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Author: Shui-Nee Chow
Publisher: Cambridge University Press
Published: 1994-07-29
Total Pages: 482
ISBN-13: 0521372267
DOWNLOAD EBOOK →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.
Author: Freddy Dumortier
Publisher: Springer
Published: 2006-12-08
Total Pages: 234
ISBN-13: 3540384332
DOWNLOAD EBOOK →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.
Author: Robert Roussarie
Publisher: Springer Science & Business Media
Published: 2013-11-26
Total Pages: 215
ISBN-13: 303480718X
DOWNLOAD EBOOK →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)
Author: Robert Roussarie
Publisher: Springer Science & Business Media
Published: 1998-05-19
Total Pages: 230
ISBN-13: 9783764359003
DOWNLOAD EBOOK →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)
Author: Daniel Panazzolo
Publisher: American Mathematical Soc.
Published: 2002
Total Pages: 122
ISBN-13: 0821829270
DOWNLOAD EBOOK →This work aims to prove a desingularization theorem for analytic families of two-dimensional vector fields, under the hypothesis that all its singularities have a non-vanishing first jet. Application to problems of singular perturbations and finite cyclicity are discussed in the last chapter.
Author: Victoriano Carmona
Publisher: Springer
Published: 2018-09-15
Total Pages: 424
ISBN-13: 3319667661
DOWNLOAD EBOOK →This book is part of a two volume set which presents the analysis of nonlinear phenomena as a long-standing challenge for research in basic and applied science as well as engineering. It discusses nonlinear differential and differential equations, bifurcation theory for periodic orbits and global connections. The integrability and reversibility of planar vector fields and theoretical analysis of classic physical models are sketched. This first volume concentrates on the mathematical theory and computational techniques that are essential for the study of nonlinear science, a second volume deals with real-world nonlinear phenomena in condensed matter, biology and optics.
Author: Jean-Pierre Francoise
Publisher: Springer
Published: 2006-11-14
Total Pages: 404
ISBN-13: 354046722X
DOWNLOAD EBOOK →Author: Christiane Rousseau
Publisher: Springer Science & Business Media
Published: 2004-02-29
Total Pages: 548
ISBN-13: 9781402019296
DOWNLOAD EBOOK →Proceedings of the Nato Advanced Study Institute, held in Montreal, Canada, from 8 to 19 July 2002
Author: Maoan Han
Publisher: Springer Science & Business Media
Published: 2012-04-23
Total Pages: 408
ISBN-13: 1447129180
DOWNLOAD EBOOK →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.
Author: Robert H. Roussarie
Publisher: Birkhauser
Published: 1998
Total Pages: 232
ISBN-13:
DOWNLOAD EBOOK →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.
