We have presented a detailed analysis o f the bifurcational behaviour o f the parametrically excited pendulum, from the relatively simple hanging behaviour, to the more complex oscillatory motions, inverted solutions, rotating orbits, and tumbling chaos. Two approaches have been used; a numerical attack based on cell-mapping, path following, and bifurcation following, and a variety o f analytical techniques from harmonic balance, the method o f strained parameters, to the more unusual concepts from braid and knot theory. Both approaches revealed details that the other missed, but there was also a great deal o f agreement between the two which served to reinforce the results. Some physical experimentation was carried out, and the findings gave qualitative agreement with some o f the numerical results.
The main points revealed by this investigation regarding non-rotating orbits centred around the approach o f reducing the system to an escape from a potential well problem. As w e saw in chapter 3, the parametrically excited pendulum is a generic example o f a system which permits escape from a symmetric potential well under parametric excitation, and this allowed much o f the bifurcational behaviour to be predicted. By considering the horseshoe formed by the invariant manifolds o f the hill
Inverted Zone 3
P
2.5 2 1.5 0.5 Q l - 0.5 1.5 2 2.5 0) .3Figure 11.1: B ifurcation diagram for the param etrically excited pendulum. H is w here the equilbrium loses stability at a pitchfork bifurcation which is sub-critical to the left o f point c. S is the locus o f a sym m etry breaking bifurcation, and F represents the end o f a period- doubling cascade although only the initial period-doubling is shown since the cascade is very rapid.The period-2 fold line A associated with the sub-critical pitchfork bifurcation is also shown, along w ith the secondary unstable zone H'. J represents the creation o f a (1,1) rotating orbit, w hich period-doubles at line B, and again at line G, w hich is close to the final bifurcation to tum bling chaos. The perio d-1 solution restabilises at line I. Tw o period-3 subharm onic orbits are also shown in w indow s (3,1) and (3,3). T he pendulum is stable in the inverted position in the 'inverted zone’.
over small parameter regimes. Braid and knot theory allowed pseudo-Anosov orbits to be located, which imply that the parametrically excited pendulum has chaotic dynamics, and by developing a crossings algorithm some bifurcational precedence relationships were obtained which were not immediately obvious from the numerical results.
For rotating orbits, there is still much work to be done, but the results in chapter 9 represent an important start in classifying rotating orbits without resorting to further braid and knot theory, and a wide range o f subharmonic solutions have been located numerically k
There are many possible extensions to this work, which could include the effect o f damping on the system, a more complete description o f the rotating orbits based on braid and knot theory, or on the more practical side, the physical realisation o f more o f the numerically located orbits could be attempted together with some nonlinear control techniques to stabilise the subharmonic orbits. However, this would involve considerable modifications to be made to the experimental apparatus by way o f added measuring sensors and finer control o f the forcing frequency. There is also scope to apply the results obtained here to some physical systems which can be modelled by the parametrically excited pendulum, and to put to use the expertise
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