Via an history of the IMPA (Instituto de Matematica Pura e Aplicada of Rio deq Janeiro) research activities beginning with 1962 (when the author started his research), this paper draws up a list of some basic problems related to the qualitative theory of ordinary differential equations (dynamical systems).
This theory approach can be identified noting that the solutions to equations of nonlinear dynamic systems are in general nonclassical transcendental functions of the mathematical analysis. The qualitative theory “strategy” is of the same type as the one used for the characterization of a function of a complex variable by its singularities: zeros, poles, and essential singularities. Here, the complex transcendental functions are defined by their singularities (equilibrium points, periodical solutions, invariant manifolds, homoclinic and heteroclinic situations, \(\ldots\)). Qualitative methods consider the nature of these singularities in the phase space and their evolutions when the equation parameters (or the equation functions) are submitted to small continuous variations. This leads to an identification of the bifurcation sets in a parameter space, or in a function space. In such a framework, the notion of structural stability is of highest importance (roughly speaking, a dynamical system is structurally stable if its qualitative behavior is not modified by small perturbations of its equation). The notion is related to the fact that a model of a “physical system” must have such a property. The foundations of the corresponding theory are due to Andronov-Pontrjagin (1937), who gave basic theorems in the case of autonomous ordinary differential equations in the plane.
Structural stability studies with generalizations of the Andronov-Pontrjagin results were a central research axis of the IMPA researches, which began with Peixoto the author’s thesis supervisor. The paper under review describes the contribution of IMPA members and visitors (Sotomayor, Melo, Palis, Kupka, Newhouse, Smale, \(\dots\)). The last section is devoted to a comment on the psychology of invention in mathematics. It shows how the collaboration of a pure mathematician (Pontrjagin) and an engineer (Andronov) led to fruitful results.
It is worth noting that the reference works used by the IMPA were essentially the American ones and the first Russian ones (Andronov, Pontrjagin, Leontovitch, \(\ldots\)) translated in English. So, before the systematic translation of the Russian texts dealing with dynamical systems, the IMPA activity was led without knowledge of the subsequent Gorki school contributions (Afraimovitch, Shilnikov, Bukov, Gavrilov, Neimark, \(\ldots\)).