Note.

An English translation by Anthony K Manning (University of Warwick) with title Geometric Theory of Dynamical Systems: An Introduction was published in 1982 and information about this translation appears below.

2.1. Acknowledgements.

This book grew from courses and seminars taught at IMPA and several other institutions both in Brazil and abroad, a first text being prepared for the Xth Brazilian Mathematical Colloquium. With several additions, it later became a book in the Brazilian mathematical collection Projeto Euclides, published in Portuguese, A number of improvements were again made for the present translation.

We are most grateful to many colleagues and students who provided us with useful suggestions and, above all, encouragement for us to present these introductory ideas on Geometric Dynamics. We are particularly thankful to Paulo Sad and, especially to Alcides Lins Neto, for writing part of a first set of notes, and to Anthony Manning for the translation into English.

2.2 Introduction.

We present in this book a view of the Geometric Theory of Dynamical Systems, which is introductory and yet gives the reader an understanding of some of the basic ideas involved in two important topics: structural stability and genericity.

This theory has been considered by many mathematicians starting with Poincare, Liapunov and Birkhoff. In recent years some of its general aims were established and it experienced considerable development.

More than two decades passed between two important events: the work of Andronov and Pontryagin (1937) introducing the basic concept of structural stability and the articles of Peixoto (1958-1962) proving the density of stable vector fields on surfaces. It was then that Smale enriched the theory substantially by defining as a main objective the search for generic and stable properties and by obtaining results and proposing problems of great relevance in this context. In this same period Hartman and Grobman showed that local stability is a generic property. Soon after this Kupka and Smale successfully attacked the problem for periodic orbits.

We intend to give the reader the flavour of this theory by means of many examples and by the systematic proof of the Hartman-Grobman and the Stable Manifold Theorems (Chapter 2), the Kupka-Smale Theorem (Chapter 3) and Peixoto's Theorem (Chapter 4). Several of the proofs we give are simpler than the original ones and are open to important generalisations. In Chapter 4, we also discuss basic examples of stable diffeomorphisms with infinitely many periodic orbits. We state general results on the structural stability of dynamical systems and make some brief comments on other topics, like bifurcation theory. In the Appendix to Chapter 4, we present the important concept of rotation number and apply it to describe a beautiful example of a flow due to Cherry.

Prerequisites for reading this book are only a basic course on Differential Equations and another on Differentiable Manifolds the most relevant results of which are summarised in Chapter 1. In Chapter 2 little more is required than topics in Linear Algebra and the Implicit Function Theorem and Contraction Mapping Theorem in Banach Spaces. Chapter 3 is the least elementary but certainly not the most difficult. There we make. systematic use of the Transversality Theorem. Formally Chapter 4 depends on Chapter 3 since we make use of the Kupka-Smale Theorem in the more elementary special case of two-dimensional surfaces.

Many relevant results and varied lines of research arise from the theorems proved here. A brief (and incomplete) account of these results is presented in the last part of the text. We hope that this book will give the reader an initial perspective on the theory and make it easier for him to approach the literature.

Rio de Janeiro, September 1981.
Jacob Palis, Jr., Welington De Melo

2.3. Review by: Russell B Walker.
Mathematical Reviews MR0669541 (84a:58004).

This graduate-level text is a beautiful treatment of the foundations of dynamical systems, a large and thriving area of modern mathematics. It is a translation of a book first published in Portuguese as part of the Brazilian mathematics collection, "Projeto Euclides". Though it is for the most part self-contained, an elementary background in differential equations and some knowledge of the basic concepts of differential topology (manifolds, diffeomorphism, tangent bundles, etc.) would smooth the way.

Particularly appreciated by the reviewer is the selection and ordering of material. As a preparatory text for students interested in doing research in dynamical systems this text is to be highly recommended.

2.4. Review by: Joel Robbin.
The American Mathematical Monthly 91 (7) (1984), 448-449.

[After giving many technical definition Robin writes the following.]

The work of Smale and his school in the sixties and seventies greatly advanced our understanding of differentiable dynamics ... . In particular, it emerged that structurally stable systems are (if suitably defined) generalisations of the map of the interval considered above and a wealth of examples and theories beyond structural stability were discovered. It also emerged that structurally stable systems are not as ubiquitous as first thought and much current research is devoted to the study of systems which are definitely not structurally stable.

