Closed-Form Orbits for a Six-Parameter 3D Dynamical System Using the Multistage Optimal Homotopy Perturbation Method

Numerous systems in electrical engineering, biology, and mechanical structures can be modeled using dynamical systems theory. This paper examines the behavior of a 3D dynamical system with six parameters, specifically its damped or periodic oscillations and asymptotic properties as functions of six physical parameters. The system is integrated explicitly through a smooth solution of a third order nonlinear differential equation, yielding exact parametric expressions that describe a heteroclinic orbit. To analyze parameter influence, we apply the Multistage Optimal Homotopy Perturbation Method (MOHPM). Its main advantage is the small number of iterations required, due to the effective choice of auxiliary convergence control functions. The MOHPM solutions agree closely with numerical results, demonstrated qualitatively through figures and quantitatively through tables. Accuracy is further assessed by comparison with the Optimal Homotopy Perturbation Method (OHPM). A qualitative analysis of errors is also provided.