Applications of Mamadu–Njoseh Polynomials in the Analysis of Dynamical System Fluctuations: Nonlinear Duffing Oscillator

Duffing oscillator, being a fundamental model for investigating nonlinear vibrations, resonance, and complex dynamics, is a key component of nonlinear dynamical systems. However, the challenge remains in obtaining accurate numerical solutions to its governing equation, especially in capturing the transient response and understanding long-lasting oscillatory motions. In this study, a Mamadu–Njoseh Polynomial (MNP) based spectral collocation method is formulated and investigated for the numerical solution and dynamical analysis of the nonlinear Duffing oscillator. The main goal of this study is to evaluate the accuracy, convergence, numerical stability, and computational efficiency of the proposed method. Specifically, the proposed method utilizes Mamadu–Njoseh polynomials as a basis to approximate the solution to the governing nonlinear differential equation through a spectral collocation framework. The accuracy and convergence properties of the proposed method are investigated by comparing the numerical approximations with the exact solution (when available). The method is also applied to capture the system’s transient response and study resonance characteristics, bifurcation behavior, and long-lasting oscillatory motions to further assess its accuracy and convergence properties. The numerical results demonstrate that the proposed method offers accurate approximations with fast convergence to the exact solution for the cases considered. In addition, the method is found to exhibit improved numerical stability and computational efficiency as a small number of basis functions are sufficient to achieve a reasonable level of accuracy. Thus, the MNP based spectral collocation method appears to be a computationally efficient and accurate method that can effectively capture various aspects of the dynamics of nonlinear Duffing oscillators. The method’s accuracy and efficiency can potentially benefit the study of nonlinear vibration dynamics and complex motions of other nonlinear dynamical systems, though a proper validation of the proposed method is essential for each specific case.