Dynamical characterisation of fractional-order Duffing-Holmes systems containing nonlinear damping under constant simple harmonic excitation

The dynamics of a fractional-order Duffing-Holmes system with a nonlinear damping term is investigated under a combined constant-simple harmonic excitation. The harmonic balance method is used to derive the approximate analytical solution of the system, focusing on the effects of constant excitation, simple harmonic excitation, and fractional-order coefficients and orders on the dynamical properties of the system; the analytical necessary conditions for chaotic motion are obtained by applying Melnikov theory, and the effects of various parameters of the system on chaotic motion are further analysed; the bifurcation diagrams and the maximum Lyapunov Exponential maps are calculated for different simple harmonic excitation amplitudes; the dynamics of the system under specific excitation frequencies and amplitudes are investigated by using time-domain maps, spectrograms, phase-plane maps, and Poincare cross sections; the static bifurcation of the system is investigated by applying the singularity theory; and global bifurcation characteristics are calculated by using the cellular mapping algorithm. It is shown that the amplitude-frequency curves of the system can be transformed from purely stiff to coexisting soft and stiff characteristics, or even purely soft, with the increase of the constant excitation for a certain amplitude of the simple harmonic excitation; the coefficients and orders of the fractional-order differential terms in the approximate solution affect the amplitude, resonance frequency, and stability of the system in the form of the equivalent linear damping and linear stiffness. The bifurcation topology shows that the system has a jumping phenomenon; with the increase of the amplitude of the constant excitation, the stability of the system in the stable state with small amplitude becomes stronger gradually until it reaches the strongest.