The book under review provides a good introduction to this theory. It treats the $(C^{0})$ linearisation problem mentioned above but in $n$-dimensions. It also discusses the theory of structurally stable (continuous time) systems on 2-manifolds as developed by Peixoto. An indication of the more advanced theory is given, and there is a bibliography with 122 entries. The book contains ample detail, plenty of examples, and 62 figures illustrating the text. Both authors have themselves made important contributions to the subject, and it is to be hoped that their effort will attract new graduate students to a beautiful subject.

2.5. Review by: Adelina Georgescu.
Bulletin mathématique de la Société des Sciences Mathématiques de la RépubliqueSocialiste de Roumanie 32 (80) (1) (1988), 92.

At the time we are writing these lines this remarkable book of the well-known Brazilia mathematicians Palis and de Melo is already a reference paper for specialists in particular-sciences applied mathematics and undergraduate students in mathematics interested into an easy but rigorous introduction to two main topics of the dynamical systems theory: structural stability and genericity. By now, the book is available in Portuguese, English and Russian. It contains four chapters; the first one summarises the most relevant results on differential equations and differentiable manifolds, Chapter 2 deals with local stability, Chapter 3 is devoted to the Kupka-Smale theorem and the last one is concerned with genericity and stability of Morse-Smale vector fields. To a wider accessibility many treated examples, comments orientative figures, remarks are included. In publishing this book Springer-Verlag accomplished a very useful job.

3.1. From the Publisher.

Dynamic consequences of a transverse homoclinic intersection. Homoclinic tangencies: Cascade of bifurcations, scaling and quadratic maps. Cantor sets. Homoclinic tangencies, cantor sets, measure of bifurcation sets. Infinitely many sinks. Hyperbolicity. Markov partitions. Heteroclinic cycles. On the shape of some strange attractors.

3.2. From the Foreword.

Perhaps one of the most striking (and still somewhat puzzling) ways of performing substantial change in the dynamical structure (bifurcation) of a system is through the creation and unfolding of a cycle, in particular a homoclinic cycle. Poincaré first noticed the existence of homoclinic orbits in his prize essay on the 3-body problem.  Subsequently, in "Les méthods nouvelles de la Mécanique Céleste", he expressed amazement about the complexity of the orbit structure of a diffeomorphisms in the presence of a transverse homoclinic orbit. More than forty years latter, Birkhoff showed that any such homoclinic orbit is accumulated by periodic ones and in the sixties Smale put this fact into the framework of (persistent) hyperbolic sets with dense subsets of periodic orbits. In the last twenty years or so several results were obtained concerning the dynamics of a parametrised family of diffeomorphisms going through a homoclinic bifurcation.

3.3. Review by: Sebastean van Strien.
Mathematical Reviews MR0953789 (90a:58143).

This monograph gives an account of some recent developments related to the dynamics of one-parameter families of diffeomorphisms going through homoclinic bifurcations. It starts by proving some well-known facts. First a transverse homoclinic bifurcation is shown to give rise to the existence of hyperbolic horseshoes. Next it is shown that, provided some mild conditions are satisfied, the only way a horseshoe can be created is through a cascade of period doubling. ...

The style of the monograph is clear and the reviewer can definitely recommend it. In the last chapter the authors announce an expanded version of this monograph with a complete proof of Newhouse's result that homoclinic bifurcations give rise to intervals of parameter values for which the corresponding diffeomorphisms have an infinite number of periodic sinks.

4.1. From the Foreword.

Homoclinic bifurcations, which form the main topic of this monograph, belong to the area of dynamical systems, the theory which describes mathematical models of time evolution, like differential equations and maps. Homoclinic evolutions, or orbits, are evolutions for which the state has the same limit both in the "infinite future" and in the "infinite past".

Such homoclinic evolutions, and the associated complexity, were discovered by Poincaré and described in his famous essay on the stability of the solar system around 1890. This associated complexity was intimately related with the breakdown of power series methods, which came to many, and in particular to Poincaré, as a surprise.

The investigations were continued by Birkhoff who showed in 1935 that in general there is near a homoclinic orbit an extremely intricate complex of periodic solutions, mostly with a very high period.

The theory up to this point was quite abstract: though the inspiration came from celestial mechanics, it was not proved that in the solar system homoclinic orbits actually can occur. Another development took place which was much more directed to the investigation of specific equations: in order to model vacuum tube radio receivers Van der Pol introduced in 1920 a class of equations, now named after him, describing nonlinear oscillators, with or without forcing. His interest was mainly in the periodic solutions and their dependence on the forcing. In later investigations of this same type of equations, around 1950, Cartwright, Littlewood and Levinson discovered solutions which were much more complicated than any solution of a differential equation known up to that time.

