A simpler example where you can see that numerical approximations predict the correct statistics, regardless of computing errors, is the logistic map x -> 4*x*(1-x). If you iterate this on the interval [0,1], you get a chaotic system, but one that is not too hard to analyze mathematically. You can explicitly calculate the distribution of a chaotic dense orbit as it fills up the interval [0,1] and verify that a numerical simulation gives the same distribution.

In my dynamical systems class, I learned that the Lorenz system actually having a strange attractor (as opposed to a ton of closed loops that somehow combine to look like the attractor) was only proven rigorously by mathematicians in the 90s!

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Looking at graph b of figure 2 (tropical all sky OLR anomaly), what would be a good way to model this as some kind of stochastic process?

This is the graph in question:

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