Theory and applications of high codimension bifurcations

The study of bifurcation of high codimension singularities and cyclicity of related limit periodic sets has a long history and is essential in the theory and applications of differential equations and dynamical systems. It is also closely related to the second part of Hilbert's 16th problem.

In 1994, Dumortier, Roussarie and Rousseau launched a program aiming at proving the finiteness part of Hilbert's 16th problem for the quadratic vector fields. For the program, 125 graphics need to be proved to have finite cyclicity. Since the launch of the program, most graphics have been proved to have finite cyclicity, and there are 40 challenging cases left. Among the rest of the graphics, there are 4 families of HH-graphics with a triple nilpotent singularity of saddle or elliptic type. Based on the work of Zhu and Rousseau, by using techniques including the normal form theory, global blow-up techniques, calculations and analytical properties of Dulac maps near the singular point of the blown-up sphere, properties of quadratic systems and the generalized derivation-division methods, we prove that these 4 families of HH-graphics (I1/12 ), (I1/13), (I1/9b) and (I1/11b) have finite cyclicity. Finishing the proof of the cyclicity of these 4 families of HH-graphics represents one important step towards the proof of the finiteness part of Hilbert's 16th problem for quadratic vector fields.