Bifurcation Trees

Abstract

In this chapter, the possible bifurcation trees are presented for analytical predictions of the routes of different periodic motions to chaos in the parametrically excited pendulum. The stability and bifurcations of periodic motions are also illustrated through eigenvalue analysis. The solid and dashed curves represent the stable and unstable motions, respectively. The black and red colors are for paired asymmetric motions. The acronyms “SN” and “PD” represent the saddle-node and period-doubling bifurcations, respectively. The symmetric and asymmetric periodic motions are labeled by “S” and “A”, respectively. All bifurcations trees are predicted with varying excitation frequency Ω. Other parameters are chosen as $$\alpha = \,4.0,\,\delta = 0.1,\,{Q_{0 = }}\,5.0.$$ ((4.1))