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I'm doing some numerical experiments to test an integrator, and I got this plot, for the motion of 5 pendula, whose initial displacement differ by $10^{-6}$ radians away from straight up ($\theta_0 = \pi - d\theta$, where $d\theta\in(1,2,3,4,5)\times10^{-6}$). It seems like they're diverging from each other exponentially. I'm wondering if this is real, or if it's due to accumulated error in my integrator. this particular integrator has passed all the other tests I've thrown at it (energy is being conserved to within one part in $10^5$, and this behavior is not replicated for pendula that begin below $\theta_0 = \frac{\pi}{2}$).enter image description here

I'd be a little surprised if this is real, because I thought that the single pendulum has only negative or null lyapunov exponents. Is there something special about inverted pendula, maybe because their effective potential is an inverted hill at the top of their swing?

Red Five
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David
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  • The displacement is due to different periods, so in time the phase difference grows linearly. The numerical error will also have an influence, but it is less over short time frames. – Lutz Lehmann May 08 '24 at 04:31
  • but isn't that also true of pendulums whose initial condition is not inverted? when I ran the same experiment, but where they all start a $d\theta$ away from $\pi/2$, then they hardly displace at all. – David May 08 '24 at 04:37
  • For small amplitude the period grows quadratically in the amplitude, so yes, near zero amplitudes will have almost the same period. You need to start farther away. – Lutz Lehmann May 08 '24 at 04:41

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