1.1. From the Preface.

Bifurcation theory is the study of small solutions of non linear equations, depending on a number of parameters, when for certain values of these parameters more than one solution may appear.

The scope of applications of this theory is thus extremely large, ranging from Biology to Physics, using tools as varied as functional analysis, topology, partial differential equations or numerical analysis.

Because of the difficulties encountered in obtaining explicit solutions to such problems, one tries to approximate the initial equation by a simpler one and from the properties of the solutions to the latter gain some insight for those of the former. If the nonlinear equation is a smooth function of its variables, a standard approximation (when justified) is to linearise the equation.

Among the techniques used to study such problems, the topological method is particularly attractive because it gives quick qualitative results, such as existence of a solution, even if for computational purposes their usefulness is rather limited. Many successful applications of topology have been made in this context, some dating back to the beginning of this century with work by Poincare, Schauder, Leray, Lusternick, Schnirelman, Morse and others, and some quite recent in the setting of global analysis and fixed point theory.

The object of this paper is to try to give a new look at one of the most basic ideas in this topological approach, that is deformation of maps or homotopy theory, and by doing so include the well known Leray-Schauder degree theory in the setting of generalised degree theories.

For this reason, the general philosophy of this paper is to give simple, direct, proofs of some very often used results as well as new ones. So, except for the Hopf bifurcation problem, almost no example of applications are included here and one is referred to the very extensive literature on the subject.

Chapter one is concerned with the study of bifurcation for equations with only one parameter, real or complex, entering. The type of equation studied is of the form: $Ax - T(\lambda)x = g(x, \lambda)$ where $A$ is a Fredholm operator of non-negative index, $T(A)$ is a continuous linear operator analytic in $\lambda$ and $g(x, \lambda)$ is a small non-linearity. In section I, the basic definitions and assumptions are made; section II considers the main topological tools used in the chapter, i.e. degree theory and the Hopf map. The homotopy class of a particular map is computed. Section III is concerned with the special case of an eigenvalue of $A$ which has finite multiplicity and which gives a splitting of the Banach space considered. In section IV, the main result for this type of bifurcation is proved, that is, after a reduction à la Liapounov-Schrnidt, natural extensions of Krasnosel'skii theorem, [22], are given for the real and complex cases and operators of index zero. Also the existence of a family of solutions for the positive index case is established. Section V is rather technical in nature but enables one to see the relationship between eigenvalues which are isolated and eigenvalues which have finite multiplicity. Lastly, section VI extends all the previous results to the case where $T(\lambda)$ replaces $\lambda I$.

The setting of section VI leads naturally to chapter two and the study of bifurcation theory when more than one parameter are involved. In this case, it is impossible to give simple criteria to prove the existence of a solution, and the attempt is to give a philosophical viewpoint for the study of such problems. So, section I is concerned with the case where there are enough parameters in the bifurcation equation to use degree theory, section II gives some sort of recipe to study the general case, and section III applies these ideas to the bifurcation of periodic orbits for autonomous system.

Chapter three goes from the local problem to study what happens to a bifurcating solution in the large. Thus as light sharpening of the important result of Rabinowitz' global theorem is given for the real case and, in the complex case, the notion of stable cohomotopy and its characterisation due to Geba and Granas is used to prove a similar result. Finally the global version of the Hopf bifurcation problem, due to Alexander and Yorke, is treated.

At last, chapter four returns to the local problem, but with new conditions on the nonlinearity. Hence, generalisations of results of Sather and Kirchgässner are proved. Here again no attempt at the greatest generality has been made, but rather a method and a complement to Dancer's articles are given.

1.2. Acknowledgements

This paper is an extended version of the author's doctoral thesis written at the Courant Institute of New York University. The author would like to express his gratitude to Professor Louis Nirenberg for his advice, for his insight and his kindness. Thanks are also due to the Consejo Nacional de Ciencia y Tecnologia of Mexico which provided a partial support for the author's doctoral studies and in particular for this research.

2.1. Extract from the review by: J E Marsden.
Mathematical Reviews MR0551656 (82c:58013).

The author generalises the result of J C Alexander and J A Yorke on global Hopf bifurcation to infinite dimensions. The idea is to have an analogue of Rabinowitz' theorem on global bifurcation of stationary solutions for periodic orbits. The general theory proceeds via stable homotopy theory, but can also be based on the Fuller index and its modifications due to S N Chow, J Mallet-Paret and Yorke. The theory is applied to mildly quasilinear parabolic equations and to the Navier-Stokes equations in particular.

