At the moment I want to estimate the fractal dimension of the Roessler 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)).
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 Roessler 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.
The thing is I don't know anything about algebraic convergence and how to apply it to my problem. I would be thankful if somebody could help me. I also searched for a book or an article about algebraic convergence, so I would also be happy if somebody could give me a tip about that.