Nonlinear Dynamics 4: 605-633, 1993. (~ 1993 Kluwer Academic Publishers. Printed in the Netherlands. Bifurcations in the Dynamics of an Orthogonal Double Pendulum S. SAMARANAYAKE and A. K. BAJAJ School of Mechanical Engineering, Purdue Universi~., West Lafayette, IN47907, U.S.A. (Received: 27 January 1992; accepted: 22 September 1992) Abstract. The weakly nonlinear resonant response of an orthogonal double pendulum to planar harmonic motions of the point of suspension is investigated. The two pendulums in the double pendulum are confined to two orthogonal planes. For nearly equal length of the two pendulums, the system exhibits 1 :1 internal resonance. The method of averaging is used to derive a set of four first order autonomous differential equations in the amplitude and phase variables. Constant solutions of the amplitude and phase equations are studied as a function of physical parameters of interest using the local bifurcation theory. It is shown that, for excitation restricted in either plane, there may be as many as six pitchfork bifurcation points at which the nonplanar solutions bifurcate from the planar solutions. These nonplanar motions can become unstable by a saddle-node or a Hopf bifurcation, giving rise to a new branch of constant solutions or limit cycle solutions, respectively.The dynamics of the amplitude equations in parameter regions of the Hopf bifurcations is then explored using direct numerical integration. The results indicate a complicated amplitude dynamics including multiple limit cycle solutions, period-doubling route to chaos, and sudden disappearance of chaotic attractors. Key words: Nonlinear dynamics, amplitude equations, orthogonal pendulum, bifurcation analysis. 1. Introduction Nonlinear multi-degree-of-freedom structural and dynamical systems under harmonic excita- tions have been studied by several researchers. It is well known that nonlinearities may lead to multiple solutions, jump phenomenon, limit cycles, natural frequency shifts, subharmonic and superharmonic resonances, period-multiplying bifurcations and chaotic motions. It is also known that the systems with quadratic and/or cubic nonlinearities, in the presence of internal resonances, may experience nonlinear periodic, quasi-periodic and chaotic motions (Nayfeh and Balachandran [1], Tousi and Bajaj [2]). One-to-one internal resonances occur in several physical systems, such as the spherical pendulum (Miles [3]), the stretched string (Miles [4], Johnson and Bajaj [5]), surface waves in a nearly square container (Feng and Sethna [6]), forced response of a simply supported beam (Maewal [7]), forced response of a nearly square plate (Yang and Sethna [8]) and orthogonal planar pendulum (Bridges [9]). Yang and Sethna [8], and Bridges [9] have included the effect of the detuning of the natural frequencies in their work. Bridges [9] analyzed the multiple periodic solutions of the orthogonal double pendulum, which is a system with Z2 ® Z2 symmetry. He studied the system in the presence of a planar harmonic excitation that breaks the Z2 ® Z2 symmetry, when the two natural frequencies are equal or nearly equal and the excitation frequency is nearly equal to the natural frequency. In his work Lyapunov-Schmidt method was used to generate a set of bifurcation equations, leading to the basic normal form for coupled equations with Z2 ® Z2 symmetry. The external force, which breaks the Z2 ® Z2 symmetry, provides an unfolding of the normal form. Known results for the normal forms were then used to determine the nature of the periodic solutions in a neighborhood of the 1:1 resonance. In the present work we revisit the orthogonal double pendulum that exhibits 1:1 internal