Structured Chaos and Multistability in a Driven van der Pol Oscillator with Cubic Damping

We investigate a novel nonautonomous dynamical system derived from the van der Pol oscillator by introducing a sinusoidal forcing term and a cubic nonlinearity. The resulting model is governed by two first-order differential equations with six parameters. We explore its behavior in the parameter plane defined by the amplitude and frequency of the forcing term. Dynamical characterization is performed using the largest Lyapunov exponent and the number of local maxima per period of a system variable. Our results reveal regions of multistability with coexisting periodic and chaotic attractors, as well as the presence of quart and quint points indicating parameter values where four or five distinct periodicities coalesce. Moreover, we identify organized periodic structures such as shrimp-like patterns in chaotic domains and Arnold tongue-like structures embedded in quasi-periodic domains, highlighting the system’s complex dynamics.