% plot attractive sets of complex mappings %define viewing window xmin = -2; xmax = 2; ymin = -2; ymax = 2; %define mapping c = -.12256117 + .74486177*sqrt(-1); map = inline( 'Z.^2 +c','Z','c'); %remember array ops .*, .^ , ./ %number of iterations of mapping N = 40; %%%%%%%%%%%%%%%%%%%%%%%% dx = (xmax-xmin)/400; dy x = [xmin:dx:xmax]; y = [X,Y] = meshgrid(x,y); iterate, graph = (ymax-ymin)/400; [ymin:dy:ymax]; Z = X + Y*sqrt(-1); %iterate map for j = 1:N, Z = map(Z,c); Z = (abs(Z) >1e20).*1e20 + (abs(Z)<1e20).*Z; big end; %plot colormap bone; pcolor( X,Y,abs(Z) ); shading interp; %array of complex values %prevent Z from getting too Project: Complex mappings and basins of attraction This project uses the MATLAB code complexmap.m , which displays the basin of attraction (i.e. the set of all initial conditions which lead to a bounded trajectory, see text) for a particular complex mapping z_(n+1) = f(z_n). Like strange attractors, their geometry is complicated, and the boundary of the basin of attraction is typically a fractal. In the first few lines of the MATLAB code, the mapping and parameters are specified along with the viewing window specified by (xmin,xmax,ymin,ymax). The code iterates the mapping N times, and plots the initial point z_0 only if z_N is fairly small (``small" is determined by some prescribed tolerance). A. Consider the complex mapping f(z) = z^2 + c. Argue that if the modulus of z_0 is large enough, the modulus of z_n will always tend toward infinity as n gets large. This means not every point will be in the basin of attraction. (Hint: |z_n+1| = |z_n^2 + c| > |z_n|^2 - |c| = (|z_n| + |c| ^(1/2))(|z_n| - |c|^(1/2)) > 2 |z_n| as long as |z_n|> 2 + |c| ^(1/2). Justify each inequality, and say what happens for large n). Considering the complex mapping f (z)=z 2 +c , as n gets larger, the value of summation that contributes to the value z c + c increases as well. As seen in the formula where 1 ( ) 1 ( ) z n c 2 >2z n as long asz n 2+ c 2 2 n 2 1 2 , each variable represents the value z n +1z + c|z n| c(|z n|+|c| ) of each iteration. When n increases, the component in each increase by a small factor after exceeding the 'threshold' value 2 since 1 ( ) z n 2+ c 2 . Hence, when the value increases, the iteration will eventually be iterated into the attractor where every point in the phase space for a stable linear system is in the basin of attraction. For part B, use c = -.12256117 + .74486177 i. B1. Run the code for the complex mapping f(z) = z^2 +c. You should (after a few seconds) see the famous "Douady's Rabbit". Plot this. B2. Change (xmin,xmax,ymin,ymax) so that you are looking at a region of radius ~0.1 near the point (-.5, .3). You should see quasi-self similarity. Plot this as well. % Douady's Rabbit % Quasi-self-similarity C. There appears to be symmetry in the basins of attraction. Prove the following: if z_0 is in the basin of attraction (so {z_n} stays bounded) so is -z_0. (hint: do these two initial points create the same eventual sequence?) As the value of z 0 increases, the value of z n approaches a bound . Since the graph is proven to have quasi-self-similarity z 0 approaches the same bound as z 0 . For Part D, set the constant c=i. D1. Make the viewing window large enough to see the whole basin of attraction, which looks like lightning. With N=40, you'll probably see nothing. This is because the basin of attraction is very "thin", and almost every initial condition will explode. Plot using N = 20 to capture initial conditions that are close enough to the basin of attraction that their iterates don't get too large. D2. Determine (using, say, the MATLAB command line) the first several iterates starting at z=0. Is z=0 in the basin of attraction? Why? D3. Zoom in on zero by changing (xmin,xmax,ymin,ymax) and adjust N so that you capture a basin structure that looks self-similar. % zooming in by zero attempting to capture the basin structure Math 454 Home Other courses