On the index of an isolated equilibrium point for piecewise polynomial differential systems

The index of an equilibrium point is an integer that characterizes the topological structure of the equilibrium. There are two primary methods for calculating the index of an isolated equilibrium point in smooth differential systems. One approach, derived by Poincaré and Bendixson, states that the index can be determined by the number of hyperbolic and elliptic sectors in a neighborhood of the equilibrium point. This is known as the Poincaré-Bendixson formula. Alternatively, various studies have proposed an algebraic method for computing the index, often referred to as Cauchy index. In this paper, we extend Cauchy index method to calculate the index of equilibrium points in piecewise polynomial differential systems. We also apply these results to piecewise smooth quadratic quasi-homogeneous differential systems.