# How do we know chaotic systems are actually chaotic and not periodic?

The usual way to calculate how chaotic a system is would be to measure the divergence rate using the Maximal Lyapunov exponent, but it requires you to wait until infinity, measure the divergence, then calculate the exponent. Isn't that impossible?

For a system like the double pendulum, we know that two double pendulums at a small angle apart diverge quite quickly. But who's to say they wouldn't converge and repeat themselves after a really long finite time? If so the separation between two trajectories would decrease and result in non-chaotic motion.

So how is it possible to prove that a specific system is chaotic when we have to wait till infinity to measure the Maximal Lyapunov exponent?

• By Poincaré recurrence theorem the best you can say is that for bounded systems, you can get to a point that is infinitely close to the starting conditions as you want. But if you have a very chaotic system, it just does not matter how close you get, if the conditions are not exactly the same, you will get a different trajectory after that point. In the case that you get exactly to the initial conditions, then it is by definition not a chaotic trajectory, but a periodic one. – Mauricio Nov 17 '20 at 19:04
• Indeed, chaotic trajectories don’t intersect, but how can one prove a specific trajectory doesn’t? Usually chaotic systems are formed from differential equations or recurrence relations and can’t be solved analytically, so there seems to be no good way to tell if they are chaotic – bluemystic Nov 17 '20 at 20:05
• there are many examples of chaotic systems with periodic orbits. Many billiard like systems have some simple trajectories that can be analyzed analytically. The three body problem in gravitation has also some well know periodic solutions. I guess it depends on the specific system and nature of the trajectories. – Mauricio Nov 17 '20 at 20:41
• @bluemystic If you discard waiting until infinity as a real life impossibility, then you also should discard perfectly obtained initial data. The best you can get is some probability-distribution around your intended/measured value. And though many chaotic systems have some periodic orbits, there are so few that the probability of picking one of them is zero. (Which is something that one often can prove.) – mlk Nov 18 '20 at 14:44

Well, on the one hand, yes, chaos is a mathematical abstraction so, for instance, there will never be an experimentally or numerically measured Lyapunov exponent, only finite Lyapunov exponents (FLEs) - the same way there won't ever be a sphere, or any exponential growth (the universe seems finite), etc. They are idealized constructs that only approximate the physical reality. On the other hand, these idealizations have been proven very useful inummerous times.

As for

who's to say it wouldn't converge and repeat itself after a really long finite time?

there are rigorous mathematical proofs for some systems, and we can consider a simple one such as $$x_{n+1} = 10\cdot x_n \mod 1,$$ which, by multiplying by $$10$$ and truncating to bellow $$1$$, produces trajectories such as: $$0.123456 \to 0.23456 \to 0.3456 \to 0.456 \to \ldots$$ which, for almost all$$^1$$ initial points, leads to infinite, never-repeating trajectories. And arbitrarily close initial conditions diverge exponentially fast: actually, if two random initial conditions coincide for the first 40 digits only, after $$n=40$$ iterations of the map above, their trajectories bear no relation to each other any more. These are certainly chaotic trajectories.

$$^1$$ This is a rigorous statement, in the sense that the irrational numbers have measure 1 on $$[0,1)$$.

• I like that example! It’s simple but chaotic. – bluemystic Nov 17 '20 at 20:00
• @bluemystic Yes, it's a very neat and useful example. It's called "Bernoulli shift". – stafusa Nov 17 '20 at 21:15

A mathematical model can be chaotic. A real life system can exhibit chaotic behaviors. That minor word shift helps capture the idea that a system is unpredictable in the same way that a chaotic system is, but which may not be completely chaotic. This is rather an important detail because no system exists in a vacuum. All real life systems interact with their environment, and this interaction can make it hard to tell whether the unpredictability is coming from inside the system or outside.

The systems you refer to which might converge at a later date are in a region known as The Edge of Chaos. This is the fuzzy boundary between order and chaos that is even more frustratingly complex than chaos itself. Evolution is a famous example of a process that is on the edge of chaos. It is not clear if it will converge to order. It is not clear if it will spiral off into chaos. Indeed, it isn't even clear that it has to do one or the other!

The usual way to calculate how chaotic a system is would be to measure the divergence rate using the Maximal Lyapunov exponent, but it requires you to wait until infinity, measure the divergence, then calculate the exponent. Isn't that impossible?

No indeed. Every conserved system (like the double pendulum you mentioned), has an energy hypersurface that its geometry is confined in the phase space. Of course, two slightly separated trajectories depart from each other on the energy hypersurface exponentially, but their distance eventually saturates to a quantity proportional to the area of energy hypersurface. So, you just have to wait for the saturation of distance function. here , here and here are some examples I just googled. Notice the saturation of distance function.

For a system like the double pendulum, we know that two double pendulums at a small angle apart diverge quite quickly. But who's to say they wouldn't converge and repeat themselves after a really long finite time? If so the separation between two trajectories would decrease and result in non-chaotic motion.

There is a recurrence time for any system with conserved energy (Poincare recurrence). But even for small systems, this time scale is enormously larger than the human average age and we should not care about them too much in lab :)

So how is it possible to prove that a specific system is chaotic when we have to wait till infinity to measure the Maximal Lyapunov exponent?

There are several ways to indicate chaotic behavior in short term. For example, the power spectrum (or Power density) of a periodic (or quasi-periodic) system is a discrete delta function series, while the spectrum of chaotic behaviors is continuous.