# Why does chaos preclude exact solutions?

It is sometimes said that the n-body problem (using the initial positions and velocities of n point masses to calculate their future paths) has no general closed-form solution because the system is chaotic: A small change in the initial conditions produces large changes in the subsequent motion. Why should that fact preclude an exact solution?

• It relates to solutions of differential equations. AFAIK, it can be proved that there is no analytical general solution for more than 2 bodies. Commented May 13 at 21:52
• For what it’s worth, for exactly solvable problems that are chaotic the solutions are deterministic. But they are still chaotic because the tiniest shift in any parameters blow everything up. Commented May 13 at 22:24
• What analytical functions could even be used to describe a chaotic system? I am not sure what such a thing even could look like
– Dale
Commented May 13 at 23:48
• There are certainly exact solutions to the 3-body problem, if your initial conditions are nice enough (e.g. all 3 bodies same mass, following the same circular orbit around their common center of mass, $120^\circ$ apart, is an exact solution). Most solutions aren't possible to write down exactly, though. But that's mostly because of how difficult differential equations in general are, not because these particular equations give chaotic behaviour. Commented May 14 at 9:26
• @dale -- you ask "What analytical functions could even be used to describe a chaotic system?". A trajectory in a chaotic system doesn't have to look tortuous and involve lots of twists and turns. It could be a smooth function. It's just that it switches fast to a very different smooth function with the smallest changes in initial conditions. Commented May 15 at 14:18

## 5 Answers

You are right about this wording being sloppy. A chaotic system is a deterministic system whose solutions at late times are exponentially sensitive to early times. Nothing about this definition precludes an exact solution. See https://www.lpthe.jussieu.fr/~viallet/solvable_chaos_pla.pdf for examples of chaotic systems that do have an exact solution.

• A chaotic system is deterministic IF we can obtain an exact measurement of starting conditions. There are limits to how much measurement precision is possible, including quantum mechanical limits. In addition, no digital computer system is capable of infinite precision, so such "sloppiness" is unavoidable. Commented May 14 at 2:20
• @DavidWhite consider (dy/dt, dx/dt) = (y, -x)*sqrt(x^2,y^2). It admits an exact solution which is a particle orbiting on a unit circle at a certain velocity I didn't bother to calculate (probably 1), but even the tiniest perturbation will make it fly off to infinite or collapse inwards. Commented May 14 at 3:52
• @DavidWhite "closed-form" and "exact solution" are not statements about the computational precision that can be achieved. They are statements about the possible nature of a solution - whether it can be directly/analytically expressed, or whether it requires an iterative numerical approach. Commented May 14 at 15:15
• @Brondahl -- "closed-form" and "exact solution" are not statements about the computational precision that can be achieved. " I believe they are. With them you can quickly find out what happens to the system a million years from now. But with a numerical approach, your error increases with time, and if you hit chaos the approach quickly becomes useless. Commented May 14 at 18:49
• @MauriceMizrahi fortunately what you believe is irrelevant :) en.wikipedia.org/wiki/Closed-form_expression Commented May 14 at 21:25

I think Connor is correct. For what it's worth, there are three different words people are using here. "Analytical" means it can be represented by a convergent power series. "Closed-form" (usually) means it is analytical AND the expression contains no infinite series, products, etc. They are both "exact."

Karl Sundman actually demonstrated that, given non-zero angular momentum, all initial conditions of the three-body problem have an analytical (so, exact) solution. It is in the form of an infinite (Puiseux) series, so it is not closed-form.

The reason people don't talk about it often is that it's not that useful. For one, it is incredibly slowly converging. For a modern introduction, see here.

• Unfortunately, an exact infinite series solution that requires the evaluation of a very large number of terms (i.e., it's VERY slow) is very little better than no solution at all, as practically the only attribute of a computer that makes it very useful is its enormous speed. Commented May 14 at 23:59
• These kinds of solutions in chaotic dynamical systems converge just fine, but they have a finite radius of convergence, and typically the singularity is in the complex plane where the differential equation might not make much sense. Also, the singularity tends to be non-simple pole, so that the Riemann sheet is infinitely branching - worse than anything you have seen in textbooks, I guarantee. Commented May 16 at 9:38
• EDIT: I think, from reading the paper you linked, Sundman constructs a Puiseux series that converges for all real time, plus a narrow strip around the real line. Unfortunately, the most important equation in the universe, the Yang-Mills classical field equation, does not have a Puiseux series with rational exponents, but instead a similar one with irrational, $X(t) = a/t + c * (t-to)^{\sqrt3}$ or some such near the singularity. Commented May 16 at 9:58

I think there might be two issues being conflated.

In a strictly mathematical sense, a system could be chaotic and yet have exactly calculable future states. This would mean that for two initial states that differ only very slightly their future states eventually are wildly different, but for each of them we can say exactly what the future states will be.

But when we're trying to do predictive physics rather than pure maths, a system being chaotic means that we cannot predict its future states even if the mathematical model we're using has exact solutions. This is because to make predictions about the evolution of a physical system we need to provide inputs to the mathematical model describing the initial state. Those inputs will be based on measurements of the real world (i.e. the masses and positions of the planets in a solar system), and those measurements are never perfect.

If the measurements are not perfect, then the initial state we provide to the model will never perfectly correspond to actual reality; there will be small differences from the true state. And if the model is chaotic those small differences matter; no matter how much we improve our measurement accuracy an arbitrarily small difference between the state that we measure and the true state will eventually lead to very different predictions of the future. It doesn't matter if we can exactly calculate the future of every initial state if we cannot tell which one of those very different exact predictions is going to be close to what happens in reality!

Non-chaotic systems can be more easily used to make predictions in physics because when our measurements provide a description of the initial state that is slightly off from true reality, the prediction will only be slightly off from the true future.

So chaos doesn't necessarily preclude exact solutions in the mathematics, it precludes using any solutions to make arbitrarily long-term predictions.

As the previous answers explained, chaos does not preclude the existence of simple analytic solutions. A simple counter is the logistic map. Say you want to solve the dynamic system: $$x_{n+1} = 4x_n(1-x_n)$$ It ticks all the usual boxes of a chaotic system: sensitivity to initial conditions (i.e. a positive Lyapunov exponent which turns out to be $$\ln2$$), dense periodic orbits etc. At the same time, you can easily solve the problem by recognising a trigonometric identity: $$x_n = \sin^2\theta_n \\ x_n = \sin^2\left(2^n\arcsin(\sqrt{x_0})\right)$$

Hope this helps.

It all depends on your definition of "solvable" and "chaos". In classical mechanics (and I believe that is the case your are interested in), the statement is true, if you are talking about Liouville integrability (as definition of solvable): see Chaos and integrability in classical mechanics