# Rössler attractor, Convergence of box counting to estimate the fractal dimension

At the moment I want to estimate the fractal dimension of the Rössler attractor. I have written a program, which is counting the boxes hitted N(ɛ) (with ɛ := side length of boxes) by a trajectory on the attractor. The trajectory is calculated numerically. So for a rising number of iterations, N(ɛ) is rising, like it is shown in the following graph (for ɛ = 1/128, horizontal axis is for iterations, vertical axis for N(ɛ)).

Graph for N(ɛ) against iterations: The thing is, that the Graph is not converging, even if I run it for 50*10^6 iterations. There is a similar work for the Lorentz attractor, written by Mark J.McGuinness called "The Fractal Dimension of the Lorentz attractor" published in the Physics Letters Volume 99A number 1 on the 14 November 1983. Since the Lorentz attractor is similar to the Rössler attractor, you could get a good orientation from this work. In the paper they have the same problem for the Lorentz attractor and they guess the convergence to be algebraic and not logarithmic.

Question What is known about the (algebraic ?) convergence of N(ɛ) (boxes bitted by a trajectory on the attractor) for strange attractors ?

• I think all that McGuinness means by algebraic convergence is that the difference to the "true value" decays as a power law, see eq. 5. I suppose to apply this to your problem you may try showing your data on a log-log plot en.wikipedia.org/wiki/Log%E2%80%93log_plot . In any case, I think this site is not quite right for your question and it would be better served on math.stackexchange.com .
– j.c.
Jan 6, 2018 at 21:49
• In my opinion, @Chopin has clearly been doing some research and he has encountered a difficulty. The given problem does not seem to be trivial, and is definitely of research level. It's not Grothendieck-like algebraic geometry, but it is still research. Jan 6, 2018 at 21:54
• Ok, thank you for the answer, this helps me. I have posted the question on math.stackexchange, but until now i've got no answer, so i posted it here. Jan 6, 2018 at 21:59
• @Chopin Perhaps you can edit your post to include a link to the post on MSE. Also, reading your question again and in light of Alex M.'s comment I think there could be an interesting research question here but you haven't made it explicit. In particular, I don't see any question marks, only a few things that you say you don't know anything about. Consider editing so that there is one or a few clear mathematical questions. For instance, you might ask something like "what is known about the convergence of box-counting algorithms for fractal dimension estimation for strange attractors?"
– j.c.
Jan 6, 2018 at 22:23
• Crossposted on MSE: math.stackexchange.com/questions/2594638/… Jan 6, 2018 at 22:43