So 5 month ago i posted this Question: Rössler attractor, Convergence of box counting to estimate the fractal dimension.

Since then i have assumed, that the rate of convergence of $n(ɛ,n)$ (sum of hitted boxes from a solution trajectory on the Attractor for $n$ iterations of the trajectory) converges algebraicly against the real values $N(ɛ)$. I have done this, because Mark J.McGuinness, had used the same convergence rate for the Lorentz attractor in his work The fractal dimension of the Lorenz attractor. His argumentation is based on the work Edward N. Lorentz: Deterministic Nonperiodic Flow. In this work Lorentz created a cusp-shaped map, by plotting successive maxima in $z$ against the previous maxima in $z$. Now McGuinness says, that this map has a invariant probability density, which fades to zero at the ends. From this on he concludes, that the ends of the probability density decay algebraicly. He than proofs, that the rate of convergence of $n(ɛ,n)$ against $N(ɛ)$ has to be algebraic and not logarithmic.

So with the assumption, that the rate of convergence would be the same for the Rössler Attractor, i have calculated the Hausdorff capacity for the Rössler Attractor and got a pretty good result.

My Problem is, that i don't know if it was the right decision, assuming that the rate of convergence would be the same for the Roessler Attractor.

Stating with McGuinness's reasoning, i would watch at successive 1D return Maps for the Rössler Attractor, and look if they have the same characteristic cusp-shape. The Problem is, they don't have the same characteristic cusp-shape, they seem to have a logistic map characteristic. Here is an example for such a logistic map. And the appropriate probability density functions don't have the characteristic bell shaped course, like the probability density functions of the cusp-shape maps have(https://arxiv.org/abs/1306.3800).

So i think i could not follow the same reasoning, McGuinness has used, because of this difference between Rössler attractor and Lorentz attractor.

Then i have searched a lot of papers, but i could not find anything on this topic.

So now i don't know how to continue arguing, that the convergence rate for the box counting for the Rössler attractor is algebraic.

Question: Is there anything known about the convergence rate of the box counting algorithm for the Rössler attractor ? Or do you have an advice for me on how to continue here ?



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