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We often talk about chaos, but is chaos an objective term or a subjective term? If it's an objective term, is there a mathematical way to determine it? Is it possible there's a threshold where something is neither chaotic or in a state of order?

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    $\begingroup$ en.wikipedia.org/wiki/Chaos_theory $\endgroup$
    – user65081
    Commented Oct 31, 2020 at 3:03
  • $\begingroup$ en.wikipedia.org/wiki/Edge_of_chaos $\endgroup$
    – user65081
    Commented Oct 31, 2020 at 3:20
  • $\begingroup$ Please look at the links Wolphram Jonny provided and let us know if they fail to answer your question. $\endgroup$
    – Cort Ammon
    Commented Oct 31, 2020 at 4:59
  • $\begingroup$ Chaos has different meanings in philosophy and mathematics. Are you asking about one of these in particular or the relationship between them? $\endgroup$ Commented Oct 31, 2020 at 9:15

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Chaos in the sense of chaos theory is a property of a dynamics. There are several competing definitions, most of which require a rather deep understanding of dynamical systems, but they yield the same result except for some pathological cases (about which very few people really care). To keep things simple, let’s consider the simple definition, that a dynamics is chaotic if it is deterministic, bounded, and has a positive Lyapunov exponent (i.e., it is sensitive to initial conditions).

This is a clear mathematical definition and allows no grey area between chaos and other dynamics. A dynamics can neither be half-deterministic, half-bounded, nor can it have a half-positive Lyapunov exponent. If a dynamics is completely accessible to me (i.e., I can simulate it for an infinite time and observe all dynamical variables), I can determine all the properties that go into the definition of chaos and that’s it. What can happen is that a dynamics is changed and transitions between chaotic and regular (think of a periodic oscillation). For example a dripping tap may switch between chaotic and regular dripping depending on the temperature.

For real systems, things get a lot more messy, as we only can only observe a part of them for a finite time and always have noise. Here you can only answer to which extent a dynamics is dominated by deterministic components and whether those are regular or chaotic. The main challenge for asserting chaos here is to distinguish it from a stochastic phenomenon, a topic about which entire textbooks exist. Very briefly, you try compare the Lyapunov exponent (or another indicator of chaos) with that for instances of an appropriate null model derived from the data. As we are dealing with reality, mathematically rigorous answers do not exist.

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There seem to be several methods of determining whether observed random behavior is truly random or the result of dynamical chaos, of course mathematics is used.

There are the methods using the Lyapunov exponent. On page 504 the exponent is defined and its use for chaos is discussed .

I found this, and it is a mathematical method that seems to be used with success.

The 0-1 Test for Chaos: A review Georg A. Gottwald and Ian Melbourne

AbstractWe review here theoretical as well as practical aspects of the 0-1 test for chaos for deterministic dynamical systems. The test is designed to distinguish between regular, i.e. periodic or quasi-periodic, dynamics and chaotic dynamics.

So yes, there are mathematical tools to define chaotic behavior in a sample.

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