Periodic Solutions of the Forced Pendulum: Exchange of Stability and Bifurcations

Abstract

We study the T-periodic solutions of the forced pendulum equation u″+cu′+a sin(u)=λf(t), where f satisfies f(t+T2)=−f(t). We prove that this equation always has at least two geometrically distinct T-periodic solutions u0 and u1. We then investigate the dependence of the set of T-periodic solutions on the forcing strength λ. We prove that under some restriction on the parameters a,c, the periodic solutions found before can be smoothly parameterized by λ, and that there are some λ-intervals for which u0(λ), u1(λ) are the only T-periodic solutions up to geometrical equivalence, but there are other λ-intervals in which additional T-periodic solutions bifurcate off the branches u0(λ), u1(λ). We characterize the global structure of the bifurcating branches. Related to this bifurcation phenomenon is the phenomenon of ‘exchange of stability’ – in some λ-intervals u0(λ) is dynamically stable and u1(λ) is unstable, while in other λ-intervals the reverse is true, a phenomenon which has important consequences for the dynamics of the forced pendulum, as we show by both theoretical analysis and numerical simulation.