Singularities of dynamical systems and their unfoldings

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The research project is focused on the study of singularities of dynamical systems, both ordinary differential equations (ODE) and difference equations, and how these singularities organize the dynamics. In the case of ODE, the project comprises two parts: 1) a first part on the bifurcation theory of planar vector fields; 2) a second part on the study of unfoldings of singularities of ODE of low codimension in real or complex finite-dimensional space. In the case of difference equations, the research project is focused on the study of unfoldings of finite codimension resonant singularities in dimension 1 and 2.Bifurcation theory is a tool to discover the phase portraits of planar vector fields, through embedding a single vector field in a family of vector fields depending on a finite number of parameters and analyzing the bifurcations in the family. The bifurcations of highest codimension in the family are the most important: they organize the bifurcation diagram and the dynamics. The main motivation for this part of the project is to advance the progress on a solution on the finiteness part of Hilbert’s 16th problem through the Dumortier-Roussarie-Rousseau program for proving the existence of a uniform bound for the number of limit cycles of a polynomial quadratic vector field. The program consists in proving that 121 graphics have finite cyclicity inside the family of quadratic vector fields. Significant progress on the program is expected from new techniques developed recently, and more are in the process of being developed. I am also interested in applying bifurcation tools to some predator-prey models in mathematical biology, mainly in students’ theses. The core part of the project deals with the study of equilibrium positions of analytic dynamical systems (either ODE, linear differential equations or difference equations) depending on parameters, more precisely with the problem of analytic classification of germs of families of dynamical systems unfolding a dynamical system with a singularity of finite codimension: when are two germs of analytic families of dynamical systems equivalent modulo an analytic change of parameters, and possibly a reparameterization of time? There are many geometric obstructions to such equivalences. The project is focused on 1) identifying a complete modulus of analytic classification for such a germ of family, and 2) identifying the moduli space for such germs. One motivation is to understand the geometric meaning of the analytic obstructions to equivalence and to describe these obstructions. The types of singularities to be studied are parabolic points of 1-dimensional germs of diffeomorphisms, saddle-nodes of germs of 2-dimensional diffeomorphisms, resonant saddles or saddle-nodes of 2-dimensional complex foliations and resonant singularities of linear differential systems.