Now we can easily interpret this as complexity caused by (the suspension of) a horseshoe, which in its turn is a consequence of the existence of one [transverse] homoclinic orbit, but that is inverting the history...

In fact, Smale, who originally had focussed his efforts on gradient and gradient-like dynamical systems, realised, when confronted with these complexities, that he should extend the scope of his investigations. Seventy years after Poincaré, Smale was again shocked by the complexity of homoclinic behaviour! By the mid 1960 he had a very simple geometric example (i.e. no formulas but just a picture and a geometric description), which could be completely analysed and which showed all the complexity found before: the horseshoe. This new prototype dynamical model, the horseshoe, together with the investigations in the behaviour of geodesic flows of manifolds with negative curvature (Hadamard, Anosov), grew, due to the efforts of a number of mathematicians, to an extension of the gradient-like theory which we now know as hyperbolic dynamics, and which in particular provides models for very complex (chaotic) dynamic behaviour.

Around 1980 this hyperbolic theory was used by Levi to reanalyse the qualitative behaviour of the solutions of Van der Pol's equation, largely extending the earlier results. He proved that, besides all the complexity we know from the hyperbolic theory, even the new and extreme complexity associated with homoclinic bifurcations, which we shall consider below, actually exists in the solutions of this equation.

Homoclinic bifurcations, or non-transverse homoclinic orbits, become important when going beyond the hyperbolic theory. In the late l960s, Newhouse combined homoclinic bifurcations with the complexity already available in the hyperbolic theory to obtain dynamical systems far more complicated than the hyperbolic ones. Ultimately this led to his famous result on the coexistence of infinitely many periodic attractors and was also influential on our own work on hyperbolicity or lack of it near homoclinic bifurcations. These developments form the main topic of the present monograph, of which we shall now outline the content.

We start with Chapter 0 that presents general background information about the hyperbolic theory and its relation to (structural) stability of systems, and discuss as well some initial aspects of chaotic dynamics; many results on stable manifolds and foliations are stated, and their proofs sketched, in Appendix 1. The later chapters, except the last one, do not depend on the results described in this chapter and are basically self-contained.

In Chapter 1, we give a number of simple examples of homoclinic orbits and bifurcations. Chapter 2 discusses the horseshoe example and shows how it is related to homoclinic orbits. Then, in Chapter 3, we consider some preliminary and more elementary consequences of the occurrence of a homoclinic bifurcation, especially in terms of cascades of bifurcations which have to accompany them.

In Chapters 4, 5, and 6, we come to our main topic: the investigation of situations where there is an interplay between homoclinic bifurcations and nontrivial basic sets, the sets being the building blocks of hyperbolic systems with complex behaviour. Since such basic sets often have a fractal structure, we start in Chapter 4 with a discussion of Cantor sets and fractal dimensions like Hausdorff dimension. In Chapter 5 the emphasis is on hyperbolicity near a homoclinic bifurcation associated with a basic set of small Hausdorff dimension. Then, in Chapter 6, we discuss types of homoclinic bifurcations which yields, in a persistent way, complexity beyond hyperbolicity. In this chapter we provide a new, and more geometric, proof of Newhouse's result on the coexistence of infinitely many periodic attractors. Finally, in Chapter 7 we present an overview of recent results, including specially Hénon-like and Lorenz-like strange attractors. We also pose new conjectures and problems which may lead to a better understanding of nonhyperbolic dynamics (the "dark realm" of dynamics) and the role of homoclinic bifurcations.

Summarising, we deal with the following rather striking collection of dynamical phenomena that take place at the unfolding of a homoclinic tangency

- transversal homoclinic orbits, which in turn are always associated to horseshoes (invariant hyperbolic Cantor sets): Chapters I and 2,

- cascades of homoclinic tangencies, i.e. sequences in the parameter line whose corresponding diffeomorphisms exhibit a homoclinic tangency: Chapter 3,

and for families of locally dissipative diffeomorphisms,

- cascades of period doubling bifurcations of periodic attractors (sinks): Chapter 3,

- cascades of critical saddle-node cycles: Chapter 7,

- residual subsets of intervals in the parameter line whose corresponding diffeomorphisms exhibit infinitely many coexisting sinks: Chapter 6 and Appendix 4,

- positive Lebesgue measure sets in the parameter line whose corresponding diffeomorphisms exhibit a Hénon-like strange attractor: Chapter 7 and Appendix 3,

- prevalence of hyperbolicity when the fractal (Hausdorff) dimension of the associated basic hyperbolic set is smaller than 1: Chapters 4 and 5 and Appendix 5,

- non-prevalence of hyperbolicity when the above fractal dimension is bigger than 1: Chapter 7.