3.1. Extract from the review by: M Shearer.
Mathematical Reviews MR0679145 (84f:58029).

The topic of this paper is topological methods in bifurcation theory. The emphasis is upon a study of linearised problems, depending upon one or more parameters. The first of the three chapters describes the Lyapunov-Schmidt reduction, briefly summarises enough homotopy theory to define generalized degree, and indicates the relationship to the ordinary degree theory. Immediate consequences for parameterised families of linear maps are listed.

The second chapter deals with local bifurcation. Krasnosel'skii's result that odd algebraic multiplicity implies bifurcation is presented, although not in the generality discussed by R Magnus. The multiparameter case is discussed in general terms, emphasising the interplay between the number of parameters and the dimensions of the spaces underlying the bifurcation equation. Chapter Three is concerned with Rabinowitz' global bifurcation result, and curiously includes a proof of the technical lemma required to set up the degree theory argument.

4.1. Extract from the review by: Richard C Swanson.
Mathematical Reviews MR0726591 (85d:58063).

The article under review provides a new topological proof that a Hopf bifurcation passes through a continuum of distinct periodic orbits, periods, and values of the parameter $\mu$. The novelty of this proof is to regard the existence of a periodic solution to an autonomous differential equation as an obstruction to the construction of an extension for equivariant mappings. Such obstructions are then realised as certain non-vanishing cohomology groups.

5.1. Extract from the review by: Norman Dancer.
Mathematical Reviews MR0784013 (86k:58023).

In this interesting paper, the author uses obstruction theory to prove global bifurcation theorems for periodic solutions. To do this, he uses an idea from some of his earlier papers to prove the existence of solutions by considering an extension problem. In particular, he proves a multiparameter version of the global Hopf bifurcation theorem due to S N Chow et al. Most of his result depends upon the primary obstruction. He first considers the local problem. There is an $S^{1}$-symmetry (due to translation invariance). He uses a trick to reduce to the case where $S^{1}$ acts freely and then takes quotients to eliminate the symmetry. He includes a short discussion of the obstruction theory he needs.

6.1. From the Abstract.

This paper is concerned with the topological dimension of global branches of solutions appearing in different problems of Nonlinear Analysis, in particular multiparameter (including infinite dimensional) continuation and bifurcation problems. By considering an extension of the notion of essential maps defined on sets and using elementary point set topology, we are able to unify and extend, in a self-contained fashion, most of the recent results on such problems. Our theory applies whenever any generalised degree theory with the boundary dependence property may be used, but with no need of algebraic structures. Our applications to continuation and bifurcation follow from the nontriviality of a local invariant, in the stable homotopy group of a sphere, and give information on the local dimension and behaviour of the sets of solutions, of bifurcation points and of continuation points.

6.2. Extract from the review by: Norman Dancer.
Mathematical Reviews MR0800246 (86k:58022).

In this paper, the authors prove some nice results on the dimension and structure of the solution set for multiparameter equations. In particular, they are interested in bifurcation problems and continuation problems on infinite-dimensional spaces. Unlike the work of Alexander and his co-workers, they use relatively elementary topological tools but obtain strong results. Their results apply to 0-epi-maps and in particular to any class of maps which have a reasonable degree theory.

7.1. From the Abstract.

This paper is devoted to some of the results in bifurcation theory obtained by topological methods in the last 25 years. The cases of one and several parameters will be reviewed, with "necessary" and sufficient conditions for bifurcation, both local and global, and the structure of the bifurcation set will be studied. The case of equivariant bifurcation will be considered, with a special application to the case of abelian groups.

7.2. Extract from the review by: Jacobo Pejsachowicz.
Mathematical Reviews MR1322327 (96e:58027).

This is a review article devoted to the development of topological methods in bifurcation theory in the past 25 years. Here bifurcation means bifurcation of solutions of a nonlinear equation from the trivial branch: assuming that a given equation $F(u) = 0$ has a known (trivial) branch of solutions depending on a parameter $\lambda \in \mathbb{R}^{n}$, a bifurcation point is any point of the trivial branch belonging to the closure of the set of nontrivial solutions.

Together with a comprehensive description of a variety of known results in one- and multiparameter bifurcation, the paper also contains some new applications of equivariant homotopy theory to bifurcation with symmetry. These results were recently obtained by the author and have not appeared previously in the literature.

8.1. From the Preface.

The present book grew out as an attempt to make more accessible to non-specialists a subject - Equivariant Analysis - that may be easily obscured by technicalities and (often) scarcely known facts from Bquivariant Topology. Quite frequently. the authors of research papers on Equivariant Analysis tend to assume that the reader is well acquainted with a hoard of subtle and refined results from Group Representation Theory, Group Actions, Equivariant Homotopy and Homology Theory (and co-counter parts. i.e. Cohomotopy and Cohomology) and the like. As an outcome, beautiful theories and elegant results are poorly understood by those researchers that would need them mostly: applied mathematicians. This is also a self-criticism.

