Closed-Form Almost Periodical Solutions for a Dynamical System Using the Optimal Auxiliary Functions Method

The main aim of our paper is concerning the damped oscillations of 3D dynamical systems, depending on a single physical parameter. This system does not admit Hamilton–Poisson structure but can be explicitly integrated, and the exact parametric solutions are built via a smooth function. The influence of the physical parameter is semi-analytically analyzed using the Optimal Auxiliary Functions Method (OAFM). One of the advantages of the applied method is the small number of iterations due to the appropriate choice of auxiliary convergence control functions. The OAFM solutions are effectively in good agreement with corresponding numerical ones, represented qualitatively by figures and quantitatively by tables. The statistical tests of residuals highlighted the accuracy of our results. The proposed method can be considered an analytical tool for nonlinear vibration analysis of numerous applications from electrical engineering or mechanical structures based on damped rotatory oscillators to the field of image encryption.