Routes of periodic motions to chaos in a periodically forced pendulum

Abstract

In this paper, with varying excitation amplitude, bifurcation trees of periodic motions to chaos in a periodically driven pendulum are obtained through a semi-analytical method. This method is based on the implicit discrete maps obtained from the midpoint scheme of the corresponding differential equation. Using the discrete maps, mapping structures are developed for specific periodic motions, and the corresponding nonlinear algebraic equations of such mapping structures are solved. Further, semi-analytical bifurcation trees of periodic motions to chaos are also obtained, and the corresponding eigenvalue analysis is carried out for the stability and bifurcation of the periodic motions. Finally, numerical illustrations of periodic motions on the bifurcation trees are presented in verification of the analytical prediction. Harmonic amplitude spectra are also presented for demonstrating harmonic effects on the periodic motions. The bifurcation trees of period-1 motions to chaos possess a double spiral structure. The two sets of solutions of period-\(2^{l}\) motions \((l=0,1,2,\ldots )\) to chaos are based on the center around \(2m\pi \) and \((2m-1)\pi (m=1,2,3,\ldots )\) in phase space. Other independent bifurcation trees of period-m motions to chaos are presented. Through this investigation, the motion complexity and nonlinearity of the periodically forced pendulum can be further understood.