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I have experimental measurements of orbits of 4 different simulated dynamical systems. Below is the log distance of initially closely separated points against time in seconds for each system (obtained through simulation by slightly perturbing initial states and averaging over many orbits). Lyapunov exponents (lambdas) indicated in the image are obtained through linear regression of the linear parts of the curves.

enter image description here

Now my questions is: for the red and blue curve, even though Lyapunov exponents are positive, the "attractor diameter" seems to be tiny, judging by the maximal separation of trajectories of around $e^{-10}$. So can these systems really be considered "chaotic"? Even though they separate exponentially fast initially, their Lyapunov time is infinite if I set the prediction error tolerance at say $e^{-8}$ (certainly not an unrealistic value). Since this is not typically the case for chaotic systems, I'm hesitant to call these two systems "truly chaotic". So which is the right measure, if any? And does this mean that in practice Lyapunov exponent alone is not sufficient to completely characterize chaoticity of a system?

Edit: I should probably add that all 4 system have the same scale, i.e. the same state space. And saying that and thinking about it some more I believe I can answer my own question (the systems for the red and blue curve are not chaotic), but I'll leave the question up for expert opinions.

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The attractor diameter isn’t determined or defined by the average value of $d$ after a certain time. It is what you call the scale of the state space, or with an equation:

$$ \text{diam} = \lim_{t→∞} \max_{t_1,t_2} \left| x(t_1) - x(t_2) \right|,$$ where $x$ is the entire state of your system. For a positive Lyapunov exponent, $d$ should first grow exponentially and then oscillate around some value that is the same order of magnitude (but smaller) as the diameter of the attractor. If it doesn’t, something has gone wrong.

Some approaches for you to consider:

  • Are your initial states near the attractor? If not, the exponential growth you observe may be a transient phenomenon.

  • What do the actual time series for the problematic states look like?

  • Apply a method based on tangent vectors (actually infinitesimal distances) such as the one by Benettin et al.. This way, you do not have to bother with the separation reaching the size of the attractor. (You still have to beware of transients.)

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  • $\begingroup$ Thanks for the reply and suggestions. As to your second bullet point, the time-series is 12-dimensional and roughly corresponds to a sine-like oscillation in [-1,1] for each dimension. So indeed then it does not make sense to call the "red system" chaotic, informally because the ball of initial states does not get "smeared" out over the entire attractor. Technically I don't know precisely what the attractor is (I don't have access to the equations), the only thing I can do is simulate sufficiently long for orbits to be on the attractor (this is what I did), and then perturb. $\endgroup$ – Matt Mar 21 '17 at 21:22
  • $\begingroup$ Some further comments/questions: 1) scanned the 2nd Benettin paper thanks, but seems that method is more work than just looking at the attractor diameter 2) You say "something has gone wrong". Do you claim that it is impossible for a non-chaotic system to have an initial exponential divergence and then stop diverging? 3) It seems to me that diam(A) != diam(S), since typically diam(A) < diam(S), where A=atractor and S=state space. $\endgroup$ – Matt Mar 21 '17 at 21:54
  • $\begingroup$ 1) The attractor diameter doesn’t play into this method – that’s the point. If it helps, I wrote a software that mostly automatises this method. 2) Not impossible, there are such things as chaotic transients after all. But they are, well, transients. It’s really hard to be any more specific without having access to the details. 3) Depends on how you define $S$. You can consider it infinite, but most of these states are not of interest to you. What is of interest to you is probably the attractor. $\endgroup$ – Wrzlprmft Mar 22 '17 at 7:01
  • $\begingroup$ 1) Got that :) But my point was that at this point (no pun intended), it seems less work for me to take into account attractor diam than to start using Benettin's method. 2) Interesting. Chaotic transients happen to be exactly what I'm after, in a sense. More details are difficult to provide through this medium, is it OK if I PM you? $\endgroup$ – Matt Mar 22 '17 at 10:58
  • $\begingroup$ @Matt: Okay. Please create a Physics Chat account (if you do not already have one) and invite me to a new room. It may take a while till I am available, however. $\endgroup$ – Wrzlprmft Mar 22 '17 at 11:57

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