Analytical threshold for chaos in a Duffing oscillator with delayed feedbacks

In this paper, the bifurcation and chaotic behaviors of a Duffing oscillator with delayed displacement and velocity feedbacks under harmonic excitation are investigated. The analytically necessary condition for the chaos in the sense of Smale horseshoes is established based on Melnikov method. It could be found that the delayed displacement and velocity feedbacks affect the necessary condition respectively, so that the analysis on the effects of the two kinds of feedbacks is presented individually. The effects of displacement feedback coefficient, velocity feedback coefficient and their time delays on the analytically necessary condition are analytically discussed. Furthermore, some typical numerical results, including the bifurcation diagrams, Poincaré maps and the largest Lyapunov exponents of the delayed oscillator are also presented and compared. The satisfactory qualitative agreement between the analytical and numerical results verify the correctness of the analytically necessary condition.