Chaos analysis of a class of fractional-order Duffing–Rayleigh oscillators with unilateral impact

Abstract

This paper investigates the chaotic threshold of a class of fractional-order Duffing–Rayleigh oscillators with unilateral impact based on the classical Melnikov method. The fractional derivative is formulated using the Caputo definition, and the fractional-order term in the system is equivalently transformed into an effective stiffness and damping. The homoclinic orbit of the unperturbed system is derived using Hamiltonian theory. By applying the Melnikov approach, the necessary parameter conditions for the onset of chaos in the impact system are analytically obtained. The analytical results are validated by comparisons with numerical simulations, including time histories, Poincaré sections, phase portraits, and bifurcation diagrams. Furthermore, the influences of system parameters—such as linear stiffness, linear damping, fractional order and its coefficient, nonlinear stiffness, nonlinear damping, and the coefficient of restitution—on the emergence of chaotic behavior are comprehensively analyzed.