Nonlinear Dynamics 13: 131–170, 1997. c 1997 Kluwer Academic Publishers. Printed in the Netherlands. Global Dynamics of a Rapidly Forced Cart and Pendulum S. WEIBEL Department of Aerospace and Mechanical Engineering, Boston University, Boston, MA 02215, U.S.A. T. J. KAPER Department of Mathematics, Boston University, Boston, MA 02215, U.S.A. J. BAILLIEUL Department of Aerospace and Mechanical Engineering, Boston University, Boston, MA 02215, U.S.A. (Received: 29 November 1995; accepted: 4 December 1996) Abstract. In this paper, we study emergent behaviors elicited by applying open-loop, 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 points of the forced system. We use the method of successive approximations to establish the existence of hovering motions, as well as compute analytical approximations of their locations, for the cart and pendulum on an inclined plane. Moreover, when small-amplitude dissipation is added, we show that the hovering motions are asymptotically stable. We compare the results for all of the local analysis with results of simulating Poincar´ e 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 orbits also yields the existence of periodic orbits near saddle equilibria of the averaged system. These latter periodic orbits are shown to be saddle periodic orbits, and in turn they have stable and unstable manifolds 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 small damping is studied. This global analysis is useful in the design of open-loop control laws. Key words: Open-loop control, averaging, global bifurcations, Melnikov theory, exponentially small separatrix splitting. 1. Introduction The subject of open-loop oscillatory control has received increasing attention in recent years, as researchers have begun developing designs based on the geometric features of noncommuting vector fields. Much of this work has been aimed at so-called drift-free systems, for which the central application is kinematics-level motion control of autonomous mobile robots, multi- fingered robotic hands, and so forth. See, e.g. [1–6] for examples of work treating stabilization and control of drift-free systems. Work has also been reported on controlling the dynamics of mechanical systems using high-frequency oscillatory forcing [7, 8]. In this case also, there is a geometric mechanism underlying the observed effects of oscillatory forcing, but because the systems do have drift, study calls for somewhat different mathematical techniques, and so far, much of the progress in understanding these systems has involved analytical tools from the theory of dynamical systems and averaging theory. An expository introduction to the subject of open-loop oscillatory control in various settings is given in [9]. In the present paper, we undertake a detailed study of a periodically forced mechanical system consisting of a free swinging simple pendulum mounted on a single degree of freedom Contributed by S. W. Shaw.