In our presentation we mainly restrict ourselves to diffeomorphisms in dimension 2 (which is the proper context to investigate classical equations e.g. the forced Van der Pol equation), although extensions to higher dimensions are mentioned; also we concentrate mainly on the general theory as opposed to the analysis of specific equations. Consequently a number of topics like Silnikov's bifurcations and the Melnikov method are not discussed.

We hope that by putting this material together, rearranging it to some extent and pointing to recent and possible future directions, these results and their proofs will become more accessible, and will find their central place in dynamics which we think they merit.

We wish to thank a number of colleagues from several different institutions as well as Ph.D. students from the Instituto de Matematica Pura e Aplicada (IMPA) who much helped us in writing this book. Among them we mention M Benedicks, L Carleson, M Carvalho, L Diaz, P Duarte, R Mané, L Mora, S Newhouse, M J Pacífico, J Rocha, D Ruelle, R Ures, J C Yoccoz and most especially M Viana. Thanks are also due to Luiz Alberto Santos for his fine typing of this text.

4.2. Review by: Roger L Kraft.
SIAM Review 38 (2) (1996), 348-349.

The study of homoclinic points has been central to dynamical systems since they were first discovered by Poincare, who showed that the existence of one transverse homoclinic point implied the existence of an infinite number of them. Then Birkhoff showed that the existence of a transverse homoclinic point implied the existence of an infinite number of periodic points. Finally, Smale showed that the existence of a transverse homoclinic point implied the existence of a hyperbolic invariant Cantor set (now called a Smale horseshoe). This much about homoclinic points is well known and can be found in anyone of the (now many) first year graduate texts on dynamical systems, and even in some of the more elementary texts. However, the other consequences ... of a homoclinic tangency are not as well known and are rarely mentioned in textbooks.

The book under review is a well thought out response to this situation, appropriately written at the level of a second graduate course in dynamical systems. This book reviews the basic dynamical systems theory needed to understand homoclinic phenomena, presents proofs of the above consequences of a homoclinic tangency, and then outlines a research program "concerning homoclinic bifurcations and their relations to chaotic dynamics" (to use the author's words) that would try to answer the last question posed above. The authors conjecture that the answer is yes, which indicates the central role they believe homoclinic tangencies play in the global bifurcations of dynamical systems. Another interesting aspect of this book, which is hinted at by its subtitle, Fractal Dimensions and Infinitely Many Attractors, is that an important tool in developing this material is the study of fractal dimension theory, in particular, dimension theory for Cantor subsets of the real line.
...
The way this book is put together, first with a section that reviews what is important from the basic theory, then with a section that develops significant recent results in order to demonstrate useful techniques, followed by a section of open problems that can hopefully be attacked using the techniques just learned, makes one want to describe this text as "a thesis adviser in a book."

4.3. Review by: Michael Hurley.
Mathematical Reviews MR1237641 (94h:58129).

As its title indicates, this book is a study of the dynamical complexity that occurs when a family of diffeomorphisms undergoes a homoclinic bifurcation; that is, the creation or destruction of an orbit that is homoclinic to a hyperbolic saddle point.
...
The book brings together results due to many people (including the authors) over the last 30 years; much of this material was heretofore available only in the original journal articles. The text seems appropriate for a second graduate course in dynamical systems. To quote the authors, "We hope that by putting this material together, rearranging it to some extent and pointing to recent and possible future directions, these results and their proofs will become more accessible, and will find their central place in dynamics which we think they merit."

5.1. Preface.

These two volumes collect original research articles submitted by participants of the International Conference on Dynamical Systems held at IMPA, Rio de Janeiro, in July 19-28, 2000 to commemorate the 60th birthday of Jacob Palis.

These articles cover a wide range of subjects in Dynamics, reflecting the Conference's broad scope, itself a tribute to the diversity and influence of Jacob's contributions to the mathematical community worldwide, and most notably in Latin America, through his scientific work, his role as an educator of young researchers, his responsibilities in international scientific bodies, and the efforts he has always devoted to fostering the development of Mathematics in all regions of the globe.

