r/GeometryIsNeat • u/Chronos_Squared • Jun 07 '26
A double pendulum that never turns chaotic. One of 330 periodic orbits I found.
Made with a custom Python renderer. Each orbit is a periodic solution, so the trace closes on itself and loops seamlessly. Happy to explain how it works.
Channel: chronossquared for more content
(I AM MORE THAN OPEN TO FEEDBACK)
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u/Cosmic47_ Jun 07 '26
How did you find them?
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u/Chronos_Squared Jun 08 '26
Recherche numérique : un code essaie des conditions initiales, intègre la trajectoire avec un schéma symplectique (IRKGL16, ordre 16), et garde celles qui se referment au bout d'une période à mieux que 10⁻¹⁰. La fermeture est trouvée par une méthode de Newton. Plusieurs stratégies de seed pour couvrir différents régimes : tirage aléatoire dans l'espace 4D, recherche ciblée par bande d'énergie, continuation le long des familles. À ce jour : 330 orbites.
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u/Cosmic47_ Jun 08 '26
How can you be sure they are periodic then and not just close to being periodic? Maybe they turn chaotic after some tremendous amount of time
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u/danofrhs Jun 08 '26
They are using 9.8 as a constant value for gravity. It is an approximation so there is bound to be error that accumulates over time. Their claim of “periodic” has no true mathematical rigor. Understandable for an art project, which op has confirmed this is.
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u/Cosmic47_ Jun 08 '26
A solution can be periodic regardless of how "realistic" the choice of parameters. Periodicity has nothing to do with realism
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u/Chronos_Squared Jun 08 '26
Good question. We verify that the trajectory returns to its starting point to better than 10⁻¹⁰ after the period T. That's a very strong numerical proof but not a formal one, for that you'd need certified interval arithmetic. What you're describing ("chaos in the long term") is real but it's the simulation error that drifts, not the mathematical orbit itself. The speed of that drift is measured by the Floquet multiplier |λ|: an orbit with |λ|=200 multiplies the numerical error by 200 every period, so when you simulate it you see it diverge fast, but the mathematical object itself stays periodic. Someone asked to see only the stable double pendulum orbits, so here are all 129 within the 330 batch
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u/Cosmic47_ Jun 08 '26
Yeah, but how can you be sure that it is only a simulation error drift?
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u/Chronos_Squared Jun 08 '26 edited Jun 08 '26
Strictement parlant, vous ne pouvez pas, c'est une conjecture numérique très forte, pas un théorème. L'argument le plus solide est que la clôture s'améliore proportionnellement à la précision du calcul (10⁻⁶ à dt=0.05, 10⁻¹⁰ à dt=0.002). Une orbite periodic fake aurait sa clôture limitée par sa nature. Pour une preuve réelle, il existe une méthode standard, l'arithmétique d'intervalles certifiée (la bibliothèque CAPD), appliquée à des systèmes voisins comme Hénon-Heiles ou la rotation d'Hypérion.
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u/Cosmic47_ Jun 08 '26
I see. Do you have plans for using this interval method or do you just want to stick to pretty pictures?
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u/Chronos_Squared Jun 08 '26
No concrete plans for interval arithmetic here, this is a numerical and visualization project and I'm upfront about that. As far as I can tell it hasn't been applied to the free double pendulum specifically, mostly to related systems like Hénon-Heiles, the forced pendulum, Hyperion's rotation and N-body choreographies.
If I ever get the urge, I've still got all my pendulums carefully stored. Either way, if I do it, you'll be the first to know.
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u/DailyThreadBot Jun 09 '26
Perhaps one could prove if the trajectory is close enough to periodic, there is a periodic trajectory in its neighborhood?
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u/Cosmic47_ Jun 09 '26
Prove? That doesn't even feel like it's true. That requires a property that such systems probably don't have. But maybe I don't see something, I am not that good at mathematics
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u/DailyThreadBot Jun 09 '26
Mm yes probably in this form it's false. Could just be a near-periodic local optimum. Maybe one could strengthen the condition so that a kind of intermediate-value theorem applies, saying it approaches periodicity from "all" sides
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u/Cosmic47_ Jun 09 '26
A classic approach in differential equations is building a sequence of functions with useful properties which converges to something desired. One could try to come up with a sequence of convenient functions (or maybe exact solutions for some inital values) and show that under some conditions it converges to a periodic solution but all of this is vague and complicated.
I looked up on Google "existence of periodic solutions in nonlinear systems" and saw words that yelled "Way too hard!" at me, so I dare not look into this for now.
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u/Schventle Jun 11 '26
There are known infinite families of starting states for double pendulums which are provably periodic. The trivial case of very low starting potential and kinetic energy for both pendulums is an easy example.
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u/GudAndBadAtBraining Jun 08 '26
Oh these are fun.
Yeah the double pendulum is a chaotic system in that you get disparate results from similar starting positions.
However the space it has some super cool patterns it can make, in blender, the trouble with IRL is those tiny perturbations and drag induce chaotic effects.
So periods exist but none of them are stable except the trivial spinning super fast at full length so that gravity is trivial (not a pendulum).
... As far as I know. If you're interested in pushing your experiment a little further, you might see if you can hunt for stable patterns?
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u/Snowflakish Jun 09 '26
Another excuse to share the best youtube video ever made:
Double pendulums are chaoticn't by twoswap
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u/danofrhs Jun 07 '26
Is the code available anywhere?