Comparative analysis of homotopy perturbation and power series solutions for strongly nonlinear oscillators with cubic restoring forces

We studied the behavior of a strongly nonlinear oscillator characterized by cubic nonlinearity, which commonly occurs in engineering, mechanical, and physical systems. Additionally, we also utilized two semi-analytical approaches, namely the Homotopy Perturbation Method (HPM) and the Power Series Method (PSM), to obtain approximate analytical solutions to the nonlinear differential equation governing the oscillator motion. To evaluate the accuracy and convergence of the HPM and PSM solutions, we compared them with each other and with the numerical solution obtained using the fourth-order Runge–Kutta (RK4) method. The comparisons indicate that the HPM solution is very similar to the RK4 solution throughout the entire time domain. On the other hand, the PSM solution only provides accurate solutions for short times, after which the difference increases significantly. The phase portraits illustrate that the motion of the system is stable and, therefore, periodic. In summary, the study demonstrates that the Homotopy Perturbation Method is a practical, reliable, and very effective technique for solving strongly nonlinear oscillators with cubic restoring forces, and consequently may be employed in a broader array of topics in applied mathematics, nonlinear mechanics, and physics.