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?
Best Answer
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
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 $[0,1)$.