I'm not really sure that I have the required knowledge to pose this question properly so I'll just pose it poorly.

The rise of numerics has allowed us to study systems of differential equations for which no general solution had previously been found (i.e. gravitational systems with >=3 particles).

My question is if these systems of equations even actually have a solution that can be represented by functions or whatever. Could it be that these systems don't have solutions in the traditional sense?

In line with this question, can a system that demonstrably shows chaos have a closed form mathematical solution? Are there any functions that show chaotic behavior?


1 Answer 1


It seems like your question can be divided into two primary lines of inquiry:

  1. Do all differential equations (given some initial/boundary conditions) possess a solution?

  2. Can a chaotic system possess a closed-form solution?

The answers are no and yes, respectively.

On the first topic, the kind of mathematical results that guarantee (or don't guarantee) solutions to ODE/PDE's are called existence theorems; one famous example is the Picard-Lindelöf theorem, which states that for some ODE initial-value problem of the form

$$\frac{dy}{dt} = f(y,t),\quad y(t_0) = y_0$$

a solution always exists and is unique around some neighborhood of the point $y_0$ if $f$ is Lipschitz-continuous.

That doesn't mean that there's a closed-form version of the solution, though; the differential equation $\frac{dy}{dt} = \sin(xy)$ is guaranteed to have a unique solution locally around some initial value $y(0) = y_0$, but no one's found a way to describe it in closed-form with elementary functions.

An example of a simple ODE with no solution is the following boundary-value problem:

$$\frac{d^2 y}{d t^2} + y = 0,\quad y(0) = 0,\quad y(\pi) = 1$$

Feel free to try and find a solution; none will satisfy those conditions (you can prove it using the Fredholm alternative). In short, existence is usually something you have to prove for a given differential equations problem, and there is an entire sub-field of math dedicated to it. One of the biggest open problems in math is proving the existence of specific kinds of solutions to a particularly nasty set of PDE's called the Navier-Stokes equations!

On the second topic, there are (very few) systems with closed-form solutions that demonstrate chaotic behavior, but they certainly exist; it's just usually the case that, because those systems are so sensitive to the initial conditions, those solutions really aren't very useful. For that reason, chaos theorists are much more interested in finding things like what the solution converges to after infinite time or if the kind of solution changes after some critical change in an initial value condition.

Examples of closed-form solutions to chaotic systems can be found, for example, in the logistic map:

$$x_{n+1} = r x_{n} (1 - x_{n})$$

where $r$ is some constant. When $r = 4$, you find a closed-form solution for $x_n$:

$$x_n = \sin^2(2^{n}\theta\pi),\quad \theta = \frac{1}{\pi} \sin^{-1} (\sqrt{x_0})$$

You can also find a closed-form solution when $r = 2$, but there are no closed-form solutions for general $r$, which is what you expect in chaotic systems. (If you could get such a solution for all cases, there isn't really anything you can't predict, which is precisely the hallmark of chaos.)

If you really wanted to, one way of generating a "closed-form" solution for a chaotic system defined over some finite interval with square-integrable solutions is to create a generalized Fourier series which will approximate it to arbitrary accuracy; but that's basically cheating and isn't very instructive (sometimes)!

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    $\begingroup$ Nice answer! But for "there are (very few) systems with closed-form solutions that demonstrate chaotic behavior, but they certainly exist", could you give or link to an example? Any that are not series solutions? $\endgroup$
    – jnez71
    Jul 5, 2018 at 15:23
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    $\begingroup$ @jnez71 You got it, will add an example; it is not "very chaotic" but is hopefully a good illustration. $\endgroup$ Jul 5, 2018 at 15:38
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    $\begingroup$ It's perfectly chaotic even when $r=2$: "unpredictability" is usually understood in the sense of sensitive dependence to initial conditions. Which of course is present here. $\endgroup$
    – Evgeny
    Jul 7, 2018 at 21:59

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