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Unformatted text preview: 14 Fractals We now proceed to quantify the “strangeness” of strange attractors. There are two processes of interest, each associated with a measurable quantity: • sensitivity to initial conditions, quantified by Lyaponov exponents. • repetitive folding of attractors, quantified by the fractal dimension. Now we consider fractals, and defer Lyaponov exponents to the next lecture. We shall see that the fractal dimension can be associated with the effective number of degrees of freedom that are “excited” by the dynamics, e.g., • the number of independent variables; • the number of oscillatory modes; or • the number of peaks in the power spectrum 14.1 Definition Consider an attractor A formed by a set of points in a p-dimensional space: etc ε We contain each point within a (hyper)-cube of linear dimension π . Let N ( π ) = smallest number of cubes of size π needed to cover A . Then if N ( π ) = Cπ − D , as π ∗ , C = const. then D is called the fractal (or Hausdorf ) dimension. 132 Solve for D (in the limit π ∗ 0): ln N ( π ) − ln C D = . ln(1 /π ) Since ln C/ ln(1 /π ) ∗ as π ∗ 0, we obtain the formal definition ln N ( π ) D = lim . ν ln(1 /π ) 14.2 Examples Suppose A is a line segment of length L : L Then the “boxes” that cover A are just line segments of length π , and it is obvious that N ( π ) = Lπ − 1 = D = 1 . ∞ Next suppose A is a surface or area S . Then N ( π ) = Sπ − 2 = D = 2 . ∞ But we have yet to learn anything from D ....
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- Fall '06