He was an intelligent young man, with very clear ideas about mathematics. It was a very important event, because it changed my life, in a certain sense. ... and I was blown away by his teaching; at that time I decided that I would study something that involved mathematics.

As a career as a mathematician did not exist, I went to study the subject in the only place where it was taught, that is, at the School of Engineering.

During the five years of the engineering course, Leopoldo and I were inseparable companions. ... Marília's influence was very strong during these "golden years" ... Both Marília Chaves and Leopoldo Nachbin were important influences in my becoming a mathematician. This in the sense of trying to live for and by mathematics.

... there is presented a brief discussion of the basic theory of convex functions as developed by Jensen and others, followed by a more extensive treatment of generalised convex functions. The latter development contains contributions which the author himself has made.

It was a State Department grant, which cost us almost daily trips to Ministry of Foreign Affairs for about a year; the aid came from the Buenos Aires Convention, through which the American government gave scholarships to Latin Americans.

I never regretted having dropped out of the doctorate, because a professorship at the University of Brazil appeared very infrequently, since it was for life; it was not an opportunity to be wasted.

The encounter, apparently accidental, followed by the active browsing through it, had launched Peixoto's insightful mathematical intuition. At home, he worked on the article until late hours. Afterwards he also discussed it with his wife Marília C Peixoto, also an engineer, mathematician and his assistant in the Chair of Mechanics. They decided that, in due time, he should prepare a presentation on Structural Stability in the Seminar he directed, which had the participation of Marília, selected students and teaching assistants of the Chair of Mechanics.

Father Henry DeBaggis, an Assistant Professor of Mathematics at Notre Dame and a PhD. under Karl Menger at Notre Dame also, joined our Project in 1949 and was a member for two years 1949-51. While his thesis was on Hyperbolic Geometry, he had little taste for that subject and wished to change over to Differential Equations. A reading of an appendix in Andronov and Chaikin 1949 recently appeared, awoke his interest in structural stability.

Lefschetz offered me a table in his spacious office and we talked at length about mathematics, mathematicians and everything else under the sun. We became true friends as much as his eminence could allow, and also the fact that, at seventy-three, he was more than twice my age. Lefschetz was very fond of Pontryagin and told me that at the age of 14, Pontryagin had lost both his eyes in an accident with a stove. Lefschetz himself had been in an accident in 1910, aged twenty-six, and had both arms amputated just below the elbows. He then decided to become a mathematician. In fact, before the summer of 1958, Lefschetz had met Pontryagin only on a few occasions during a conference in Moscow in the early 1930s. They talked at length about mathematics and Lefschetz was tremendously impressed. He referred to the young Pontryagin of those days in hyperbolic terms: "Everything he touched turned to gold."

It is only fair to acknowledge here my indebtedness to Professor S Lefschetz, who followed closely the development of this work, and to Professor H Whitney for his kind advice. Both read a first draft of the manuscript and suggested many valuable improvements ...

During the International Congress of Mathematicians in Edinburgh in 1958, Lefschetz took the initiative to arrange a meeting between me and Pontryagin and offered to serve as an English-Russian interpreter. The idea was that I would communicate to Pontryagin the content of 'On structural stability' that had not yet been published. From my point of view, the meeting was not very productive. Pontryagin was not interested in the matter ...

... the theorem of Peixoto on the structural stability in two-dimensional varieties inspired the mathematician S Smale to create the general theory of dynamic systems.

Another mathematician who believed in structural stability, and who would become the central figure in the qualitative theory of ordinary differential equations, was Steve Smale. I was introduced to him by Elon Lima at Princeton in the summer of 1958 shortly after I returned from Edinburgh. Before that, some time ago, and through Lima himself, I was already aware that Smale had great promise. Later, in evocative notes, Smale wrote about our meeting at Princeton and describes his reaction upon learning of the results that were supposed to appear in my Annals article: "Through Lefschetz, Peixoto became interested in structural stability and showed me his own results on structural stability in the disk (in an article due to appear in the "Annals of Mathematics, 1959"). I was immediately excited, not only by what he was doing, but by the possibility that, using my experience in Topology, I could extend his work to the n th dimension."

For his pioneering and fundamental study of the structural stability of dynamical systems, in particular, for proving that flows on surfaces are generally structurally stable.

I would like to point out that this work on structural stability was basically carried out in several articles mentioned below, one of which was in collaboration with my first wife Marília [Structural Stability in the plane with enlarged boundary conditions (1959)], who did not live to see the end of this adventure. However, her influence in those golden days, decisive and already distant, was great.

We survey here a certain line of developments of bifurcation theory of dynamical systems on a compact differentiable manifold $M^{n}$. The word bifurcation was introduced by Poincaré in connection with differential equations of celestial mechanics depending on a real parameter $\lambda$. He focused on a certain feature of the equation, say an isolated closed orbit, present for $\lambda = \lambda_{0}$. If when $\lambda$ varies past $\lambda_{0}$ this closed orbit disappears or undergoes a sudden topological change, $\lambda_{0}$ is said to be a bifurcation value; otherwise it is an ordinary value of the parameter. Traditionally bifurcation theory has been developed from this local point of view. After the emergence of the generic theory of dynamical systems via the coming of age of the concept of structural stability it became natural to take a more general and global approach to bifurcation theory ...