TITLE:
Primary Resonance and Stability Analysis of a Duffing-Type Oscillator with Fractional-Order Damping and an Effective Periodic-Medium Stiffness Correction
AUTHORS:
Taha H. El-Ghareeb
KEYWORDS:
Fractional Calculus, Duffing Oscillator, Fractional-Order Damping, Primary Resonance, Frequency Response, Stability Analysis, Multiple Scales, Periodic Medium
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.14 No.9,
September
15,
2026
ABSTRACT: This study investigates the primary-resonance response and local stability of a reduced Duffing-type oscillator with fractional-order damping and an effective stiffness correction representing the dominant influence of a spatially periodic medium. The model includes cubic nonlinear stiffness, weak fractional damping of order
g
, harmonic excitation, and a linear stiffness-modification parameter. The fractional derivative is interpreted in the Caputo sense and is represented under the long-time harmonic approximation for the steady-state analysis. Using the method of multiple scales, amplitude and phase modulation equations are derived near primary resonance. The resulting algebraic frequency-response equation is solved numerically to obtain the positive real steady-state amplitudes, while the eigenvalues of the Jacobian matrix of the modulation system are used to distinguish stable and unstable branches. The results show that positive cubic nonlinearity produces hardening-type bending of the resonance response. Variations in
g
modify both the effective dissipative and effective detuning contributions through the factors
sin(
πg/2
)
and
cos(
πg/2
)
, respectively. The effects of the fractional order, damping coefficient, and excitation amplitude on the steady-state response are examined. The analytical frequency-response relation is also compared with direct fourth-order Runge-Kutta integration in the classical limiting case
g=1
, where the fractional derivative reduces to the ordinary first derivative. This comparison is used only as a consistency check for the classical limit and is not a direct time-domain validation of the fractional model for
0