His own mathematical work, which extends for more than 80 publications, is described in Sheldon Newhouse's opening article. It is, perhaps, best summarised by the following quotation from Jacob's recent nomination for the French Academy of Sciences: "sa vision, en constante évolution, a considérablement élargi le sujet".
As Jacob does not seem willing to slow down, we should expect much more from him in the years to come...

Rio de Janeiro and Paris, 20 May 2003.

6.1. From the publisher.

The Theory of Dynamical Systems was first introduced by the great mathematician Henri Poincaré as a qualitative study of differential equations. For more than forty years, Jacob Palis has made outstanding contributions to this area of mathematics. In the 1970s, following in the wake of Stephen Smale, he became one of the major figures in developing the Theory of Hyperbolic Dynamics and Structural Stability.

This volume presents a selection of Jacob Palis' mathematical contributions, starting with his PhD thesis and ending with papers on what is widely known as the Palis Conjecture. Most of the papers included in the present volume are inspired by the earlier work of Poincaré and, more recently, by Steve Smale among others. They aim at providing a description of the general structure of dynamical systems.

Jacob Palis, whose work has been distinguished with numerous international prizes, is broadly recognised as the father of the Latin American School of Mathematics in Dynamical Systems and one of the most important scientific personalities on the continent. In 2010 he was awarded the Balzan Prize for his fundamental contributions in the Mathematical Theory of Dynamical Systems, which has been the basis for many applications in various scientific disciplines.

6.2. Table of contents

- A Short Summary of my Scientific Life: J Palis.
- On Morse-Smale Dynamical Systems: J Palis.
- Structural Stability Theorems: J Palis and S Smale.
- A Note on $\Omega$-Stability: J Palis.
- Neighborhoods of Hyperbolic Sets: M Hirsch, J Palis, C Pugh and M Shub.
- Hyperbolic Non wandering Sets on Two-Dimensional Manifolds: S Newhouse and J Palis.
- The Topology of Holomorphic Flows with Singularity: C Camacho, N H Kuiper and J Palis.
- Topological Equivalence of Normally Hyperbolic Dynamical Systems: J Palis and F Takens.
- Moduli of Stability and Bifurcation Theory: J Palis.
- Characterising Diffeomorphisms with Modulus of Stability One: W de Melo, J Palis and S J van Strien.
- Bifurcations and Stability of Families of Diffeomorphisms: S Newhouse, J Palis and F Takens.
- Stability of Parametrized Families of Gradient Vector Fields: J Palis and F Takens.
- A Note on the Inclination Lemma ($\lambda$-Lemma) and Feigenbaum's Rate of Approach: J Palis.
- Cycles and Measure of Bifurcation Sets for Two-Dimensional Diffeomorphisms: J Palis and F Takens.
- Hyperbolicity and the Creation of Homoclinic Orbits: J Palis and F Takens.
- On the $C^{1}\Omega$ -Stability Conjecture: J Palis.
- Centralizers of Anosov Diffeomorphisms on Tori: J Palis and J C Yoccoz.
- Homoclinic Tangencies for Hyperbolic Sets of Large Hausdorff Dimension: J Palis and J-C Yoccoz.
- High Dimension Diffeomorphisms Displaying Infinitely Many Periodic Attractors: J Palis and M Viana.
- On the Arithmetic Sum of Regular Cantor Sets: J Palis and J-C Yoccoz.
- A Global View of Dynamics and a Conjecture on the Denseness of Finitude of Attractors: J Palis.
- Fers a cheval non uniformement hyperboliques engendres par une bifurcation homocline et densite nulle des attracteurs [Non-Uniformly Hyperbolic Horseshoes Generated by Homoclinic Bifurcations and Zero Density of Attractors]: J Palis and J-C Yoccoz.
- Homoclinic Tangencies and Fractal Invariants in Arbitrary Dimension: C G Moreira, J Palis and M Viana.
- A Global Perspective for Non-Conservative Dynamics: J Palis.
- Non-Uniformly Hyperbolic Horseshoes Arising From Bifurcations of Poincare Heteroclinic Cycles: J Palis and J-C Yoccoz.
- List of Publications of Jacob Palis Junior.
- List of Ph.D. Students of Jacob Palis Junior at IMPA.
- Acknowledgements.