GLOBAL DYNAMICS OF A RAPIDLY FORCED CART AND PENDULUM

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