Singular impasse points of planar constrained differential systems

Abstract

Planar analytic constrained differential systems (or impasse systems) are given by $A(x)\dot{x} = F(x), \quad x\in\mathbb{R}^{2}$, where $F$ is a vector field and $A$ is a matrix valued function. This class of systems differs from classical ODE's due to existence of the so called impasse set $\Delta=\{x:{\rm det} A (x) = 0\}$. The dynamics near smooth impasse points is well known in the literature. In this paper, we use Newton polygon and weighted blow ups in order to study its phase portrait near singular points of $\Delta$. Moreover, we apply our results in electric circuit problems.