Nonlinear dissipation and parametric forcing govern multistability and parameter-space structures in a driven van der Pol oscillator

We investigate how the form of nonlinear dissipation influences the global organization of stability and multistability in a harmonically driven van der Pol oscillator, a prototypical model for self-sustained oscillators in engineering and applied physics. The proposed formulation can be equivalently interpreted as a parametrically forced van der Pol oscillator, where the external excitation generates both an additive term and a state-dependent multiplicative contribution (nonlinear parametric forcing). Using Lyapunov exponents, the count of local maxima per oscillation period of the variable x(t), and bifurcation diagrams, we reveal structured stability architectures including Arnold tongues-like structures, stability rings formed by identical shrimp-shaped domains, and eye-of-chaos structures embedded within chaotic regions. A central finding is the systematic occurrence of quint points, at which five distinct periodic domains coalesce in parameter space. We show that quint points persist under both piecewise and smooth dissipation laws, indicating that they are robust geometric features associated with the combined action of external forcing and nonlinear dissipation, rather than artifacts of a specific dissipation model. Although their presence is robust, their spatial distribution and geometric arrangement depend sensitively on the dissipation form and saturation level. Overall, our results demonstrate that nonlinear dissipation plays a key role in shaping multistable response landscapes in driven oscillatory systems, with potential implications for engineering applications such as electronic circuits, mechanical resonators, and nonlinear energy-harvesting devices.