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I suspect that you cannot prove it purely from experimentally/observational, and I think that approach is useless. If you think that the whole universe as a single state in state space, then there is no way you can compare with any other state. You may think that there might be a phase space with (macroscopic) parameters $(V,\{\alpha_i\})$, so we can have a ...


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A dynamical system is a system that evolves by a rule over time. As opposed to modeling a system with a PDE, the system is modeled by some type of iterated function, $f^t(x)$. Smooth implies that the function being iterated is differentiable, so we are not talking about an iterated system like cellular automata.



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