Dynamic Analysis and Accurate Analytical Solution of the Hybrid Duffing-Van der pol Oscillator via a Modified Homotopy Perturbation Method

Abstract

 In this paper, the Modified Homotopy Perturbation Method (MHPM) is proposed to develop closed-form analytical solutions for the hybrid Duffing–Van der Pol oscillator  which is prompted by piezoelectric vibration energy harvesting under aerodynamic galloping. The new order provides a previously unaccounted frequency included in the linear homotopy operator, enabling self-consistent frequency–amplitude determination and systematic removal of secular terms. Analytical expressions for the limit cycle amplitude  and nonlinear frequency  are obtained for the autonomous system. The equations of the amplitude–frequency response are derived for the forced system, and stability of the multi-valued branches is proved by means of a Jacobian/Routh–Hurwitz analysis. Dense Poincaré sections and bifurcation diagrams map the transition to chaos via period-doubling cascades. Relative errors of moduli are no greater than 1% in moderately nonlinear regimes, and no greater than 8% in strongly nonlinear ones, verified against fourth-order Runge–Kutta solutions. The effectiveness of this new approach in handling self-excited systems is confirmed by comparative analysis against the standard HPM and the He–Laplace method.