I was reading that Chaos (in the mathematical sense) is deterministic, but not predictable. Is this unpredictability an intrinsic property of Chaos, or is it a practical matter of computing power and/or mathematical knowledge? In other words, given the most powerful computer that is physically possible in the universe, as well as a full understanding of mathematics (and Chaos in particular), would Chaos be predictable, or would it still be unpredictable, since its unpredictability is an intrinsic property?

I would greatly appreciate it if the more knowledgeable members could please take the time to clarify this.

EDIT: If we presume that predictability is in reference to the equations' sensitive dependence on initial conditions, then I would change the question into a more sensical one. My question, then, is not whether or not such a computer would make Chaos insensitive (in other words, eliminate sensitivity) to initial conditions, since my understanding is that the sensitivity of the equations to initial conditions is an intrinsic property of Chaos (by definition); rather, my question would be that, assuming the most powerful computer physically possible in the universe, as well as a complete knowledge of mathematics (and Chaos in particular), could we predict Chaotic phenomena exactly, as we can with non-Chaotic phenomena, since Chaos is deterministic? Or is there some intrinsic property of Chaos that makes its (full (continued?) ?) evolution (in the mathematical sense) unknowable, given the possible computing power and mathematical knowledge available in the universe?

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    $\begingroup$ What do you mean by "predictable"? $\endgroup$ – Servaes Mar 11 at 13:25
  • $\begingroup$ @Servaes that’s a good question. The lecture slide just has the dot-point deterministic, but not predictable, so I’m not sure. $\endgroup$ – The Pointer Mar 11 at 13:28
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    $\begingroup$ The "unpredictability" of chaos usually refers to the phenomenon of sensitive dependence on initial conditions. $\endgroup$ – Rahul Mar 11 at 13:34
  • $\begingroup$ @Rahul Yes, that sounds sensible. $\endgroup$ – The Pointer Mar 11 at 13:36
  • $\begingroup$ @Rahul In light of this definition of unpredictability, I have posed a more sensical question (see my edit). $\endgroup$ – The Pointer Mar 11 at 13:47

The definition of chaos is still not completely established. The following conditions are, however, commonly accepted:

  1. deterministic dynamics: given perfect knowledge of the initial conditions and perfectly accurate integration, a future state can be known exactly.
  2. sensitivity to initial conditions: two initial conditions that are arbitrarily close together, yet not the same, will result in wildly different behaviors over time.

The second condition is what makes chaotic systems intrinsically unpredictable. This is not because we do not have enough compute or imperfect models, but rather that we never have perfectly accurate initial conditions. Even if we had a perfect model, and an analytical solution (or more to the point, perfectly accurate integration), if we had a tiny error in the initial state, our prediction could be wildly off. The prediction would, however, be perfectly accurate for the initial condition we actually used.

Lack of compute power, or an less-than-perfect model (I assume that by "full understanding of mathematics", you mean a perfect model), are of also issues. But in essence, imperfect integration (whether due to numerical errors or an imperfect model) are generating tiny errors at each integration-step, which can be viewed as restarting the integration with a slightly incorrect initial condition. So, the "intrinsic" reason is still the same: sensitivity to initial conditions.

Good references imho are Nonlinear Dynamics and Chaos by Steven Strogatz (very intuitive and applied), and the inaugural lecture of Will Merry

  • $\begingroup$ Great answer! Thanks for taking the time to clarify. $\endgroup$ – The Pointer Mar 12 at 12:00

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