Bifurcation analysis of Duffing oscillator with a fifth-order nonlinear factor

This paper aims to investigate and analyze the dynamic behavior of a Duffing equation with a non-linear factor of the fifth-order, which is similar to the one developed by Zhao and Zhang but with a more general bifurcation analysis. While Zhao and Zhang use numerical methods to explore the behavior of the model, this paper employs both analytical and numerical approaches to analyze the bifurcation behavior. The paper determines normal form coefficients of codim-1 bifurcation points, including fold, pitchfork, and Hopf, as well as codim-2 bifurcation points such as Bogdanov-Takens and cusp. The sub-criticality or super-criticality of these bifurcation points is also calculated analytically. The analytical results are then confirmed through numerical simulation and continuations, which reveal more dynamic behaviors of the system. Furthermore, the paper investigates the external forcing effect on the model and extracts its dynamic behaviors.