Bifurcation of a Limit Cycle for Planar Piecewise Smooth Quadratic Differential System via Averaging Theory

In this article, the focus is on exploring planar piecewise smooth quadratic systems, a significant class of dynamical systems that exhibit changes in behavior under different conditions but with smooth transitions between these states. The main objective is to introduce a second-order averaged method designed specifically to identify limit cycles, repeating trajectories in a system's phase space indicative of periodic behavior. This innovative method not only allows for the detection of these cycles but also quantifies their number, providing a deeper understanding of the system's long-term behavior. The paper highlights its applicability by demonstrating the maximum number of limit cycles that can exist in two distinct systems, offering valuable insights into the dynamics of such systems and contributing to the broader field of mathematical modeling and analysis.