Polynomial normal forms of Constrained Differential Equations with three parameters

We study generic constrained differential equations (CDEs) with three parameters, thereby extending Takens's classification of singularities of such equations. In this approach, the singularities analyzed are the Swallowtail, the Hyperbolic, and the Elliptic Umbilics. We provide polynomial local normal forms of CDEs under topological equivalence. Generic CDEs are important in the study of slow-fast (SF) systems. Many properties and the characteristic behavior of the solutions of SF systems can be inferred from the corresponding CDE. Therefore, the results of this paper show a first approximation of the flow of generic SF systems with three slow variables.Comment: This is an updated and revised version. Minor modifications mad

Model order reduction and composite control for a class of slow-fast systems around a non-hyperbolic point

In this letter we investigate a class of slow-fast systems for which the classical model order reduction technique based on singular perturbations does not apply due to the lack of a Normally Hyperbolic critical manifold. We show, however, that there exists a class of slow-fast systems that after a well-defined change of coordinates have a Normally Hyperbolic critical manifold. This allows the use of model order reduction techniques and to qualitatively describe the dynamics from auxiliary reduced models even in the neighborhood of a non-hyperbolic point. As an important consequence of the model order reduction step, we show that it is possible to design composite controllers that stabilize the (non-hyperbolic) origin

Stabilization of a planar slow-fast system at a non-hyperbolic point

In this document we study the stabilization problem of a planar slow-fast system at a non-hyperbolic point. At these type of points, the classical theory of singular perturbations is not applicable and new techniques need to be introduced in order to design a controller that stabilizes such a point. We show that using geometric desingularization (also known as blow up), it is possible to design, in a simple way, controllers that stabilize non-hyperbolic equilibrium points of slow-fast systems. Our results are exemplified on the van der Pol oscillator