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I was simulating a really complicated dynamical system $\dot x = f(x(t))$ in 3D and the solution curve resulted in something like this

(After 10 time steps - Showing starting points) enter image description here

(After 50 time steps) enter image description here

(After 500 time steps)

enter image description here

I also have a plot of the norm of $x(t)$ at 500 time steps enter image description here

Note that the trajectory is constrained on a plane described by $\{x \in \mathbb{R}^n| \sum_i x_i = 1\}$, it is not protruding out. Can anyone describe this type of chaotic behavior?

Guess: strange attractor. Can we say more? I feel like we can because this type of chaotic behavior is always around some circle/polygon shape.

Shamisen Expert
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    Yes we can. But for that we have to see the equations :) also, are there any other evidences that you are dealing with chaotic attractor? For me right now it just looks like irrational winding of a torus. Have you tried computing Lyapunov exponents? Also, there is a strange moment with your observation. If you perform simulation in 3d and trajectory stays on some plane, then this plane seems to be invariant and all usual Poincaré-Bendixson stuff applies to it and forbids chaos. – Evgeny Apr 28 '17 at 07:33
  • @Evgeny Thanks for mentioning that to me. I can't produce the equations easily because it generated through a simulation software (drag-and-drop type / black box) – Shamisen Expert Apr 28 '17 at 07:43
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    Too bad( The problem is that pictures never tell the whole truth. The only way to understand what kind of attractor you have is to know what bifurcational mechanism gives birth to it. And that of course requires knowledge of equations to perform bifurcational analysis on them. Also, have in mind the problem with Poincaré-Bendixson theorem: the combination of facts in your OP is a bit weird and contradictory. – Evgeny Apr 28 '17 at 08:00
  • If this is in a plane, then you have crossing solutions, which indicates with high probability that it is not a first order system, most likely the projection from the 6-dimensional phase space of a second order conservative system. For more, as already said, one would need more information on the system, if not the equations. – Lutz Lehmann Apr 28 '17 at 08:04
  • I suspect a (projection of, as by the comment above) Liouville torus, so my bet is that you would have an integrable Hamiltonian system. Where does the system comes from? – D. Thomine Apr 28 '17 at 11:37

1 Answers1

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From a visual inspection, I would guess that your time series is quasiperiodic: It has a periodic envelope with non-periodic contents. The phase-space structure looks like a slightly warped torus. If your plot of the time series suffers from considerable aliasing, it may also be periodic with a high period length.

As you do not have access to the differential equations, I suggest the following analyses:

  1. You can look at the autocorrelation function. For chaotic dynamics, it decays; for quasiperiodic systems, it doesn’t.

  2. You can calculate the Lyapunov exponents, preferably by analysing how small perturbations of your state grow over time. If you cannot even perturb your system, you can still apply pure time-series analysis. If your maximum Lyapunov exponent is zero (within statistical fluctuations), the dynamics is not chaotic. If there is a further zero Lyapunov exponent, you have a quasiperiodic dynamics; otherwise it’s periodic.

  3. Usually, you can exclude a periodic dynamics (with a reasonably short period length) by a close look at the time series. If you cannot and you need to know whether it’s periodic or quasiperiodic, you can apply more sophisticated statistical tests.

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