classification of flow behavior near generalized equilibrium points in piecewise smooth systems

The aim of this paper is to classify the various states of flow behavior for piecewise smooth systems near generalized equilibrium points. seven categories are introduced based on the sign of the vector field across the discontinuity boundary, each encompassing distinct dynamical configurations. we investigate how a small perturbation parameter $mu$ influences the existence, type, and stability of generalized singular points in planar piecewise linear systems. starting with a one-dimensional example to illustrate core mechanisms, we extend the analysis to two dimensions, providing a detailed classification grounded in the signs of the system's components. our results yield a comprehensive framework for understanding how generalized singular points govern local dynamics, including bifurcations induced by parameter variation. this work contributes to the theoretical foundation for analyzing discontinuity-induced phenomena such as sliding modes and non-smooth bifurcations.