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Correct me if I am wrong, but I take it that people actually think the classic Lorenz system is fully characterized, i.e. we know what the attractor looks like and the various fractal dimensions of it, and it's always (with all parameters fixed; no bifurcations) going to be deterministically chaotic (with a thin fractal dimension). If this is not the case, please let me know what is a better way to describe the classic Lorenz attractor.

If my understanding is true, then I am super confused about how do people know this at infinite time? I think the techniques for finding the fractal dimensions, e.g. the Lyapunov dimension or the correlation dimension, both have a subtle assumption that the attractor is sufficiently sampled. Just because Lyapunov exponents have converged or we see the leveling off when estimating the correlation dimension doesn't mean the attractor is sufficiently sampled. Do we actually have rigorous proofs, even for the most well-studied classic Lorenz equations, that the trajectories will never fall onto some periodic orbit or get on to another smaller/bigger attractor with a very different fractal dimension? Basically, how do we know we have integrated for long enough?

I know the above is not a problem in the phase plane (2D), because of Poincare-Bendixon Theorem.

Last question: if I integrated for some time for an unknown (high-dimensional) system, and calculated its Lyapunov or correlation dimension, what can I actually say about the system?

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    $\begingroup$ This question is not about computational science. It would make more sense to ask on math.SE or perhaps physics.SE. $\endgroup$
    – David Ketcheson
    May 30 at 19:51
  • $\begingroup$ I do apologize. I intended this for the Math SE but I guess I just mis-clicked and ended up here. $\endgroup$
    – Axel Wang
    May 30 at 21:40
  • $\begingroup$ I think the second part of the question leans computational enough that it is fine even though the first part is basically lit review. Numerical estimation of any properties of chaotic systems is interesting and difficult IMO $\endgroup$
    – whpowell96
    May 31 at 2:04
  • $\begingroup$ OP is specifically asking for non-computational evidence/proof. $\endgroup$
    – David Ketcheson
    Jun 1 at 3:33

2 Answers 2

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If I am remembering correctly, periodic orbits are known to be dense and countable in the Lorenz attractor, but are also known to be unstable (The first return map has $|F'| >1$ everywhere). A rigorous proof of the existence of a strange attractor for the Lorenz attractor was given by Warwick Tucker. This was done by constructing a Sinai–Ruelle–Bowen measure on the attractor, which is like a generalization of an ergodic measure in the case where volume is hard to characterize (like on fractal dimension attractors). This ensures that the dynamics properly saturate the attractor in the limit of infinite time.

As for general dynamical system, it is difficult to determine when your simulated trajectories have sufficiently saturated the attractor. Eckmann and Ruelle provide a heuristic rule for determining the number of data points on an attractor needed to meaningfully estimate Lyapunov exponents. The heuristic says that given the attractor diameter $L$ and attractor correlation dimension $D$ , to ensure good esimates of Lyapunov exponents, one needs a sufficient number of neighbors in a small ball with radius $r = \rho L$, where $\rho \ll 1$. To achieve this, one should take $N \approx \rho^{-D}$ samples on the entire attractor. This heuristic can also be inverted to place maximum limits for the attractor dimensionality one can accurately estimate using $N$ total samples, so this can be used as a measure of sufficient integration time, but anything more precise is likely difficult to state in general.

Tucker, Warwick, The Lorenz attractor exists, C. R. Acad. Sci., Paris, Sér. I, Math. 328, No. 12, 1197-1202 (1999). ZBL0935.34050.

Eckmann, J.-P.; Ruelle, D., Fundamental limitations for estimating dimensions and Lyapunov exponents in dynamical systems, Physica D 56, No. 2-3, 185-187 (1992). ZBL0759.58030.

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  • $\begingroup$ Very nice references! Computationally, I was wondering how does one then constrain the distribution of these N points to somehow uniformly cover the entire attractor, in stead of just having them all concentrated at a corner by e.g. interpolation. I think the Eckmann&Ruelle paper (the paragraph following Eq. (6), starting with "In case $(u_i)$...") is essentially saying this, but still how this can be done rigorously in a simulation is not well defined (for an unknown attractor)? $\endgroup$
    – Axel Wang
    Jun 1 at 0:12
  • $\begingroup$ You could estimate a characteristic frequency of the dynamics via a fourier transform $\omega_0$ then integrate until $T = N/\omega_0$ to ensure you get $N$ characteristic "periods." Another way to do this would be to simulate the dynamics for different time lengths and perform a time-delay embedding on the outputs of each simulation. You could then estimate the dimension of the embedding via SVD or something and the integration time when that starts to level off might indicate it is long enough to learn sufficient information about the attractor. $\endgroup$
    – whpowell96
    Jun 1 at 1:26
  • $\begingroup$ Generally, the best way to populate the attractor is to integrate the system forward in time, but then you have to be careful sampling points from the output as samples can be highly correlated. Long integration times can get around this since points will be distributed better around the attractor. If your system is such that it is either too stiff or expensive to integrate for long times, then nondimensionalization/renormalization or dimensionality reduction via projection or something might ease the burden $\endgroup$
    – whpowell96
    Jun 1 at 1:29
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The argument may be the other way round. When considering dynamical systems there are a couple of well studies dynamics that can happen.

  1. Your dynamic converges to a stable fixed point.
  2. Your dynamic converges into some form of repeated oscillation.
  3. Your dynamic is fixed from the beginning (trivial).
  4. your dynamic runs of to infinity.

The conditions for these are known and can be proven for given dynamical systems. Then one can prove that the Lorentz attractor does not meet any of the requirements for these dynamics. You first prove, that your dynamic can't run of to infinity. Then you prove that there is no stable fixed point to converge to. Then you prove that no simple oscialltory solutions are possible.

That means something else entirely has to be going on.

Edit: This of course dependes on the region of phase space you are in. It is possible to have multiple regions.

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    $\begingroup$ It is worth noting that, besides chaos being difficult to prove, one of the main hurdles in proving systems were chaotic is that it took many years to define chaos in a rigorous way other than "It's doesn't fall into these other categories." $\endgroup$
    – whpowell96
    Jun 1 at 16:19

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