Diff. Equa. & Dyna. Sys.
Applied Equivariant Degree
By Zalman Balanov, Wieslaw Krawcewicz and Heinrich Steinlein
The book is a self-contained comprehensive exposition of the equivariant degree theory and its applications to a variety of problems arising in physics, chemistry, biology and engineering. This monograph presents the theoretical foundations, construction, and the fundamental properties of the equivariant degree and its practical variations, which are applied to a series of examples from (functional) differential equations. It contains.
1. the first thorough and complete introduction up to the present state of art to equivariant degree theory including non-abelian actions, and
2. provides for the first time several computer routines allowing an effective practical computation of the degree, illustrated by numerous concrete examples and charts.
The exposition of the material is mainly addressed to experienced researchers and graduate students interested in applications of equivariant topological methods, or working with differential equations and their applications, like physicists, biologists, chemists and engineers dealing with nonlinear dynamics with symmetries.
To View /Download Contents and Chapter1 (Introduction)
Professor William F. Langford
This is a beautiful book and a valuable contribution to the mathematical literature. It is a worthy successor to the famous classic of Krasnosel'skii, "Topological Methods in the Theory of Nonlinear Integral Equations", that I have kept on my shelf since my graduate student days. It reflects the enormous progress that has been made in degree theory for nonlinear equations, since that early classic.
This book represents a significant advance for degree theory in the equivariant setting; that is, to systems of equations with symmetry. For example, it introduces the notion of G-equivariant twisted degree, which is essential for the study of equivariant Hopf bifurcations of dynamical systems. It presents a very complete treatment of this important case.
A major strength of this book is that it includes explicit applications of equivariant degree theory to examples of dynamical systems with symmetry that are important in physical applications. As promised in the title of the book, it is written to facilitate applications of equivariant degree theory. In addition, the book is written in a friendly and accessible style that is most pleasant to read. I recommend this book to all who are working in nonlinear analysis. It is a must-have acquisition for any university mathematics library.
Professor Jean Mawhin
Equivariant degree theory is both powerful and difficult. It is a blend of group theory, algebraic topology, and its ultimate computational aspects may require computer science. In order to obtain global information on nonlinear differential equations with symmetries, it is almost unavoidable. For example, when dealing with periodic solutions of autonomous ordinary differential equations through a functional analytic approach, like in the study of Hopf bifurcation, Leray-Schauder degree can only detect the equilibria, and remains blind to nontrivial orbits. The right topological tool to overcome the difficulty is equivariant degree, and the price to pay is dealing with an object somewhat more complicated than an integer.
The original approach to equivariant degree developed in the substantial treatise ‘Applied Equivariant Degree?of Z. Balanov, W. Krawcewicz and H. Steinlein intends not only to offer a beautiful theory, but also an approach which can be used in applications. This explains their special care in giving explicit computations of this degree, even if the result of the ‘computation?can be surprising to mathematicians familiar with classical degrees. But, after all, fourty year ago, the classical degree also looked quite exotic to mathematicians only exposed to Brouwer or Schauder fixed point theorem. The readers who will make the valuable effort of reading the book will be helped by the detailed presentation of the used tools of group theory, as well as by the paradigmic case of S1-equivariant maps.
The applications occupy half of the volume, with special emphasis upon Hopf bifurcation for ordinary and functional differential equations, symmetric bifurcation in parabolic systems, systems of van der Pol equations, and variational problems. Eighty pages of appendices develope Maple routines for computing equivariant degrees, algebraic methods for the study of bounded solutions of homogeneous ordinary differential equations, and some complements on equivariant homogeneous maps.
This monograph has all the ingredients to become rapidly a reference book for all mathematicians interested in the construction, computation and application of equivariant degree. It undoubtly will provide the fram for inspiring many other applications of this powerful topological tool to problems with symmetries.
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