
Category : Mathematics
ISBN : 8524400412

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Lectures on One-dimensional Dynamics by Welington de Melo Pdf

Author : Ana Rodrigues
Publisher : CRC Press
Page : 119 pages
File Size : 43,7 Mb
Release : 2021-08-10
Category : Mathematics
ISBN : 9781000427974

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One-Dimensional Dynamical Systems by Ana Rodrigues Pdf

• Example-driven approach • Suitable as supplementary reading for a graduate or advanced undergraduate course in dynamical systems

Author : Karen M. Brucks,Henk Bruin
Publisher : Cambridge University Press
Page : 316 pages
File Size : 52,8 Mb
Release : 2004-06-28
Category : Mathematics
ISBN : 0521547660

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Topics from One-Dimensional Dynamics by Karen M. Brucks,Henk Bruin Pdf

Publisher Description

Author : A.N. Sharkovsky,S.F. Kolyada,A.G. Sivak,V.V. Fedorenko
Publisher : Springer Science & Business Media
Page : 268 pages
File Size : 49,9 Mb
Release : 2013-06-29
Category : Mathematics
ISBN : 9789401588973

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Dynamics of One-Dimensional Maps by A.N. Sharkovsky,S.F. Kolyada,A.G. Sivak,V.V. Fedorenko Pdf

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.

Author : Edson de Faria,Welington de Melo
Publisher : Cambridge University Press
Page : 192 pages
File Size : 52,8 Mb
Release : 2008-10-02
Category : Mathematics
ISBN : 9781139474849

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Mathematical Tools for One-Dimensional Dynamics by Edson de Faria,Welington de Melo Pdf

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.

Author : Louis S. Block,William A. Coppel
Publisher : Springer
Page : 251 pages
File Size : 54,7 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540470236

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Dynamics in One Dimension by Louis S. Block,William A. Coppel Pdf

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.

Author : Karen M. Brucks,Henk Bruin
Publisher : Cambridge University Press
Page : 312 pages
File Size : 50,8 Mb
Release : 2004-07-12
Category : Mathematics
ISBN : 0521838967

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Topics from One-Dimensional Dynamics by Karen M. Brucks,Henk Bruin Pdf

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.

Author : Yoshio Kuramoto,Yusuke Kato
Publisher : Cambridge University Press
Page : 487 pages
File Size : 46,5 Mb
Release : 2009-08-06
Category : Mathematics
ISBN : 9780521815987

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Dynamics of One-Dimensional Quantum Systems by Yoshio Kuramoto,Yusuke Kato Pdf

A concise and accessible account of the dynamical properties of one-dimensional quantum systems, for graduate students and new researchers.

Author : Maria José Pacifico,Pablo Guarino
Publisher : Springer Nature
Page : 333 pages
File Size : 47,8 Mb
Release : 2019-12-14
Category : Mathematics
ISBN : 9783030168339

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New Trends in One-Dimensional Dynamics by Maria José Pacifico,Pablo Guarino Pdf

This volume presents the proceedings of the meeting New Trends in One-Dimensional Dynamics, which celebrated the 70th birthday of Welington de Melo and was held at the IMPA, Rio de Janeiro, in November 2016. Highlighting the latest results in one-dimensional dynamics and its applications, the contributions gathered here also celebrate the highly successful meeting, which brought together experts in the field, including many of Welington de Melo’s co-authors and former doctoral students. Sadly, Welington de Melo passed away shortly after the conference, so that the present volume became more a tribute to him. His role in the development of mathematics was undoubtedly an important one, especially in the area of low-level dynamics, and his legacy includes, in addition to many articles with fundamental contributions, books that are required reading for all newcomers to the field.

Author : Tim Langen
Publisher : Springer
Page : 146 pages
File Size : 47,6 Mb
Release : 2015-05-22
Category : Science
ISBN : 9783319185644

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Non-equilibrium Dynamics of One-Dimensional Bose Gases by Tim Langen Pdf

This work presents a series of experiments with ultracold one-dimensional Bose gases, which establish said gases as an ideal model system for exploring a wide range of non-equilibrium phenomena. With the help of newly developed tools, like full distributions functions and phase correlation functions, the book reveals the emergence of thermal-like transient states, the light-cone-like emergence of thermal correlations and the observation of generalized thermodynamic ensembles. This points to a natural emergence of classical statistical properties from the microscopic unitary quantum evolution, and lays the groundwork for a universal framework of non-equilibrium physics. The thesis investigates a central question that is highly contested in quantum physics: how and to which extent does an isolated quantum many-body system relax? This question arises in many diverse areas of physics, and many of the open problems appear at vastly different energy, time and length scales, ranging from high-energy physics and cosmology to condensed matter and quantum information. A key challenge in attempting to answer this question is the scarcity of quantum many-body systems that are both well isolated from the environment and accessible for experimental study.

Author : Eugene M. Izhikevich
Publisher : MIT Press
Page : 459 pages
File Size : 52,6 Mb
Release : 2010-01-22
Category : Medical
ISBN : 9780262514200

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Dynamical Systems in Neuroscience by Eugene M. Izhikevich Pdf

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.

Author : Yunping Jiang
Publisher : World Scientific
Page : 327 pages
File Size : 49,7 Mb
Release : 1996-09-20
Category : Science
ISBN : 9789814500173

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Renormalization And Geometry In One-dimensional And Complex Dynamics by Yunping Jiang Pdf

About one and a half decades ago, Feigenbaum observed that bifurcations, from simple dynamics to complicated ones, in a family of folding mappings like quadratic polynomials follow a universal rule (Coullet and Tresser did some similar observation independently). This observation opened a new way to understanding transition from nonchaotic systems to chaotic or turbulent system in fluid dynamics and many other areas. The renormalization was used to explain this observed universality. This research monograph is intended to bring the reader to the frontier of this active research area which is concerned with renormalization and rigidity in real and complex one-dimensional dynamics. The research work of the author in the past several years will be included in this book. Most recent results and techniques developed by Sullivan and others for an understanding of this universality as well as the most basic and important techniques in the study of real and complex one-dimensional dynamics will also be included here.

Author : Lluís Alsedà,Jaume Llibre,Michal Misiurewicz
Publisher : World Scientific Publishing Company
Page : 432 pages
File Size : 52,9 Mb
Release : 2000-10-31
Category : Science
ISBN : 9789813105591

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Combinatorial Dynamics and Entropy in Dimension One by Lluís Alsedà,Jaume Llibre,Michal Misiurewicz Pdf

This book introduces the reader to the two main directions of one-dimensional dynamics. The first has its roots in the Sharkovskii theorem, which describes the possible sets of periods of all cycles (periodic orbits) of a continuous map of an interval into itself. The whole theory, which was developed based on this theorem, deals mainly with combinatorial objects, permutations, graphs, etc.; it is called combinatorial dynamics. The second direction has its main objective in measuring the complexity of a system, or the degree of “chaos” present in it; for that the topological entropy is used. The book analyzes the combinatorial dynamics and topological entropy for the continuous maps of either an interval or the circle into itself.