Fractals Some and Fractal Dimensions
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.
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).
D = Lim(log(Nr)/log(1/r)) = log(2) / log(3)
r 1 1/3 1/3^2 1/3^3 1/3^n Nr 1 2 2^2 2^3 2^n
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.
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.
D = Lim(log(Nr)/log(1/r)) = log(4) / log(3)
r 1 1/3 1/3^2 1/3^3 1/3^n Nr 3 3*4 3 * 4^2 3 * 4^3 3 * 4^n
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.
D = Lim(log(Nr)/log(1/r)) = log(8) / log(3)
r 1 1/3 1/3^2 1/3^3 1/3^n Nr 1 8 8^2 8^3 8^n
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 :-)
D = Lim(log(Nr)/log(1/r)) = log(3) / log(2)
r 1 1/2 1/2^2 1/2^3 1/2^n Nr 1 3 3^2 3^3 3^n
We can also remove other shapes.
D = Lim(log(Nr)/log(1/r)) = log(4) / log(4) = 1
r 1 1/4 1/4^2 1/4^3 1/4^n Nr 1 4 4^2 4^3 4^n
D = Lim(log(Nr)/log(1/r)) = log(3) / log(3) = 1
r 1 1/3 1/3^2 1/3^3 1/3^n Nr 1 3 3^2 3^3 3^n
Relating this to nonlinear dynamical systems
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 . . .
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 . . .
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 . . .
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 . . .
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.
Now let's take a Poincar section (space slice) through the system
Etc.
Now let's take a Poincar section (space slice) through the system
Etc.
We are building a Cantor set (actually, a slight generalization, a Cantor dust . . .)!
Etc.
This sort of behavior will be generic for the stretching and folding of the state/phase space for (hyperbolic) dissipative systems . . .
Etc.
So, we should expect to see Cantor dusts in many Poincar sections of these sorts of systems . . .
Etc.
This also suggests that the fractal dimension of an attractor, or of a Poincar 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.
Fin