Smoothness property for bifurcation diagrams
This paper deals with the qualitative approach related to the study of continuous dynamical systems. Such an approach, developed after Poincaré by the Andronov' School from 1935, consists in the identification of the properties of two spaces: the phase space, and the parameter space. Here the study is made in the framework of the ``abstract theory'' of dynamical systems, i.e. it concerns the class of two-dimensional autonomous ordinary differential equations (``planar vector field''), defined by continuous functions with infinitely many continuous derivatives. It concerns the properties of the parameter space, where the bifurcation set is the set of values separating qualitatively different topological structures (or ``phase portraits'') of the phase space. There are situations for which strata of bifurcation sets related to singular points nature, or to connections between hyperbolic saddles in smooth families of planar vector fields, are smoothly equivalent to sub-analytic sets. The author shows that it is no longer true when the bifurcation is related to transition near singular points of bifurcation sets, these points corresponding to codimension higher or equal to two (structural instability, or ``inertness'' of second degree in the language of the Andronov' School), and may correspond to the contact of two different types of bifurcation sets. Then the contact is flat. It is possible to prove that the flatness is smooth and to compute its asymptotic properties. In the framework of the ``abstract theory'', by their generality such results bring some interesting points. Nevertheless it is regrettable that they do not mention the fundamental contribution of the Andronov's School, the basis being given by Pontryagin, Andronov, Leontovitch, Gordon, Mayer, Bautin,\dots.
two-dimensional autonomous ordinary differential equations
scientific article; zbMATH DE number 1057645