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I have what I think is a very simple doubt, but one that I've never seen explicitly addressed.

It's a classic coding activity to simulate a double pendulum. You can code this simulation up more or less carefully, but you end up calculating what state the double pendulum should be in at discrete timesteps.

However, it's a famous fact about the double pendulum that it's chaotic. Small changes to initial conditions lead to large changes in the evolution of the system.

My question: don't floating point errors and errors introduced by discretization constitute the sort of small changes that lead to large changes later on? Why, then, should I put any faith in the state the system spits out several timesteps later?

Is there any sense in which a numerical simulation of a double pendulum can actually show me what the system will do over large timescales? Or is it more that the simulation shows me a sequence of moment-to-moment reasonable evolutions that nevertheless don't accurately simulate the system over large timescales?

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    $\begingroup$ It's still a fairly stable system over small timescales, the point of the simulation isn't to accurately simulate the movement of a double pendulum over long periods of time (which would likely be impossible due to numerical imprecision as you've pointed out) but to point out that it's indeed chaotic. Funnily enough your simulation might still be more accurate than a real life model, simply because it leaves out external factors like wind. $\endgroup$
    – TC159
    Dec 31, 2022 at 2:37
  • $\begingroup$ Models in physics are anyway just approximations of the reality. The point is that even if our model would be actually exact (and totally correct describe the process) , and if we would know the starting parameters with an accuracy that cannot be imagined , we could not predict the configuration in near future (probably not even a few minutes in the future). This makes a simulation utterly worthless for practical purposes. $\endgroup$
    – Peter
    Dec 31, 2022 at 10:08
  • $\begingroup$ Of course, in reality we have too many factors disturbing the process we cannot consider , so the situation is even worse. $\endgroup$
    – Peter
    Dec 31, 2022 at 10:11
  • $\begingroup$ @TC159: Funnily enough your simulation might still be more accurate than a real life model, simply because it leaves out external factors like wind. – How can a simulation be more accurate than real life? What is your reference here? The point of the real-life double pendulum is not to model the differential equations of a double pendulum. $\endgroup$
    – Wrzlprmft
    Dec 31, 2022 at 12:13
  • $\begingroup$ A Hamiltonian system can have under some circumstances, chaotic behavior. Such is the case. See Boris Chirikov papers. $\endgroup$
    – Cesareo
    Dec 31, 2022 at 12:28

1 Answer 1

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My question: don't floating point errors and errors introduced by discretization constitute the sort of small changes that lead to large changes later on?

Yes.

Why, then, should I put any faith in the state the system spits out several timesteps later?

Well, it depends on what you want from your simulation:

  • Do you want to predict the individual motions of a given double pendulum with given initial conditions? In this case, you have much bigger problems that numerical inaccuracies, namely your limitations of knowing or preparing the initial conditions and all things that do not factor into your model (friction, detailed shape of the pendulum, air flow, etc.).

    In a more practical example, consider weather prediction: Again, uncertainties about the current state of the system and modelling inaccuracies are the limiting factors, not numerical accuracy.

  • Do you want to have a model that reproduces the general dynamics of the double pendulum so you can investigate the influence of parameters, etc.? In that case, your numerical errors are not a problem because they do not relevantly affect the general dynamics. For example, you can investigate the rate of flips depending on the lengths of the pendulum arms, weights, initial energy, etc. If your model is sufficiently close to your real pendulum (which usually boils down to reducing friction in the latter as much as possible), you will get decently accurate results from this.

    Most mathematical models (of anything, not only physics) are like this, even without chaos coming into play. For example, we can model simple pendulums very well and, e.g., predict their frequency with reasonable modelling effort. However, it’s much more difficult to predict the precise motion of a given pendulum and answer questions such as: Will the pendulum swing to the left or right in exactly one hour? (Even for precision pendulum clocks this will be impossible after a few days, and those are the culmination of centuries of technological development on optimising this on the experimental side.)

  • Do you want to prove that the double pendulum is chaotic by means of simulation? In that case, you can simply simulate initial conditions that differ by a small amount that is yet orders of magnitude higher than your numerical accuracy¹. To ensure that you do not see effects by numerical noise, a good check is to increase the numerical accuracy and see whether it affects any indicators of chaos. Finally, there is the shadowing lemma, which states that you can find a true trajectory (without numerical errors) close to any numerically obtained trajectory.


¹ There are more sophisticated techniques than this, but they would go beyond the scope of the question.

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  • $\begingroup$ This was the answer I was looking for. Am I right to say that it boils down to "it can be useful if you ask the right questions of it," but that determining exact behavior in the far future is essentially hopeless? $\endgroup$ Dec 31, 2022 at 17:59
  • $\begingroup$ @CharlesHudgins: Yes. Though bear in mind that determining the exact behaviour of the far future is hopeless primarily due to incomplete knowledge of initial conditions and imperfect models rather than numerical errors. $\endgroup$
    – Wrzlprmft
    Dec 31, 2022 at 21:33

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