Nonlinear response analysis of the Duffing-Van der Pol oscillator with external excitation using a modified Lindstedt-Poincaré method

Abstract

The Van der Pol oscillator is well known for generating a self-sustained limit cycle in its unforced state; however, when an external periodic force is applied, its dynamical responses may deviate sharply away from this natural rhythm. The existing Lindstedt-Poincaré method becomes cumbersome in the presence of strong nonlinearity, damping, and external forces. To overcome this limitation, an alternative modified Lindstedt-Poincaré method is introduced to obtain the steady-state response of the Duffing-Van der Pol oscillator. Excellent accuracy is achieved by comparing the approximate results with those obtained using the Runge-Kutta fourth-order (RK4) method and the harmonic balance method (HBM). The proposed modified Lindstedt-Poincaré method offers a simple solution procedure and demonstrates broad applicability to a wide range of nonlinear oscillatory problems in applied sciences and engineering.