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Correlation Dimension d c

Understand the correlation dimension (dc) in fractals, comparing it with the box-counting dimension (dF), common in multifractals like the Henon map. Learn how dc and dF diverge in complex structures like strange attractors.

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Correlation Dimension d c

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  1. Correlation Dimensiondc • 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=dc

  2. Example • Henon Map - xn+1=a-xn2+byn yn+1=xn dc=1.21 • Notice that the strange attractor in the Henon Map while it has structure at all length scales is not exactly self-similar

  3. Definitions of dimension • Two definitions so far • dF - the number of boxes need to cover fractal • dc - number of points within a given distance on fractal • Question: • is dc=dF ? • Very often no!

  4. When do the two dimensions agree ? • For exactly self-similar fractals like the Sierpinski triangle dC=dF

  5. When do they not - strange attractors • Eg. The logistic map at ac x=ax(1-x) • dc=0.498 • dF=0.537 • So, in this case these two dimensions are not equal! • Same is true for Henon.

  6. MultiFractals • For most “real world” fractals dc is not equal to dF ! • 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

  7. 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 dF=1.7.. • i.e mass=L1.7

  8. 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 ...

  9. 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 dF=1.52! • Scales from 30,000 km to 2500 km • Bronchial tree.

  10. 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!

  11. Fractal dimensionfor multifractals • For exact fractal NrD=1 N=number of pieces r=length of each • Generalize: • eg. At each iteration split into 2 pieces but with different lengths r1 and r2 r1D+r2D=1

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