Hildeberto Jardón Kojakhmetov (Johann Bernoulli Institute)
Polynomial normal forms of Constrained Differential Equations with three parameters
In this presentation we discuss the problem of the classification of singularities of Constrained Differential Equations (CDEs) Our work is based on F. Takens’s 1975 work on the theory of CDEs. The study of such type of equations is motivated by the so called Slow-Fast systems, which are used to model phenomena occurring in two or more time-scales.
The classification of singularities of CDEs is closely related with Catastrophe Theory. This close relationship allows to provide local normal forms, under topological equivalence, in polynomial form. In his work, Takens classified singularities of CDEs with 1 or 2 parameters. When three parameters are involved, new generic singularities appear: the Swallowtail, the Hyperbolic, and the Elliptic Umbilics. Hence, we extend Takens's investigation by presenting the analysis of Constrained Differential Equations which have 3 parameters and so, the mentioned singularities may appear. We finally emphasise the importance of Normal Forms of CDEs in the study of slow-fast systems.