Analytical investigation of fractionally damped Duffing and Van der Pol nonlinear oscillators using adapted direct normal forms technique

Abstract

In this paper, the Direct Normal Forms (DNF) framework is extended to nonlinear oscillators systems with fractional-order damping, namely, Fractionally Damped Duffing (FDD) and Van der Pol (FDV) oscillators. Analytical frequency-amplitude relations are derived using an adapted DNF formulation based on the Davison–Essex fractional derivative. Numerical validation is then performed for both FDD and FDV oscillators using Grunwald–Letnikov time-domain simulations. The steady-state response is obtained, and the fundamental harmonic is extracted using Fourier analysis. The compared analytical and numerical results demonstrated good agreement away from resonance (typically within 5–10%), while larger deviations occurred near resonance, due to neglected higher-order nonlinear effect. For FDD oscillator, the analytical prediction showed close agreement when compared with averaging-based numerical solutions. Similarly, for FDV oscillator, analytical results demonstrated close agreement when compared with solutions obtained using Lucas wavelets method. The comparisons indicate that the steady-state response in both systems is principally governed by the fundamental harmonic component. In general, the implemented DNF formulation provides a practical analytical approximation for fractionally damped systems within the considered range, nevertheless, its accuracy decreases close to strongly nonlinear resonance conditions.