Periodic solutions of the forced pendulum: Exchange of stability and bifurcations

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:182

Article

Katriel, G

关键词: forced pendulum;    periodic solutions;    bifurcation;    stability;   

DOI  :  10.1006/jdeq.2001.4091

来源: Elsevier

【 摘 要 】

We study the T-periodic solutions of the forced pendulum equation u + cu' + a sin(u) = lambdaf(t), where f satisfies f(t + (T)/(2)) = -f(t). We prove that this equation always has at least two geometrically distinct T-periodic solutions u(0) and u(1). We then investigate the dependence of the set of T-periodic solutions on the forcing strength lambda. We prove that under some restriction on the parameters a, c, the periodic solutions found before can be smoothly parameterized by lambda, and that there are some lambda-intervals for which u(0)(lambda), u(1)(lambda) are the only T-periodic solutions up to geometrical equivalence, but there are other lambda-intervals in which additional T-periodic solutions bifurcate off the branches u(0)(lambda), u(1)(lambda). We characterize the global structure of the bifurcating branches. Related to this bifurcation phenomenon is the phenomenon of 'exchange of stability' - in some lambda-intervals u(0)(lambda) is dynamically stable and u(1)(lambda) is unstable, while in other lambda-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. (C) 2002 Elsevier Science (USA).

【 授权许可】

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