We felt that an overturn was badly needed. This is what we try to do here. If you keep in mind these few strokes you most probably will understand our strenuous efforts in keeping the mathematical background to a minimum. Surprisingly enough, this is at the same time an easy and very difficult task. Once we look the decision of expressing a given mathematical fact in as elementary as possible terms, then the easy part of the game consists in letting ourselves to go down to ever simpler terms. This way one swiftly enters the realm of stop and go procedures, the difficult part being when and where to stop. In our case, we felt relatively at ease only when we arrived at the safe harbour of matrices. Of course, you have to buy a ticket to enter. The fair price is to become a jingler with them. After all, nothing is given for free. We have enjoyed (and suffered) with the fact that so many beautiful results can be obtained with so little mathematics. Our hope is that you will enjoy (and not suffer) reading this book.

8.2. Extract from the review by: Wieslaw Z Krawcewicz
Mathematical Reviews MR1984999 (2004i:47122).

The book under review is the first of its kind on equivariant degree theory and should be considered as an important contribution to modern nonlinear analysis. The importance of symmetry in every aspect of life cannot be underestimated. Unfortunately, studying nonlinear problems with symmetries is not simple and involves advanced mathematical methods and techniques that make this area difficult to access by non-specialists.

The book by Ize and Vignoli is an honest and substantial effort to open the area of equivariant analysis to applied mathematicians interested in nonlinear equations with symmetries. It requires minimal mathematical background, yet cannot be considered as elementary and easy to read. This is a serious mathematical monograph that can be used by researchers and students in nonlinear analysis as a valuable resource for equivariant methods and techniques. In fact it is a gold mine of interesting ideas and new approaches which make the reading stimulating, especially for those who are interested in applications of these methods to concrete mathematical models with symmetries.

Let us briefly explain what the equivariant analysis presented here is about. It deals with the impact of symmetries, represented by a certain group $G$ and translated as the equivariance of the corresponding operators, on the existence, multiplicity, stability and topological structure of solutions of nonlinear equations, bifurcation phenomena (local and/or global, with one or more parameters), the applicability of different kinds of approximation schemes, etc. In particular it involves natural symmetries in mathematical models, such as systems of ODE's with symmetries (including delay equations), among them the symmetric Hamiltonian systems, PDE's on symmetric domains, phase transition and symmetry breaking.

Degree theory turned out to be one of the main tools in the study of nonlinear problems. It was extensively used in a large variety of settings to conduct qualitative analysis of nonlinear equations. In the form presented in the book under review, the equivariant degree is an alternative to other topological methods, such as variational methods (minimax theory, Morse theory, the Conley index, Morse-Floer complexes), singularity theory, and cohomological obstruction theory dealing with the problems of equivariant analysis. We have to point out that the presentation by Ize and Vignoli of the related concepts and methods of homotopy theory, which should be called "geometric obstruction theory'', avoids the algebraic formalism of cohomology theory. This "simplification'' may possibly lead to a widespread usage of the equivariant degree methods in applied areas of mathematics. However, technicalities, scrupulous but elaborated computations and sometimes confusing notations can cause a lot of frustration to a reader new to this area of mathematics. Nevertheless, this book is the first attempt at a systematic and detailed exposition of equivariant degree theory in a setting relevant to nonlinear analysis.

8.3. From: Jorge Ize: A Tribute to his Mathematical Work, by Z Balanov, W Krawcewicz and J Pejsachowicz.
Boletín de la Sociedad Matemática Mexicana (3) 18 (2) (2012), 89-112.

A special consideration should be given to the monograph "Equivariant Degree Theory" by Ize and Vignoli - the first book written on the topic of the equivariant degree theory and its applications to differential equations with symmetries. This pioneering work constitutes a significant contribution to the area of nonlinear analysis. Although it requires only minimal mathematical background, it is a serious work which is not easy to read. One should remember that this was just the first attempt to open the stream of ideas related to the equivariant degree theory to the wide public. Therefore, one should not be surprise to find out that many important topics were only briefly outlined or presented in very technical way. Nevertheless, a reader will discover there a multitude of interesting ideas and new approaches that will give an inspiration to conduct further research. We believe that all specialists in the field of nonlinear analysis should appreciate this book, which is an excellent source of information and ideas related to the equivariant degree and its applications.