Rotating orbits of a parametrically-excited pendulum Xu Xu a , M. Wiercigroch a, * , M.P. Cartmell b a Centre for Applied Dynamics Research, Department of Engineering, Fraser Noble Building, King’s College, University of Aberdeen, Aberdeen, AB24 3UE, Scotland, UK b Department of Mechanical Engineering, James Watt Building, University of Glasgow, Glasgow, G12 8QQ, Scotland, UK Accepted 22 June 2004 Abstract The authors consider the dynamics of the harmonically excited parametric pendulum when it exhibits rotational orbits. Assuming no damping and small angle oscillations, this system can be simplified to the Mathieu equation in which stability is important in investigating the rotational behaviour. Analytical and numerical analysis techniques are employed to explore the dynamic responses to different parameters and initial conditions. Particularly, the param- eter space, bifurcation diagram, basin of attraction and time history are used to explore the stability of the rotational orbits. A series of resonance tongues are distributed along the non-dimensionalied frequency axis in the parameter space plots. Different kinds of rotations, together with oscillations and chaos, are found to be located in regions within the resonance tongues. Ó 2004 Elsevier Ltd. All rights reserved. 1. Introduction The parametrically excited pendulum has been of scientific interest for a considerable amount of time. It has been found that this system can generate various types of motion, from simple periodic oscillation to complex chaos. The system motion depends on the initial condition, the excitation frequency and amplitude, and the parameters of the pen- dulor such as the length of the pendulor rod. The parametric pendula, as a kind of parametric vibration system, can be analysed by a means of various methods. Application of perturbation techniques, originally emanating from orbited mechanics, to problems in parametric and nonlinear vibration [1] have been popularised by fevered key researchers, in particular Nayfeh [2]. A good example of using the perturbation method of multiple scales for analysis of the externally loaded parametric oscillator is in the work by Watt and Cartmell [3], in which numerical and experimental investigations were also discussed for a system with power take-off capable of doing measurable work. Analytical studies of the parametrically-excited pendulum can be traced back to 1981 and the work of Koch and Leven [4], in which chaotic behaviour of the pendulum system was identified. In their later work [5], subharmonic and homoclinic bifurcations were analysed by using the Melnikov and averaging methods. More recently Bishop and Clifford [6,7,8,9,10,11,12] have been heavily involved in research into 0960-0779/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2004.06.053 * Corresponding author. Chaos, Solitons and Fractals 23 (2005) 1537–1548 www.elsevier.com/locate/chaos