Complex Dynamics in the Simplest Nonlinear RLC Circuit Under Biharmonic Driving: Vibrational Resonance, Chaos and Multistability

Abstract

This study investigates the vibrational resonance and complex dynamics of a nonpolynomial Van der Pol oscillator driven by biharmonic excitation. This specific class of nonpolynomial oscillators models a nonlinear RLC series circuit where inductance and resistance are current-dependent. Using the method of direct separation of fast and slow motions, we analyze how various system parameters including inertial and impure cubic damping nonlinearities, linear damping, and the frequencies of the biharmonic signals influence the system’s behavior. Our analysis reveals the existence of single and double resonances, showing that parameter variations significantly affect the frequency response amplitude and the critical resonance point. The system’s performance, evaluated by its gain factor, identifies the weak signal frequency as a critical control parameter for signal amplification. The analytical solution is validated through excellent agreement with numerical results. A global analysis of the system’s dynamic changes, performed using a 4th-order Runge-Kutta algorithm, reveals complex behaviors such as periodic, quasiperiodic, and chaotic oscillations, including a notable period-one route to chaos. These behaviors are further confirmed by phase portraits and time series. Furthermore, the system’s sensitivity to initial conditions highlights the coexistence of multiple attractors, a phenomenon validated through bifurcation diagrams, Lyapunov exponents, and phase portraits.