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I know the strength of attraction for a single attractor might vary from place to place. Say, if I calculated the Lyapunov exponents for a small portion of the attractor and they have already converged, is it possible that they could diverge again had I followed that same trajectory longer such that it has traversed more portions of the attractor?

Basically, my question is, assuming my system is governed by one single attractor, can I characterize its fractal dimension (to detect degree of chaos) as soon as I have a set of converged Lyapunov exponents, even though my time-integrated trajectory hasn't fully traced out the shape of the attractor yet?

Any ideas or references would be really appreciated!

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Consider the system from this paper of mine. It exhibits chaotic transitions between several dynamical patterns. An exemplary temporal evolution of the local largest Lyapunov exponent looks like this:

local largest Lyapunov exponents for example system

As you can see, the local Lyapunov exponents behave characteristically different in the green and blue sections. An estimator for the global Lyapunov exponent will certainly converge to some extent in the initial large green segment, but then diverge again once the dynamics switches to the blue pattern.

And that’s not even the end of the story: If you wait for several pattern switches to occur, you will see the estimator converge yet again (to a positive value) for an average comprising all patterns. But this still would be wrong, since the entire pattern-switching dynamics is only an extremely long chaotic transient leading to a periodic attractor. (To be accurate, there are probably multiple periodic attractors for the dynamics studied in the paper, but I am pretty confident, I could craft an example that only has a single attractor but retains all other properties.)

Now you might say that this dynamics is high-dimensional and so on and this doesn’t apply to whatever application you have. However, then you are making an educated human decision and quantitative techniques such as Lyapunov exponents exist to avoid those and all their problems.

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  • $\begingroup$ Very nice example! Say if you don't know the chaos was only transient and the final state is periodic, how do you use a quantitative technique such as Lyapunov exponents to determine this? Sounds like you are saying you just have to calculate the exponents for long enough such that it sees more on the trajectory, but even then it seems you can only be sure for the time<(time you have integrated) and still can't say anything about the final state of the system? What I am looking for here is really how to be certain about a final characterization of the system (e.g. fractal dim.) $\endgroup$
    – Axel Wang
    Commented May 17, 2023 at 21:56
  • $\begingroup$ Also I would be interested to learn how you can make a single attractor that has a chaotic transient and then converge to a limit cycle. What would be the fractal dimension of this one attractor? Would the system has a final largest Lyapunov exponent <0? I really think this situation can only happen in systems with multiple attractors and a trajectory just switched from following a strange attractor first (the chaotic transient) to another periodic attractor, which would need to be characterized separately (chaotic with lyapunov exponents>1 and vice versa for the periodic). $\endgroup$
    – Axel Wang
    Commented May 17, 2023 at 22:07
  • $\begingroup$ And for my second question above, I am asking the situation when we keep the system parameters exactly the same and only let it evolve in time (i.e. no bifurcations.) $\endgroup$
    – Axel Wang
    Commented May 17, 2023 at 22:14
  • $\begingroup$ What I am looking for here is really how to be certain about a final characterization of the system (e.g. fractal dim.) – I don’t think there is a way for this. This is about numerical estimates, not mathematical proofs after all. Of course, at the end of the day the question is whether the difference matters in application. $\endgroup$
    – Wrzlprmft
    Commented May 18, 2023 at 9:36
  • $\begingroup$ Also I would be interested to learn how you can make a single attractor that has a chaotic transient and then converge to a limit cycle. What would be the fractal dimension of this one attractor? – The attractor, like any limit cycle, has a dimension of 1. The transient itself happens within the vicinity of what is called a chaotic saddle (not an attractor). If it were an attractor, the dynamics could not escape from it. Of course you can characterise the saddle separately. $\endgroup$
    – Wrzlprmft
    Commented May 18, 2023 at 9:43

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