Global Dynamics of a Rapidly Forced Cart and Pendulum

摘要: In this paper, we study emergent behaviors elicited by applying open-loop, high-frequency oscillatory forcing to nonlinear control systems. First, hovering motions, which are periodic orbits associated with stable fixed points of the averaged system not forced system. We use method successive approximations establish existence as well compute analytical their locations, for cart and pendulum on an inclined plane. Moreover, when small-amplitude dissipation is added, show that motions asymptotically stable. compare results all local analysis simulating Poincare maps. Second, perform a complete global Toward end, same iteration scheme also yields near saddle equilibria These latter shown be orbits, in turn they have unstable manifolds form homoclinic tangles. A quantitative these tangles carried out. Three distinguished limiting cases analyzed. Melnikov theory applied one case, extension recent result about exponentially small splitting separatrices developed another case. Finally, influence damping studied. This useful design open-loop laws.