Bifurcation Of Planar Vector Fields And Hilbert S Sixteenth Problem
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Bifurcations Of Planar Vector Fields And Hilbert S Sixteenth Problem
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Author : Robert Roussarie
language : en
Publisher: Springer Science & Business Media
Release Date : 1998-05-19
Bifurcations Of Planar Vector Fields And Hilbert S Sixteenth Problem written by Robert Roussarie and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 1998-05-19 with Mathematics categories.
In a coherent, exhaustive and progressive way, this book presents the tools for studying local bifurcations of limit cycles in families of planar vector fields. A systematic introduction is given to such methods as division of an analytic family of functions in its ideal of coefficients, and asymptotic expansion of non-differentiable return maps and desingularisation. The exposition moves from classical analytic geometric methods applied to regular limit periodic sets to more recent tools for singular limit sets. The methods can be applied to theoretical problems such as Hilbert's 16th problem, but also for the purpose of establishing bifurcation diagrams of specific families as well as explicit computations. - - - The book as a whole is a well-balanced exposition that can be recommended to all those who want to gain a thorough understanding and proficiency in the recently developed methods. The book, reflecting the current state of the art, can also be used for teaching special courses. (Mathematical Reviews)
Bifurcations Of Planar Vector Fields And Hilbert S Sixteenth Problem
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Author : Robert Roussarie
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-11-26
Bifurcations Of Planar Vector Fields And Hilbert S Sixteenth Problem written by Robert Roussarie and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-11-26 with Mathematics categories.
In a coherent, exhaustive and progressive way, this book presents the tools for studying local bifurcations of limit cycles in families of planar vector fields. A systematic introduction is given to such methods as division of an analytic family of functions in its ideal of coefficients, and asymptotic expansion of non-differentiable return maps and desingularisation. The exposition moves from classical analytic geometric methods applied to regular limit periodic sets to more recent tools for singular limit sets. The methods can be applied to theoretical problems such as Hilbert's 16th problem, but also for the purpose of establishing bifurcation diagrams of specific families as well as explicit computations. - - - The book as a whole is a well-balanced exposition that can be recommended to all those who want to gain a thorough understanding and proficiency in the recently developed methods. The book, reflecting the current state of the art, can also be used for teaching special courses. (Mathematical Reviews)
Global Bifurcation Theory And Hilbert S Sixteenth Problem
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Author : V. Gaiko
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-11-27
Global Bifurcation Theory And Hilbert S Sixteenth Problem written by V. Gaiko and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-11-27 with Mathematics categories.
On the 8th of August 1900 outstanding German mathematician David Hilbert delivered a talk "Mathematical problems" at the Second Interna tional Congress of Mathematicians in Paris. The talk covered practically all directions of mathematical thought of that time and contained a list of 23 problems which determined the further development of mathema tics in many respects (1, 119]. Hilbert's Sixteenth Problem (the second part) was stated as follows: Problem. To find the maximum number and to determine the relative position of limit cycles of the equation dy Qn(X, y) -= dx Pn(x, y)' where Pn and Qn are polynomials of real variables x, y with real coeffi cients and not greater than n degree. The study of limit cycles is an interesting and very difficult problem of the qualitative theory of differential equations. This theory was origi nated at the end of the nineteenth century in the works of two geniuses of the world science: of the Russian mathematician A. M. Lyapunov and of the French mathematician Henri Poincare. A. M. Lyapunov set forth and solved completely in the very wide class of cases a special problem of the qualitative theory: the problem of motion stability (154]. In turn, H. Poincare stated a general problem of the qualitative analysis which was formulated as follows: not integrating the differential equation and using only the properties of its right-hand sides, to give as more as possi ble complete information on the qualitative behaviour of integral curves defined by this equation (176].
Bifurcations Of Planar Vector Fields
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Author : Freddy Dumortier
language : en
Publisher: Springer
Release Date : 2006-12-08
Bifurcations Of Planar Vector Fields written by Freddy Dumortier and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-12-08 with Mathematics categories.
The book reports on recent work by the authors on the bifurcation structure of singular points of planar vector fields whose linear parts are nilpotent. The bifurcation diagrams of the most important codimension-three cases are studied in detail. The results presented reach the limits of what is currently known on the bifurcation theory of planar vector fields. While the treatment is geometric, special analytical tools using abelian integrals are needed, and are explicitly developed. The rescaling and normalization methods are improved for application here. The reader is assumed to be familiar with the elements of Bifurcation and Dynamical Systems Theory. The book is addressed to researchers and graduate students working in Ordinary Differential Equations and Dynamical Systems, as well as anyone modelling complex multiparametric phenomena.
Normal Forms And Bifurcation Of Planar Vector Fields
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Author : Shui-Nee Chow
language : en
Publisher: Cambridge University Press
Release Date : 1994-07-29
Normal Forms And Bifurcation Of Planar Vector Fields written by Shui-Nee Chow and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1994-07-29 with Mathematics categories.
This book is concerned with the bifurcation theory, the study of the changes in the structures of the solution of ordinary differential equations as parameters of the model vary.
Normal Forms Melnikov Functions And Bifurcations Of Limit Cycles
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Author : Maoan Han
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-04-23
Normal Forms Melnikov Functions And Bifurcations Of Limit Cycles written by Maoan Han and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-04-23 with Mathematics categories.
