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?