A note on estimating chaotic systems

This is a short note unlike other posts but it is necessary: Recently I came across with some complex systems presentations in Youtube. Most of the scientists, especially complex system scientists, treat chaotic dynamical systems as mysterious objects. The models are beautiful (so you should be careful around them)  and they are not that difficult to deal with in real world. The typical example you would hear that a chaotic system is extremely sensitive to the uncertainty in the initial condition. That's true: If I give you the initial condition of a deterministic chaotic system with a machine epsilon uncertainty along with the exact dynamics, the trajectory you would predict will differ from the real one vastly after a while. This, then they say, can be counted as the evidence of undecidability in chaotic systems.

Well, that's not quite the case. The property of nonlinear response to initial uncertainties can be used to "explain" certain things in a supposedly interesting dinner discussion, but apart from that, you never have this fictional scenario in the practitioners' world. First, you never do such long-term predictions with small initial uncertainties  and short-term predictions with sufficiently small uncertainties are not that bad either. Second, when you have a chaotic system, the usual task is to estimate states of the system given noisy observations from it instead of predicting long horizons from an initial state. In this case, you have noisy observations, and it helps to estimate the state at a given time. More specifically, if you have the observation model and system model, it is entirely possible to estimate chaotic systems over long time horizons with particle filters (filtering, not prediction!). If you are sure of the dynamic model, you can even safely do prediction (but for short time horizons).

See, for example, the following very nice visualisation of the particle cloud of the particle filter where it tracks a Lorenz 63 chaotic model (The system is 3D but only two dimensions are in the plot).

Credit: This is from my friend Victor.

PS: This is not to say that it is easy to do prediction in the real world, it is mostly impossible. This is to say that chaotic systems which are shown as examples of unpredictability are not quite unpredictable unless you try to predict them in a senseless scenario. However, the real world systems don't behave like these toy chaotic systems, they are usually much more complicated.


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