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The definitions, as I understand them, are:

A limit Torus is characterized by two or more periods, that form an irrational fraction. And a chaotic attractor (roughly speaking) has an infinite periodicity.

Question: How would I tell a limit torus with many (say a million) periods from a truly chaotic attractor? Analytical methods generally don't apply to complex dynamical systems, and it seems incredibly computationally expensive to figure out numerically.

Context: I am asking because I often find mention of chaotic dynamics, without much justification of what makes them chaotic. I am left with a feeling that mathematicians just want to call their phenomenon chaotic because it is complex.

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  • $\begingroup$ It is not just complex; it is everywhere dense (ergodic, shall we say?) $\endgroup$ – Ivan Neretin Mar 16 '17 at 8:02
  • $\begingroup$ You can compute Lyapunov exponents for the trajectory of interest. It's not so cheap, but you have no other choice. The computation of largest Lyapunov exponent is also pretty inexpensive and is a good indicator of something non-trivial happening (I don't say chaotic attractor because there are some complications). The rule of thumb is the following: if LLE < 0 than it is an asymptotically stable motion, if LLE = 0 -- it is possibly a torus and if LLE > 0 -- the dynamic is non-trivial and maybe chaotic. $\endgroup$ – Evgeny Mar 16 '17 at 8:08
  • $\begingroup$ There are few other tools that are helpful. 1) You can compute the dimension of attractor. Attractors like nice tori should have dimension close to integer. Chaotic attractors often have non-integer dimension. 2) You can look for specific bifurcations that are known for giving rise to chaotic behaviour. One of the most known of them is Shilnikov saddle-focus loop and others resemble it very much (in the sense that Smale's horseshoe occurs and this explains complexity). $\endgroup$ – Evgeny Mar 16 '17 at 8:19
  • $\begingroup$ ... Or you can look for specific bifurcations that give rise to invariant tori, like Neimark-Sacker bifurcation of limit cycle. Finding limit cycles can also be done by means of Andronov-Hopf bifurcation. $\endgroup$ – Evgeny Mar 16 '17 at 8:22
  • $\begingroup$ @Evgeny: Why do you post all of this as comments? It should be an answer. $\endgroup$ – Wrzlprmft Mar 16 '17 at 8:23
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I am asking because I often find mention of chaotic dynamics, without much justification of what makes them chaotic. I am left with a feeling that mathematicians just want to call their phenomenon chaotic because it is complex.

The main distinction between chaotic dynamics and quasiperiodic ones is that chaotic dynamics are sensitive to initial conditions – the butterfly effect. Quasiperiodic dynamics are not: An slight perturbation of the initial conditions essentially means slightly perturbing the phases of each component, and these perturbations do obviously not grow.

The practical consequence of this is that the precise evolution of chaotic dynamics is impossible to predict after a while, while you can predict quasiperiodic dynamics forever. Two examples:

  • The movement of the moon is a quasiperiodic dynamics with three periodic processes being superimposed (rotation of the moon around the earth, nodal precession, apsidal precession). Mankind has long learnt how to predict solar eclipses caused by this movement.

  • The dynamics of a double pendulum is chaotic. No matter how well you reproduce a certain initial condition, the behaviour will be strongly different after a while. It’s impossible to predict the behaviour longer than a given time (at least until friction kicks in).

Sidenote: Chaos is not necessarily complex. There are chaotic systems which we perfectly understand.

How would I tell a limit torus with many (say a million) periods from a truly chaotic attractor?

If the limit torus is what you have, you can’t. Too have a 1000000-torus, you need a system with at least 1000001 dynamical variables (degrees of freedom) and analysing this is pretty difficult. On the other hand, chaos can be produced by model systems with just three dynamical variables (the double pendulum has four), so the case of a 1000000-torus can be excluded here.

What is usually indeed impossible to distinguish is a chaotic dynamics and a periodic or quasiperiodic one with a very high period length. But here the period length is what matters, not the number of superimposed periodic processes.

Finally, if you do not have some nasty case as your 1000000-torus, you can distinguish chaotic and (quasi)periodic dynamics by several characteristics (which you usually have to compute numerically). The most popular one is the largest Lyapunov exponent, which quantifies the divergence of nearby trajectories and thus the butterfly effect.

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