Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields

Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields PDF

Author: John Guckenheimer

Publisher: Springer Science & Business Media

Published: 2013-11-21

Total Pages: 475

ISBN-13: 1461211409

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An application of the techniques of dynamical systems and bifurcation theories to the study of nonlinear oscillations. Taking their 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 readers develop an intuitive feel for the properties involved.

Dynamics and Bifurcations

Dynamics and Bifurcations PDF

Author: Jack K. Hale

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 577

ISBN-13: 1461244269

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In recent years, due primarily to the proliferation of computers, dynamical systems has again returned to its roots in applications. It is the aim of this book to provide undergraduate and beginning graduate students in mathematics or science and engineering with a modest foundation of knowledge. Equations in dimensions one and two constitute the majority of the text, and in particular it is demonstrated that the basic notion of stability and bifurcations of vector fields are easily explained for scalar autonomous equations. Further, the authors investigate the dynamics of planar autonomous equations where new dynamical behavior, such as periodic and homoclinic orbits appears.

Dynamic Bifurcations

Dynamic Bifurcations PDF

Author: Eric Benoit

Publisher: Springer

Published: 2006-11-14

Total Pages: 224

ISBN-13: 3540464719

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Dynamical Bifurcation Theory is concerned with the phenomena that occur in one parameter families of dynamical systems (usually ordinary differential equations), when the parameter is a slowly varying function of time. During the last decade these phenomena were observed and studied by many mathematicians, both pure and applied, from eastern and western countries, using classical and nonstandard analysis. It is the purpose of this book to give an account of these developments. The first paper, by C. Lobry, is an introduction: the reader will find here an explanation of the problems and some easy examples; this paper also explains the role of each of the other paper within the volume and their relationship to one another. CONTENTS: C. Lobry: Dynamic Bifurcations.- T. Erneux, E.L. Reiss, L.J. Holden, M. Georgiou: Slow Passage through Bifurcation and Limit Points. Asymptotic Theory and Applications.- M. Canalis-Durand: Formal Expansion of van der Pol Equation Canard Solutions are Gevrey.- V. Gautheron, E. Isambert: Finitely Differentiable Ducks and Finite Expansions.- G. Wallet: Overstability in Arbitrary Dimension.- F.Diener, M. Diener: Maximal Delay.- A. Fruchard: Existence of Bifurcation Delay: the Discrete Case.- C. Baesens: Noise Effect on Dynamic Bifurcations:the Case of a Period-doubling Cascade.- E. Benoit: Linear Dynamic Bifurcation with Noise.- A. Delcroix: A Tool for the Local Study of Slow-fast Vector Fields: the Zoom.- S.N. Samborski: Rivers from the Point ofView of the Qualitative Theory.- F. Blais: Asymptotic Expansions of Rivers.-I.P. van den Berg: Macroscopic Rivers.

Numerical Methods for Bifurcations of Dynamical Equilibria

Numerical Methods for Bifurcations of Dynamical Equilibria PDF

Author: Willy J. F. Govaerts

Publisher: SIAM

Published: 2000-01-01

Total Pages: 384

ISBN-13: 9780898719543

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Dynamical systems arise in all fields of applied mathematics. The author focuses on the description of numerical methods for the detection, computation, and continuation of equilibria and bifurcation points of equilibria of dynamical systems. This subfield has the particular attraction of having links with the geometric theory of differential equations, numerical analysis, and linear algebra.

Discrete Dynamical Systems, Bifurcations and Chaos in Economics

Discrete Dynamical Systems, Bifurcations and Chaos in Economics PDF

Author: Wei-Bin Zhang

Publisher: Elsevier

Published: 2006-01-05

Total Pages: 459

ISBN-13: 0080462464

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This book is a unique blend of difference equations theory and its exciting applications to economics. It deals with not only theory of linear (and linearized) difference equations, but also nonlinear dynamical systems which have been widely applied to economic analysis in recent years. It studies most important concepts and theorems in difference equations theory in a way that can be understood by anyone who has basic knowledge of calculus and linear algebra. It contains well-known applications and many recent developments in different fields of economics. The book also simulates many models to illustrate paths of economic dynamics. A unique book concentrated on theory of discrete dynamical systems and its traditional as well as advanced applications to economics Mathematical definitions and theorems are introduced in a systematic and easily accessible way Examples are from almost all fields of economics; technically proceeding from basic to advanced topics Lively illustrations with numerous figures Numerous simulation to see paths of economic dynamics Comprehensive treatment of the subject with a comprehensive and easily accessible approach

Dynamics, Bifurcations and Control

Dynamics, Bifurcations and Control PDF

Author: Fritz Colonius

Publisher: Springer

Published: 2003-07-01

Total Pages: 300

ISBN-13: 3540456066

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This volume originates from the Third Nonlinear Control Workshop "- namics, Bifurcations and Control", held in Kloster Irsee, April 1-3 2001. As the preceding workshops held in Paris (2000) and in Ghent (1999), it was organized within the framework of Nonlinear Control Network funded by the European Union (http://www.supelec.fr/lss/NCN). The papers in this volume center around those control problems where phenomena and methods from dynamical systems theory play a dominant role. Despite the large variety of techniques and methods present in the c- tributions, a rough subdivision can be given into three areas: Bifurcation problems, stabilization and robustness, and global dynamics of control s- tems. A large part of the fascination in nonlinear control stems from the fact that is deeply rooted in engineering and mathematics alike. The contributions to this volume reflect this double nature of nonlinear control. We would like to take this opportunity to thank all the contributors and the referees for their careful work. Furthermore, it is our pleasure to thank Franchise Lamnabhi-Lagarrigue, the coordinator of our network, for her s- port in organizing the workshop and the proceedings and for the tremendous efforts she puts into this network bringing the cooperation between the d- ferent groups to a new level. In particular, the exchange and the active p- ticipation of young scientists, also reflected in the Pedagogical Schools within the Network, is an asset for the field of nonlinear control.

Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems

Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems PDF

Author: Mariana Haragus

Publisher: Springer Science & Business Media

Published: 2010-11-23

Total Pages: 338

ISBN-13: 0857291122

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An extension of different lectures given by the authors, Local Bifurcations, Center Manifolds, and Normal Forms in Infinite Dimensional Dynamical Systems provides the reader with a comprehensive overview of these topics. Starting with the simplest bifurcation problems arising for ordinary differential equations in one- and two-dimensions, this book describes several tools from the theory of infinite dimensional dynamical systems, allowing the reader to treat more complicated bifurcation problems, such as bifurcations arising in partial differential equations. Attention is restricted to the study of local bifurcations with a focus upon the center manifold reduction and the normal form theory; two methods that have been widely used during the last decades. Through use of step-by-step examples and exercises, a number of possible applications are illustrated, and allow the less familiar reader to use this reduction method by checking some clear assumptions. Written by recognised experts in the field of center manifold and normal form theory this book provides a much-needed graduate level text on bifurcation theory, center manifolds and normal form theory. It will appeal to graduate students and researchers working in dynamical system theory.