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Chaos and Predictions — Trying to predict a floating leaf

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    Even when we know all the rules governing the motion of an object, and can determine precisely the initial conditions, it turns out that some motions can be predicted and some cannot. And it is not just a matter of complexity: we can model unpredictable systems with very simple equations.

    [See the full post at: Chaos and Predictions — Trying to predict a floating leaf]


    So equilibrium as n->oo depends on an infinite number of finite sequences (the ones represented by letters of the alphabet and further subdividing the interval by an order of magnitude).

    By what process the choice of such initial conditions is accomplished and how can that be deterministic while remaining ‘unpredictable’?

    Aren’t you “borrowing” the “chaoticness” of such a process to endow this system with ‘chaos’?


    Does this not depend somewhat on what definition of prediction we are using? If we assume that we are talking about predicting the equilibria over an infinite number of steps or the state of the system after an arbitrary number of steps, then it’s true that we can’t predict the behavior of these systems. However, in many real-world systems where we want to make predictions we have initial conditions (with the fuzziness mentioned above) and a horizon (time into the future or number of subsequent steps) within which we want to make our prediction. Therefore, rather than a simple statement of some systems can be predicted and some cannot, it would seem that the predictability of a complex or chaotic system depends on how far out our horizon is and the level of sensitivity to initial conditions. Also, in real-world systems we tend to have an update rule to our measurements to get a “new” set of initial conditions to recalibrate our predictions. This is why we tend to be pretty good at predicting the weather 24 hours in advance, but dismal at predicting it a year in advance.

    I am not an expert on dynamical (chaotic or not) systems so I would be very interested in knowing if there are systems that are truly unpredictable with this model of prediction.

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