'Islands' of regularity discovered in the famously chaotic three-body problem | Hacker News Reader

'Islands' of regularity discovered in the famously chaotic three-body problem

phys.org

119 pointswglb26 comments

AFAIK, these islands of regularity/stability are a common aspect of chaotic systems, say the logistic map[1].

[1]: https://en.m.wikipedia.org/wiki/Logistic_map

I may be mistaken but I seem to recall having also read on numerous occasions that ejection of one of the bodies is an inevitability in these types of systems.

Given the limited state space of the simulation, I'm not sure I see what the big discovery is here.

It's certainly a neat result to see it visualized though.

Off topic buying haven't seen a bit of code in Wikipedia to run to see a visualisation before

The paper is at [1], and the summary chart is at [2].

The top chart shows which of the three bodies escapes the system given initial starting conditions. The bottom one makes it easier to see patterns by reducing noise. It uses the k-nearest-neighbors algorithm to find the dominant color near each pixel. More than the large pools of stability, what's interesting are the bands of stability - like electromagnetic fields.

I have no idea how one codes this up in a computer, with fixed precision math and all. Numerical methods is a black box for me.

[1]: https://doi.org/10.1051/0004-6361/202449862 [2]: https://www.aanda.org/articles/aa/full_html/2024/09/aa49862-...

I never really understood the 3 body problem. Am I correct in saying that given three bodies that attract one another, there is currently no known way of predicting, at arbitrary time, where they'll end up at, even though it is possible to simulate to reasonable degree of accuracy for a limited time horizon? If that is the case, is this an example of what Stephen Wolfram regularly calls as computationally irreducible? Also it is so interesting to me that the jump from 2 to 3 in different domains herald such unfathomable complexity. 2SAT for instance is polynomially solved whereas 3SAT is NP Hard. If I were a competent mathematician, I'd definitely investigate the link between NP Hardness and Computationally irreducible chaotic process like 3Body problem.

The problem, a bit more precisely, is that very small changes in the initial conditions can result in very large changes in result. So even if you have a perfect numerical simulation, if your inputs are even slightly off the simulation might eventually be worthless. Unless you're in one of these islands of stability, maybe, I'm fuzzier on that. (And I don't know a lot about numerical simulation techniques, but I think that sort of situation also makes a "perfect" simulation even harder than usual.)

Compare to two-body interactions, where there is a pretty small family of closed-form solutions to all the possible trajectories. You can just plug in a future time t and (in principle) get a mostly correct answer. In practice nothing is exactly two-body, the bodies aren't point masses, etc, but the small differences have proportionally small effects.

For a two-body problem computing a future location in spacetime of the two bodies is easy enough -- easy with Newtonian gravity, and a bit harder with General Relativity. For an N-body problem, with N>2 there are no formulas that can give you the answer, and instead you have to do a numerical computation using as small a dt (ideally an infinitesimal time step) as possible, with as much precision as possible, and as many steps as needed to get to that future point in time. Obviously the numerical computation will a) take time, b) have errors (due to not having infinite precision and true infinitesimals). Apart from that the N-body problem is chaotic anyways.

Yes, I believe it is the same as Wolfram's "Computational Irreducability". I'll copy paste the question I emailed him about it:

> In computers, floating point numbers have a limited precision, based on how many bits are used. Does the innate 'error' in floating point numbers have implications on your theory of 'Computational Irreducability'? Are the 'errors' introduced with each floating point operation is fundamentally computationally irreducable? Does this have a domino effect that causes basic classical mechanics to become computationally irreducible? Like the integration of velocity into position, it is a O(1) operation to compute the future position given a velocity and a starting point, but if a computer is rendering each individual frame, it will accumulate error in each frame, and the final frame may end up in a different position. Does this concept relate to the real world through things like the Planck length? Is space perhaps not purely continuous?

I have a partial answer for that. ;-) For at least some functions, iterative solutions exist which converge to correct solutions. For at least some iterative solutions the rate of convergence (the estimated error bounds) is a continuous function: it can be seen that convergence proceeds normally (I should be more careful with that word) as long as the estimated error bounds for the iteration is larger than the computationally induced error. I'll hazard a guess that with constraints some formulations which model a three body problem are solvable to the point where the estimated error bounds approaches the computationally induced error due to floating point imprecision.

The thing about the three body problem, same as here, is that there are regions in the initial position and velocity space where very small changes produce extremely large differences in function output.

There has been some work / recognition of the value of topological understanding to orbital mechanics, although I can't find whatever I read a few months ago. Best I could find was https://en.wikipedia.org/wiki/Symplectic_geometry and this DDG search looks promising: https://duckduckgo.com/?q=symplectic+integrators+solar+syste...

Those are some interesting open questions about physics, but the happenstance limitations of modern computer hardware have no bearing on them.

>Is space perhaps not purely continuous?

Pretty sure that's the entire basis of quantum mechanics

No - space and time are continuous in quantum mechanics, despite the fact that other quantities such as angular momentum and energy are discrete (quantized). Relevant comment from Luboš Motl, a string theorist who produced a proof that, under some assumptions, quantum mechanics on discrete space and time is inconsistent, echoed by Peter Shor: https://physics.stackexchange.com/a/9077

space? or energy?

Does anyone know if that visualization is representative or just an illustration? I’m struggling with how all these parameters (mass, speed, and relative direction for all three objects) could map to a fractal on a 2 dimensional plane.

The study didn't plot the entire state space. It's simply a map of the end state of a limited space of parameters (specifically, the mass of the objects is fixed, two of them are in a particular orbit, and basically only the angle and timing of when the third object approaches is varied). The plot just shows which object gets thrown out of the system for each initial configuration.

That makes a huge amount more sense. Thanks!

Rehydrate! Rehydrate!

I'd bet a thousand dollars that when graphed, these islands of regularity will correspond to the graph of the Mandelbrot set.

I've spent a good portion of my life working on fractal software, trying basically every fractal rendering algorithm there is, reading everything I can about it, managing fractal art communities and organising meetups... so I'm obviously very pro-fractals biased, and even so, I'm curious what makes you conjecture so strongly that 3-body stability islands would have anything to do with the Mandelbrot set. Why that one in particular, over the vast infinitude of possible fractal shapes?

I’m willing to take that bet.

So what do you think the math is going to look like about the starting values of these stable states? I'm speculating that they correspond to Julia sets by some fundamental relationship.

1) They're not stable, they're regular. The point is these areas of the state space have similar outcomes if they are perturbed slightly, as opposed to the result being random

2) there's literally a plot of it in the article, and while it shares some properties with the mandelbrot set (symmetry about the X axis, is a fractal), it doesn't resemble it at all.

Will it be that fractal? it seems tenuous. Will it be a fractal? That would make sense.

Why Mandelbrot, specifically?

I am going out on a limb and speculating that these stable sets of conditions correspond to Julia sets by some mathematical relationship and that the underlying fundamental principle is the same.

https://en.m.wikipedia.org/wiki/Julia_set

The Mandelbrot and Julia sets are the outcomes of specified functions and have a relationship with each other. But yeah I would speculate that something similar, in the way that a Mandelbrot set is self-similar, would be observed in at least some circumstances. Gave you a vote for that.

Edit: reading the article I can see that Julia sets are defined separately from the Mandelbrot set and the corresponding Julia set(s) are an example. So maybe you're right. I'm not really a mathematician, I just use math to get stuff done.

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