Correlation Dimension d c Another measure of dimension Consider one point on a fractal and calculate the number of other points N(s) which have distances less than s. Average over all starting points –C(s) Plot ln(C(s)) against log(s) –gradient=d c

Example Henon Map - x n+1 =a-x n 2 +by n y n+1 =x n d c =1.21 Notice that the strange attractor in the Henon Map while it has structure at all length scales is not exactly self-similar

Definitions of dimension Two definitions so far –d F - the number of boxes need to cover fractal –d c - number of points within a given distance on fractal Question: – is d c =d F ? Very often no!

When do the two dimensions agree ? For exactly self-similar fractals like the Sierpinski triangle d C =d F

When do they not - strange attractors Eg. The logistic map at a c x=ax(1-x) – d c =0.498 – d F =0.537 So, in this case these two dimensions are not equal! Same is true for Henon.

MultiFractals For most “real world” fractals d c is not equal to d F ! Strange attractors fall into this category eg. logistic map These attractors have structure at all length scales but are not exactly self-similar. Called multifractals

Examples of multifractals Diffusion limited aggregation or DLA –grow a crystal by allowing molecules to move randomly until they stick to substrate –stick preferentially near tips of growing structure –(multi)fractal –In 2D (correlation) fractal dimension DLA cluster is d F =1.7.. –i.e mass=L 1.7

Viscous fingering Similar problem: –two miscible liquids (gelatin and water), DLA-like structure appears when mixed carefully. Low surface tension –immiscible liquids (water and oil) fingers are wider tension is large For oil recovery - add soap to lower surface tension - allows water to penetrate shales and flush out oil...

Fractals in Nature Coastline of Norway –Fjords of all sizes ! –Length of coastline depends on scale at which we look –count how many boxes the outline of the coast penetrates –see d F =1.52! –Scales from 30,000 km to 2500 km Bronchial tree.

Explanation It looks fractal - but how do we know for sure... A single tube of diameter D splits into 2 tubes of diameter d 2(d/D) 3 =1 approx. Remember Cantor’s set …. D=ln(2)/ln(3) or.. 2*(1/3) D =1 fractal with dimension D=3! –Space-filling!

Fractal dimension for multifractals For exact fractal Nr D =1 N=number of pieces r=length of each Generalize: eg. At each iteration split into 2 pieces but with different lengths r 1 and r 2 r 1 D +r 2 D =1