First International Symposium on Dynamical Systems

Maurício Peixoto organised the first International Symposium on Dynamical Systems, held at the University of Bahia, Salvador, Brazil from 26 July to 14 August. He edited the conference proceeding which was published as M M Peixoto (ed.), Dynamical Systems. Proceedings of a Symposium Held at the University of Bahia, Salvador, Brazil, July 26-August 14, 1971 (Academic Press, 1973). We give below the publisher's description of the book and the preface by Maurício Peixoto.

1. Description

Dynamical Systems is a collection of papers that deals with the generic theory of dynamical systems, in which structural stability becomes associated with a generic property. Some papers describe structural stability in terms of mappings of one manifold into another, as well as their singularities. One paper examines the theory of polyhedral catastrophes, particularly, the analogues of each of the four basic differentiable catastrophes which map the line. Other papers discuss isolating blocks, the exponential rate conditions for dynamical systems, bifurcation, catastrophe, and a non-density theorem. One paper reviews the results of functional differential equations which show that a qualitative theory will emerge despite the presence of an infinite dimensionality or of a semigroup property. Another paper discusses a class of quasi-periodic solutions for Hamiltonian systems of differential equations. These equations generalize a well-known result of Kolmogorov and Arnold on perturbations of $n$-dimensional invariant tori for Hamiltonian systems of $n$ degrees of freedom. The researcher can derive mathematical models based on qualitative mathematical argument by using as "axioms" three dynamic qualities found in heart muscle fibres and nerve axons. The collection can prove useful for mathematicians, students and professors of advanced mathematics, topology or calculus.

2. Preface

These proceedings contain the contributions that were presented at the Symposium on Dynamical Systems held at Salvador.

The papers by J Mather, S Smale, R Thom, and E C Zeeman correspond to four series of special lectures given by them.

Roughly speaking, the symposium was devoted to the generic theory of dynamical systems, where one looks for properties that are true for "most" dynamical systems. This theory began to take shape a decade or so ago as an outgrowth of the case of structural stability in two dimensions, where structurally stable flows are simple enough and constitute an open and dense set in the space of all flows. Although this proved false in higher dimensions, experience has shown that one always finds some kind of structural stability associated with a generic property. This point of view, developed with great success by Smale and others, now dominates the qualitative theory of differential equations, including bifurcation theory, with repercussions in related topics, such as Hamiltonian mechanics, foliations, and Morse theory. Most papers in this book are along this line, including Smale's paper on economics.

On the other hand, the same idea of structural stability, looked on from the point of view of mappings of one manifold into another, and their singularities, was undergoing a parallel and somewhat more successful development, in the hands of Whitney, Thom, and Mather. By this we mean that structurally stable maps turned out to constitute an open dense set in the space of all maps. This is reported here in one paper by Mather.

These two points of view were instrumental in the formulation by Thom of his very deep theory - or rather method - of catastrophes, which aims to put under the sway of the mathematician a vast array of phenomena, thus far considered beyond his reach. The beautiful paper by Zeeman on heartbeats illustrates this method.

Altogether, we find in this book many important contributions to mathematics, both pure and applied, and exhibiting great conceptual unity.

M M Peixoto
Rio, February 1972

Last Updated November 2022