Nonlinear Dynamics 7: 231-247, 1995. (~) 1995 Kluwer Academic Publishers. Printed in the Netherlands. Geometric Aspects of the Parametrically Driven Pendulum DANILO CAPECCHI Dipartimento di Ingegneria delle Strutture, Acque e Terreno, University of L'Aquila, 67040 Monteluco di Roio, Italy (Received: 20 March 1993; accepted: 2 November 1993) Abstract. The behaviour of the parametrically driven pendulum is very complex. Therefore, a global study is carded out to cover all possible situations. The study is mainly numeric, though primary bifurcations of subharmonic motions, as well as the homoclinic intersection of the hilltop saddle, are evaluated according to the Melnikov theory. Extended use is made of the cell-to-cell mapping algorithm to evaluate attracting basins of the various periodic motions. Heteroclinic intersections are always present, independently of the excitation intensity, so that the boundaries of attracting basins are always very complicated, even below the homoclinic tangency of the hilltop saddle. The oscillator exhibits various kinds of rotating and oscillating motions. All these motions lead to chaos after a period doubling cascade. It is shown that chaos usually occurs at a much greater excitation level than at that which produces homoclinic tangency of the hilltop saddle; the greater the damping, the greater the difference. The oscillatory chaotic motion is associated with the first change in the period two Birkhoff signature. Key words: Simple pendulum, parametric excitation,chaoticmotion,globalbifurcations. 1. Introduction The behaviour of parametrically excited nonlinear systems is less welt known and more com- plex than that of classically excited nonlinear systems. The peculiarity of the former systems is immediately apparent from the fact that even the linear case gives rise to very complex responses and instability phenomena. Moreover, the presence of at least one equilibrium, the zero response, for any forcing amplitude brings other problems. Nowadays interest has again been aroused as a result of practical structural engineering factors [1-4]. One of the first studies on parametric excitation was made by Bolotin [5], wose book highlights the main aspects of the periodic response and stability of the Mathieu equation with nonlinear terms added, making reference to experimental results. Interesting observations are also made in [2, 6] for the same equation. In [7] the nonlinear term is multiplicative instead of additive. In all these papers the problem was studied analytically either by perturbation methods or harmonic balance and averaging. This paper is a continuation of previous work [8, 9] on the simple parametrically driven pendulum. This system is sufficiently simple to be studied and at the same time sophisticated enough to embrace the most important aspects of nonlinear parametric excitation; moreover it is easy to assemble a mechanical experimental apparatus that allows the verification of the analytical results [10-12]. Here attention is focused on the geometry of the pendulum; namely the occurrence of homoclinic or heteroclinic intersections and the evolution of the attracting basins with increasing amplitude of parametric excitation. The study is mainly numeric, though primary bifurcations of subharmonic motions as well as the homoclinc intersection of the hilltop saddle are evaluated using the Melnikov theory. Together with saddle-node bifurcations of periodic motion, in addition and independently, there are bifurcations that stem