Nonlinear Dyn (2006) 46:427–437 DOI 10.1007/s11071-006-9033-0 ORIGINAL ARTICLE Symmetry breaking bifurcations of a parametrically excited pendulum B. P. Mann · M. A. Koplow Received: 12 October 2005 / Accepted: 27 February 2006 / Published online: 29 September 2006 C  Springer Science + Business Media B.V. 2006 Abstract This paper examines the bifurcation behav- ior of a planar pendulum subjected to high-frequency parametric excitation along a tilted angle. Paramet- ric nonlinear identification is performed on the ex- perimental system via an optimization approach that utilizes a developed approximate analytical solution. Experimental and theoretical efforts then consider the influence of a subtle tilt angle in the applied parametric excitation by contrasting the predicted and observed mean angle bifurcations with the bifurcations due to excitation applied in either the vertical or horizontal direction. Results show that small deviations from ei- ther a perfectly vertical or horizontal excitation will result in symmetry breaking bifurcations as opposed to pitchfork bifurcations. Keywords Parametric excitation . Pendulum . Symmetry breaking bifurcations B. P. Mann () Nonlinear Dynamics Laboratory, Department of Mechanical and Aerospace Engineering, University of Missouri-Columbia, MO 65211, USA M. A. Koplow Nonlinear Dynamics Laboratory, Department of Mechanical and Aerospace Engineering, University of Florida-Gainesville, FL 32611, USA 1. Introduction This research grew out of the need to design and build a benchtop experiment for a children’s museum that would demonstrate counter-intuitive nonlinear behav- ior. Based upon the currently available literature on the parametrically excited pendulum, this system was chosen demonstrate several nonlinear phenomenon – such as dynamic stabilization of otherwise unstable equilibria [1–4]. In order to aid museum participants to better explore certain counter-intuitive nonlinear phenomenon, a specific goal was to experimentally and theoretically characterize the bifurcation transition points. However, upon experiment construction and ini- tial demonstration, it was observed that the experimen- tal behavior of the parametrically excited pendulum differed from numerical prediction. In particular, the experimentally observed bifurcation point for the hor- izontally shaken pendulum was delayed and the mean angle oscillations for the vertically shaken case were not about the vertical position. Previous research by Bartuccelli [2], and Scmitt [5], has shown that high frequency excitation of a horizon- tally or vertically shaken pendulum will result in oscil- lations about a non-zero mean angle. Additionally, it has been shown that for the cases of vertical and hor- izontal excitation, subcritical and supercritical pitch- fork bifurcations will occur. Other researchers have investigated period doubling oscillations and chaotic attractors [1, 6]. However, it was noticed that the aforementioned differences between experiment and Springer