Bifurcation and Dynamics Analysis of a Piecewise-Linear van der Pol Equation

In this study, we examine the bifurcations and dynamics of a piecewise linear van der Pol equation—a model that captures self-sustained oscillations and is applied in various scientific disciplines, including electronics, neuroscience, biology, and economics. The van der Pol equation is transformed into a piecewise linear system to simplify the analysis of stability and controllability, which is particularly beneficial in engineering applications. This work explores the impact of increasing the number of linear segments on the system’s dynamics, focusing on the stability of the equilibria, phase portraits, and bifurcations. The findings reveal that while the bifurcation structure at critical values of the bifurcation parameter is complex, the topology of the piecewise linear model remains unaffected by an increase in the number of linear segments from three to four. This research contributes to our understanding of the dynamics of nonlinear systems with piecewise linear characteristics and has implications for the analysis and design of real-world systems exhibiting such behavior.