On the Poincaré–Bendixson Formula for Planar Piecewise Smooth Vector Fields.

The topological index, or simply the index, of an equilibrium point of a vector field is an integer which saves important information about the local phase portrait of the equilibrium point. There are mainly two ways to calculate the index of an isolated equilibrium point of a smooth vector field. First Poincaré and Bendixson proved that the index of an equilibrium point can be obtained from the number of hyperbolic and elliptic sectors that there are in a neighborhood of the equilibrium point, which is known as Poincaré–Bendixson formula for the topological index of an equilibrium point. Second several works contributed to the algebraic method of Cauchy’s index for computing the index of an equilibrium point. In this paper, we extend the Poincaré–Bendixson formula to planar piecewise smooth vector fields. Applying this formula, we provide the index of generic codimension-0 and codimension-1 equilibrium points for piecewise smooth vector fields. [ABSTRACT FROM AUTHOR]