Dynamical system theory has developed rapidly over the past fifty years. It is a subject upon which the theory of limit cycles has a significant impact for both theoretical advances and practical solutions to problems. Hopf bifurcation from a center or a focus is integral to the theory of bifurcation of limit cycles, for which normal form theory is a central tool. Although Hopf bifurcation has been studied for more than half a century, and normal form theory for over 100 years, efficient computation in this area is still a challenge with implications for Hilbert’s 16th problem. This book introduces the most recent developments in this field and provides major advances in fundamental theory of limit cycles. Split into two parts, the first focuses on the study of limit cycles bifurcating from Hopf singularity using normal form theory with later application to Hilbert’s 16th problem, while the second considers near Hamiltonian systems using Melnikov function as the main mathematical tool. Classic topics with new results are presented in a clear and concise manner and are accompanied by the liberal use of illustrations throughout. Containing a wealth of examples and structured algorithms that are treated in detail, a good balance between theoretical and applied topics is demonstrated. By including complete Maple programs within the text, this book also enables the reader to reconstruct the majority of formulas provided, facilitating the use of concrete models for study. Through the adoption of an elementary and practical approach, this book will be of use to graduate mathematics students wishing to study the theory of limit cycles as well as scientists, across a number of disciplines, with an interest in the applications of periodic behavior.
Infinite Dimensional Dynamical Systems
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Author : John Mallet-Paret
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-10-11
Infinite Dimensional Dynamical Systems written by John Mallet-Paret and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-10-11 with Mathematics categories.
This collection covers a wide range of topics of infinite dimensional dynamical systems generated by parabolic partial differential equations, hyperbolic partial differential equations, solitary equations, lattice differential equations, delay differential equations, and stochastic differential equations. Infinite dimensional dynamical systems are generated by evolutionary equations describing the evolutions in time of systems whose status must be depicted in infinite dimensional phase spaces. Studying the long-term behaviors of such systems is important in our understanding of their spatiotemporal pattern formation and global continuation, and has been among major sources of motivation and applications of new developments of nonlinear analysis and other mathematical theories. Theories of the infinite dimensional dynamical systems have also found more and more important applications in physical, chemical, and life sciences. This book collects 19 papers from 48 invited lecturers to the International Conference on Infinite Dimensional Dynamical Systems held at York University, Toronto, in September of 2008. As the conference was dedicated to Professor George Sell from University of Minnesota on the occasion of his 70th birthday, this collection reflects the pioneering work and influence of Professor Sell in a few core areas of dynamical systems, including non-autonomous dynamical systems, skew-product flows, invariant manifolds theory, infinite dimensional dynamical systems, approximation dynamics, and fluid flows.
Proceedings Of The Sixth International Colloquium On Differential Equations
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Author : Dimitūr Baīnov
language : en
Publisher: VSP
Release Date : 1996-01-01
Proceedings Of The Sixth International Colloquium On Differential Equations written by Dimitūr Baīnov and has been published by VSP this book supported file pdf, txt, epub, kindle and other format this book has been release on 1996-01-01 with Mathematics categories.
The Sixth International Colloquium on Differential Equations was organized by the Institute for Basic Science of Inha University, the International Federation of Nonlinear Analysts, the Mathematical Society of Japan, the Pharmaceutical Faculty of the Medical University of Sofia, the University of Catania, and UNESCO, with the cooperation of a number of international mathematical organizations, and was held at the Technical University of Plovdiv, Bulgaria, from 18 to 23 August 1995. This proceedings volume contains selected talks which deal with various aspects of differential and partial differential equations.
Bifurcation Theory And Spatio Temporal Pattern Formation
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Author : Wayne Nagata
language : en
Publisher: American Mathematical Soc.
Release Date : 2006-10-03
Bifurcation Theory And Spatio Temporal Pattern Formation written by Wayne Nagata and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-10-03 with Mathematics categories.
Nonlinear dynamical systems and the formation of spatio-temporal patterns play an important role in current research on partial differential equations. This book contains articles on topics of current interest in applications of dynamical systems theory to problems of pattern formation in space and time. Topics covered include aspects of lattice dynamical systems, convection in fluid layers with large aspect ratios, mixed mode oscillations and canards, bacterial remediation of waste, gyroscopic systems, data clustering, and the second part of Hilbert's 16th problem. Most of the book consists of expository survey material, and so can serve as a source of convenient entry points to current research topics in nonlinear dynamics and pattern formation. This volume arose from a workshop held at the Fields Institute in December of 2003, honoring Professor William F. Langford's fundamental work on the occasion of his sixtieth birthday. Information for our distributors: Titles in this series are copublished with the Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada).
Dynamical Systems With Applications Using Mathematica
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Author : Stephen Lynch
language : en
Publisher: Birkhäuser
Release Date : 2017-10-12
Dynamical Systems With Applications Using Mathematica written by Stephen Lynch and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-10-12 with Mathematics categories.
This book provides an introduction to the theory of dynamical systems with the aid of the Mathematica® computer algebra package. The book has a very hands-on approach and takes the reader from basic theory to recently published research material. Emphasized throughout are numerous applications to biology, chemical kinetics, economics, electronics, epidemiology, nonlinear optics, mechanics, population dynamics, and neural networks. Theorems and proofs are kept to a minimum. The first section deals with continuous systems using ordinary differential equations, while the second part is devoted to the study of discrete dynamical systems.