I have what I think is a very simple doubt, but one that I've never seen explicitly addressed.
It's a classic coding activity to simulate a double pendulum. You can code this simulation up more or less carefully, but you end up calculating what state the double pendulum should be in at discrete timesteps.
However, it's a famous fact about the double pendulum that it's chaotic. Small changes to initial conditions lead to large changes in the evolution of the system.
My question: don't floating point errors and errors introduced by discretization constitute the sort of small changes that lead to large changes later on? Why, then, should I put any faith in the state the system spits out several timesteps later?
Is there any sense in which a numerical simulation of a double pendulum can actually show me what the system will do over large timescales? Or is it more that the simulation shows me a sequence of moment-to-moment reasonable evolutions that nevertheless don't accurately simulate the system over large timescales?