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Symmetry-breaking in the response of the parametrically excited pendulum model. (English) Zbl 1136.70324

Summary: A planar pendulum is considered which is parametrically excited by a periodic vertical force. The amplitude and frequency of the excitation are used as control parameters. The downward, hanging and the upward, inverted positions correspond to equilibrium positions if we only consider the variation in angle measured from the downward position. For moderate levels of forcing, there are zones that exist in the space of control parameters, where the downward hanging position is unstable and initial conditions that are close to the hanging position lead to steady state oscillations of period-2. To review this situation, this paper describes the development of these oscillations as the amplitude of forcing is varied. In the largest zone, a symmetry-breaking occurs which brings about a pair of asymmetric oscillations. This break in symmetry of the period-2 solution can lead to either an increase or decrease in the amplitude of the forthcoming swing and reference to the experimental significance of this angle change is noted in this paper. Typically, further increases of the parameter produce a cascade of period doubling bifurcations, before most oscillating solutions eventually lose their stability so that the system must experience a rotation. As a result, symmetry-breaking becomes an effective precursor to escape from the local potential well around the hanging position. Here we compare this behaviour with that in other resonance zones. The change of geometric structure when the symmetry-breaking bifurcation occurs is examined and graphically represented as a ’pinched’ cylinder-like shape, compared with the Möbius strip that has been associated with the period-doubling bifurcation. The paper also refers to practical problems, where the introduction of nonlinearity means that potentially all frequencies below the main zone of the control space lead to dangerous effects and in some scenarios disastrous outcomes.

MSC:

70K40 Forced motions for nonlinear problems in mechanics
70K55 Transition to stochasticity (chaotic behavior) for nonlinear problems in mechanics
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[1] Bishop, S. R., The use of low dimensional models of engineering dynamical systems, Nonlinear Phenomena Complex Syst., 3, 1, 71-80 (2000)
[2] Bishop, S. R.; Clifford, M. J., Non rotating orbits in the parametrically excited pendulum, Eur. J., Mech. A/Solids, 13, 4, 581-587 (1994) · Zbl 0809.70015
[3] Bishop, S. R.; Clifford, M. J., Zones of chaotic behaviour in the parametrically excited pendulum, J. Sound Vib., 189, 1, 142-147 (1996) · Zbl 1232.70031
[4] Bishop, S. R.; Sudor, D. J., The ‘not quite’ inverted pendulum, Int. J. Bifurcat. Chaos, 19, 1, 273-285 (1999) · Zbl 0946.70016
[5] Bryant, J.; Miles, J., On a periodically forced, weakly damped pendulum. Part 1: Applied torque, J. Aust. Math. Soc. Ser. B, 32, 1-22 (1990) · Zbl 0702.70026
[6] Bryant, J.; Miles, J., On a periodically forced, weakly damped pendulum. Part 2: Horizontal forcing, J. Austr. Math. Soc., Ser. B, 32, 23-41 (1990) · Zbl 0702.70027
[7] Bryant, J.; Miles, J., On a periodically forced, weakly damped pendulum. Part 3: Vertical forcing, J. Austr. Math. Soc. Ser. B, 32, 42-60 (1990) · Zbl 0702.70028
[8] Clifford, M. J.; Bishop, S. R., Estimation of symmetry-breaking and escape by observation of manifold tangencies, Int. J. Bifurcat. Chaos, 5, 3, 883-890 (1995) · Zbl 0885.58058
[9] Jordan, DW, Smith, P. Nonlinear ordinary differential equations, Oxford Applied Mathematics and Computing Science Series, 1987; Jordan, DW, Smith, P. Nonlinear ordinary differential equations, Oxford Applied Mathematics and Computing Science Series, 1987
[10] Kim, Y.; Lee, S.; Kim, S., Experimental observation of dynamic stabilisation in a double-well Duffing oscillator, Phys. Lett. A, 275, 254-259 (2000) · Zbl 1115.70307
[11] Mullin, T., The nature of chaos (1993), Clarendon Press: Clarendon Press Oxford · Zbl 0784.58001
[12] Szernplinka-Stupnicka, W.; Tyrkieland, E.; Zubrzycki, A., Global bifurcations that lead to transient tumbling chaos in a parametrically driven pendulum, Int. J. Bifurcat. Chaos, 10, 9, 2161-2175 (2000) · Zbl 0965.70036
[13] Thompson, J. M.T.; Stewart, H. B., Nonlinear dynamics and chaos (2002), John Wiley & Sons Ltd.: John Wiley & Sons Ltd. Chichester · Zbl 0601.58001
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