Develop an original program for the course Ordinary Differential Equations II of the IST Master Program on Applied Mathematics centered on Structural Stability and Bifurcations and aiming to prepare students to research in this area

This course was developed for lecturing it in 1986/87 and 1988/89 in the 2nd and 3rd editions of the IST Master Program in Applied Mathematics and it was further enhanced to be lectured as a Special Topics course offered in 1990/91 to finalists of the IST Undergraduate Mathematics Program. The emphasis was on the systematization of concepts and methods with wide applications, highlighting several open problems. Some of the main ideas were to privilegiate topics of demonstrated fertility by the area evolution, whenever possible exploring connections with classical Mathematics, with special attention to topics allowing to explore relationships with other areas of Mathematics, since the interaction of ideas and methods of distinct areas is highly fertile for creativity. Accordingly, it had a strong component of Several Variables Functions (Weierstrass and Malgrange preparation theorems), Functional Analysis (fixed point theorems, Fredholm operators), Differential Geometry (differential manifolds, jets, nonlinar coordinate transformations), Differential Topology (Transversality and Singularity Theory), Lie Groups (symmetries), Algebraic Topology (Conley Index).

The outline of the program was:

• Notions of topological conjugacy, structural stability and bifurcations, and Grobman-Hartman Theorem.
• Caracterization and genericity of structural stability in dimension 1 and 2, including in the 2-dimensional torus and the circle, and Sard, Andronov-Pontryagin. Peixoto and Denjoy theorems.
• Transversality of differential manifolds and of functions and differentiable manifolds, spaces of jets, splitting of degenerate singular points, and Thom Transversaty Theorem and Newton Polygon.
• Genericity of Kupka-Smale fields and existence of Kupka-Smale fields not structurally stable.
• Non-genericity of structural stable fields in dimension ≥3, Morse-Smale systems and structural stability, Smale Horseshoe, symbolic dynamics, chaos in deterministic systems, transverse homoclinic points, Silnokov theorems for saddle-focus, Conley-Moser Theory, the Lorenz strange attractor, hyperbolic sets, Axiom A diffeomorphisms, Anosov Theorem, non-genericity of Axiom A systems.
• Liapunov-Schmidt reduction method, noncritical systems relatively to bounded, periodic or almost periodic solutions, Fredholm Alternative.
• Bifurcations at equilibrium points at a simple eigenvalue, Crandall-Rabinowitz Theorem, relation with dynamics in central manifolds, sddle-node, transcritical and pitchfork bifurcations.
• Analysis of bifurcation functions, Weierstrass and Malgrange Preparation Theorems.
• Bifurcation at a conjugate pair of simple pure imaginary eigenvalues, analysis with the Liapunov-Schmidt method and relation with the dynamics in central manifolds, generic and nongeneric Hopf bifurcations.
• Averaging methods for period equations, relation of system obtained by averaging with the discrete system defined by the original equation in Poincaré sections.
• Bifurcations of of homoclinic and heteroclinic orbits to periodic orbits, transverse homoclinic points and analysis with the Liapunov-Schmidt and the Melnikov methods, bifurcations of subharmonics in the neighborhood of homoclinic bifurcations, relation with cahotic regimes, period doubling and homoclinic doubling.
• Analytic and geometric theory of Poincaré-Birkhoff normal forms, Poincaré-Dulac Theorem, small denominators problem, Poicaré and Siegel domains.
• Finite determination of singularities, blow-up of singularities and σ-process, Lojasiewicz condition, Takens Theorem for a singularity with double zero eigenvalue.
• Bifurcation codimension, versal unfolding, characterization of codimension-1 bifurcations in the plane, codimension-2 bifurcations in the plane, Bogadanov-Takens singularity, singularity with one simple zero and one pair of conjugate pure imaginary eigenvalues.
• Elements of bifurcation with symmetry, Lie groups, equivariance, Z² symmetry and degenerate Hopf bifurcation.

The course involved problem sessions where students presented their solutions to challenging proposed exercises, some requiring them to develop maturity by thinking about the problems during several days or even weeks.

Several Master Program students attended the course and pursued for PhDs abroad. The undergraduate student who followed it most deeply was Luís Barreira; it is possible to identify in several of his research papers motivations that can be traced back to the contents of this course, with the important exception of those on Ergodic Theory and Dimension Theory, which have roots in the research he initiated in his PhD at Pennsylvania State U., supervised by Yakov Pesin and highly influenced by Anatole Katok, both scientific descendents of Luzin and Egorov, through the parallel exceptional Russian lineages Anosov-Pontryagin-Aleksandrov and Sinai-Kolmogorov.