Period-3 motions to chaos in a periodically forced duffing oscillator with a linear time-delay

In this paper, bifurcation trees of period-3 motions to chaos in a periodically excited, Duffing oscillator with a linear delay are investigated through the Fourier series. The analytical solutions of period-m motions are presented and the stability and bifurcation of such period-m motions in the bifurcation trees are discussed by eigenvalue analysis. Two independent symmetric period-3 motions were obtained, and the two independent symmetric period-3 motions are not relative to chaos. Two bifurcation trees of period-3 motions to chaos are presented through period-3 to period-6 motion. Numerical illustrations of stable and unstable period-3 and period-6 motions are given by numerical and analytical solutions. The complicated period-3 and period-6 motions exist in the range of low excitation frequency. © 2014, Springer-Verlag Berlin Heidelberg.