This is the logistic map:
. Image of bifurcation diagram of the logistic map
It is a fractal, as some might know here.
It has a Hausdorff fractal dimension of 0.538.
Is it possible to calculate/measure its fractal dimension using the box counting method?
A "hand waving calculation" is good enough.

Update: I understand there is another way to calculate the logistic map using the Kaplan-Yorke Conjecture. Can anyone explain that and how it can help calculating the fractal dimension of the logistic map?

Update2: Seems like the way to go around this is not Kaplan-Yorke Conjecture (which is a unproven Conjecture still), but use the correlation dimension. There is a paper with the solution here, hope I'll know more as I read it.

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    $\begingroup$ @KaziarafatAhmed Surely you have heard about chaos theory? Well, one great importance to physics is that fractals are very closely related to chaos in many cases. So this question is very much on topic. $\endgroup$
    – Wouter
    Commented May 25, 2013 at 17:23
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    $\begingroup$ @KaziarafatAhmed Fractals are important in physics. Look up the Hofstadter Butterfly. First fractal found in physics :) I do however motion a move of this post to mathematics. Cheers. $\endgroup$
    – John M
    Commented May 26, 2013 at 0:40
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    $\begingroup$ @KaziarafatAhmed fractal dimensions are used in some intermittency models in turbulence theory, where the Hausdorff dimensions appear too. $\endgroup$
    – Dilaton
    Commented May 26, 2013 at 0:47
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    $\begingroup$ I just spoke to my physics professor today. Apparently you can do this using the correlation dimension. I am reading about it and hope to post an answer. There is a paper on how to do this here $\endgroup$
    – GuySoft
    Commented May 26, 2013 at 21:54
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    $\begingroup$ @KaziarafatAhmed: enwp.org/Causal_dynamical_triangulation $\endgroup$ Commented Sep 6, 2013 at 1:50

2 Answers 2


Yes you can estimate the dimension by box counting. It is not quite hand waving but the idea has an advantage to be intuitive.

1) You consider the logistic map attractor like an analogy to a Cantor set whose dimension you can compute by box counting.

2) You remember that when the chaotic bands double, their sizes scale like $1/a$ and $1/a^2$ where $a$ is the second Feigenbaum constant, $a \approx 2.5029$. So the procedure looks like the Cantor set production because at each doubling you "remove" a part of the previous band. The difference being that the new smaller 2 parts have not the same size like in the Cantor set.

3) You suppose that at the Nth doubling you need $2^n$ boxes of size $R_n$ to cover the bands. Then at the $N+1$ stage you will need $2^{n+1}$ boxes of average size $$ R_{n+1} = \frac{R_n}{2}\left(\frac{1}{a} + \frac{1}{a²}\right). $$ The hand waving part is the arithmetical average because there is no rigorous reason to use it but one sees the idea of approximation.

4) The box counting dimension is then $$D_b = - \frac{\log 2}{\log(1/2(1/a + 1/a²))} = 0.544$$

You will admit that the hand waving was not so bad because one is not far from the much more rigorously derived value of $0.538$.

As a particular remark, the box counting is not very practical when the attractor is not strictly self similar like f.ex the Cantor set because the results depend then on the particularities of the covering method chosen

  • $\begingroup$ The Feigenbaum constant is 4.669... Your calculations don't work. $$1/2 \cdot (1/a+1/a^2)=0.130...$$ This results in a value of $0.339...$ It's better to just use $$1/a$$ and get an estimate of $0.449$. $\endgroup$
    – Zach466920
    Commented Aug 14, 2015 at 1:42

As the number of bifurcations goes to infinity, we can make some various claims.

$$d_f={{\ln(N)} \over {\ln(s)}}$$ Where $d_f$ is the fractal dimension, N is the number of boxes, and s is the scale.

(Examine $N\sim s^{d_f}$ to get the above)

$$N=2^n$$ And $$s \sim {\delta_f}^n$$ Where $\delta_f$ is the Feigenbaum constant.

We obtain, in the limit...

$$d_f \sim { {\ln(2)} \over {\ln(\delta_f)}}=0.4498...$$

Which has about 16% error from the empirical value of 0.538...

  • $\begingroup$ 0.538... is not empirical, its calculated. Its the Hausdorff dimension. This not even a good approximation when you consider box counting. $\endgroup$
    – GuySoft
    Commented Aug 16, 2015 at 8:07
  • $\begingroup$ @GuySoft odd, that you don't have an exact value then...either way, my approximation is better than the above. $\endgroup$
    – Zach466920
    Commented Aug 16, 2015 at 14:08
  • $\begingroup$ @GuySoft btw the very link you give cites the dimension as approximate and calculated rather than exact... $\endgroup$
    – Zach466920
    Commented Aug 16, 2015 at 21:21
  • $\begingroup$ Its an approximate to any degree of accuracy you want. You could do the calculation and get a finer result if you want. $\endgroup$
    – GuySoft
    Commented Aug 18, 2015 at 10:32
  • $\begingroup$ @GuySoft but you said it was exact. It's not, so there's a contradiction. Either way, the above answer gets a more accurate answer based on a mathematical mistake, so you should take that into consideration too. $\endgroup$
    – Zach466920
    Commented Aug 18, 2015 at 14:17

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