Dynamical characterization of a Duffing–Holmes system containing nonlinear damping under constant excitation

A Duffing-Holmes system containing nonlinear damping is used to investigate some dynamical properties of the system under the combined action of constant excitation and simple harmonic excitation. The harmonic balance method is used to find the main vibration equation of the system and obtain the amplitude-frequency response relationship. In the analysis of the global characteristics of the system, the Melnikov method is applied to analyze the necessary, insufficient conditions for the analysis of the system chaos, and the correctness of the analytical solution is verified by numerical calculations; the effect of the constant excitation amplitude on the global characteristics of the system is analyzed by the cell mapping method. The paper shows: gradually increasing the constant excitation amplitude causes the system amplitude-frequency response to appear rigidly asymptotic phenomenon; the boundary curve of the system generating chaos in the sense of Smale's horseshoe, with the increase of excitation frequency ω, the curve first decreases, then rises, finally tends to infinity, and the possibility of chaos occurs most near the resonance frequency of the system; with the change of the constant excitation amplitude, The number of attractor domains and attractors in the global attraction domain of the system changes substantially with the change of the constant excitation amplitude, and the attractor domains are entangled with each other.