Can simple equations explain complex behaviors in oscillating systems? This research delves into the fascinating world of relaxation oscillations, a phenomenon observed in systems transitioning between distinct states with varying time scales. The authors explore how these oscillations, when subjected to external periodic forces, exhibit complex responses, including multiple periodicities and bistability. Focusing on the harmonically driven van der Pol equation, a prototype for these behaviors, the study analytically derives conditions for bistable orbits in a related piecewise discontinuous equation. Using **mathematical modeling** and analysis, the research validates the results across a broad parameter space, offering insights into the more complex dynamics observed in the forced van der Pol equation. The study identifies the parameter regions of such bistable orbits analytically for the closely related harmonically driven Stoker–Haag piecewise discontinuous equation. These findings have far-reaching implications, suggesting extensions to scenarios where forced relaxation oscillations are crucial components of operating mechanisms in various physical and biological systems. Future studies can apply these models to real-world applications.
Published in the _International Journal of Bifurcation and Chaos_, this article perfectly aligns with the journal's focus on exploring nonlinear dynamical systems and their complex behaviors. By providing analytical insights into the bistability of harmonically forced relaxation oscillations, the study contributes to the journal's core themes in chaos theory and bifurcation analysis.