some fractals and fractal dimensionscsustan.csustan.edu/~tom/sfi-csss/lecture-notes... · •to...
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Some Fractalsand Fractal Dimensions
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• The Cantor set: we take a line segment, and remove the middle third. For each remaining piece, we again remove the middle third, and continue indefinitely.
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• To calculate the fractal / Hausdorff / capacity / box-counting dimension, we see how many boxes (circles) of diameter 1/r^n we need to cover the set (in this case, we will use r = 3, since it fits nicely).
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D = Lim(log(Nr)/log(1/r)) = log(2) / log(3)
r Nr
1 1
1/3 2
1/3^2 2^2
1/3^3 2^3
1/3^n 2^n
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• The Koch snowflake: We start with an equilateral triangle. We duplicate the middle third of each side, forming a smaller equilateral triangle. We repeat the process.
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• To calculate the fractal / Hausdorff / capacity / box-counting dimension, we again see how many boxes (circles) of diameter (again)1/3^n we need to cover the set.
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D = Lim(log(Nr)/log(1/r)) = log(4) / log(3)
r Nr
1 3
1/3 3 * 4
1/3^2 3 * 4^2
1/3^3 3 * 4^3
1/3^n 3 * 4^n
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• The Sierpinski carpet: We start with a square. We remove the middle square with side one third. For each of the remaining squares of side one third, remove the central square. We repeat the process.
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D = Lim(log(Nr)/log(1/r)) = log(8) / log(3)
r Nr
1 1
1/3 8
1/3^2 8^2
1/3^3 8^3
1/3^n 8^n
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• The Sierpinski gasket: we do a similar process with an equilateral triangle, removing a central triangle. (Note: we could also do a similar thing taking cubes out of a larger cube -- the Sierpinski sponge -- but it’s hard to draw :-)
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D = Lim(log(Nr)/log(1/r)) = log(3) / log(2)
r Nr
1 1
1/2 3
1/2^2 3^2
1/2^3 3^3
1/2^n 3^n
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• We can also remove other shapes.
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D = Lim(log(Nr)/log(1/r)) = log(4) / log(4) = 1
r Nr
1 1
1/4 4
1/4^2 4^2
1/4^3 4^3
1/4^n 4^n
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D = Lim(log(Nr)/log(1/r)) = log(3) / log(3) = 1
r Nr
1 1
1/3 3
1/3^2 3^2
1/3^3 3^3
1/3^n 3^n
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Relating this to nonlinear dynamical
systems
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Suppose we have a dissipative dynamical system (continuous) with positive Lyapunov exponent -- for
ease of viewing, let’s suppose it is 2-dimensional . . . so it will stretch along one direction and shrink along the other (locally) -- and let’s follow the state space at
successive times . . .
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Suppose we have a dissipative dynamical system (continuous) with positive Lyapunov exponent -- for
ease of viewing, let’s suppose it is 2-dimensional . . . so it will stretch along one direction and shrink along the other (locally) -- and let’s follow the state space at
successive times . . .
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Suppose we have a dissipative dynamical system (continuous) with positive Lyapunov exponent -- for
ease of viewing, let’s suppose it is 2-dimensional . . . so it will stretch along one direction and shrink along the other (locally) -- and let’s follow the state space at
successive times . . .
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Suppose we have a dissipative dynamical system (continuous) with positive Lyapunov exponent -- for
ease of viewing, let’s suppose it is 2-dimensional . . . so it will stretch along one direction and shrink along the other (locally) -- and let’s follow the state space at
successive times . . .
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Suppose we have a dissipative dynamical system (continuous) with positive Lyapunov exponent -- for
ease of viewing, let’s suppose it is 2-dimensional . . . so it will stretch along one direction and shrink along the other (locally) -- and let’s follow the state space at
successive times . . .
Etc.
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Now let’s take a Poincaré section (space slice) through the system
Etc.
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Now let’s take a Poincaré section (space slice) through the system
Etc.
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We are building a Cantor set (actually, a slight generalization, a Cantor dust . . .)!
Etc.
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This sort of behavior will be generic for the stretching and folding of the state/phase space for (hyperbolic)
dissipative systems . . .
Etc.
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So, we should expect to see Cantor dusts in many Poincaré sections of these sorts of systems . . .
Etc.
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This also suggests that the fractal dimension of an attractor, or of a Poincaré section of the attractor, can
give us the possibility of a characteristic number, to identify the attractor, or at least to distinguish between
attractors, and thus between dynamical systems . . .
Etc.
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Fin