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Not sure if this question belongs in physics or mathematics.

Recently I have been making some computer simulations of somebody swinging on a playground swing at varying frequencies. Specifically I was interested if it is possible to cause a 1:3 resonance by moving the swinging person's centre of mass three times as fast as the frequency of the swing. By KAM theory I would expect that this should be the case.

However, when I did the computation swinging at 1:3 (or at any other rational multiple except 1:2) did not cause any resonances.

Typically KAM theory only tells you something about the stable solutions and not the resonant ones. So it is not really a contradiction. However I still would except that you can cause resonance at 1:3. Or would this resonance only be visible at very large time scales?

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closed as off topic by Sasha, rschwieb, Arkamis, Thomas, Aang Sep 24 '12 at 9:30

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This sounds like a physics question to me. –  picakhu Aug 16 '12 at 13:14
    
Voted to migrate to physics –  Sasha Aug 16 '12 at 14:30
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why? Seems perfectly fine to me... –  Thomas Rot Aug 16 '12 at 14:40

1 Answer 1

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Ah so I think the answer lies in numerical mathematics. It appears that the 1:3 resonance is very difficult to find since the resonant domain is really small.

You will see this if you look at the bifurcation diagrams indicating the resonant domains. http://ars.els-cdn.com/content/image/1-s2.0-S1874575X10003140-gr6-2.jpg Then you observe that the resonant domains for all frequencies except the 1:2 resonance is small if the epsilon is small (very informally the size of epsilon would correspond in the swing to the magnitude of your initial velocity), hence numerically these are probably very hard to find. However if I increase this epsilon then the resonance domains will increase their size and indeed you can verify this very easy numerically.

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