Fractals from the Lorenz Equations My Solutions I Determine the fractal that arises from using Newton's method to comput
Posted: Mon Mar 21, 2022 4:39 pm
- Fractals From The Lorenz Equations My Solutions I Determine The Fractal That Arises From Using Newton S Method To Comput 1 (44 KiB) Viewed 82 times
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Fractals from the Lorenz Equations My Solutions I Determine the fractal that arises from using Newton's method to compute the fixed-point solutions of the Lorenz equations. Use the parameter values r = 28, o= 10 and B = 8/3. Initial values (xo, zo) are taken on a grid in the r-z plane with always yo = 3 V2. For assessment purposes, the computational grid and the graphics code will be given in the Learner Template. To pass the assessment, every pixel in your figure needs to be colored correctly. (Hint: Some grid points may require as many as 33 Newton iterations to converge while others may require as few as three. Unfortunately, if you uniformly use 33 Newton iterations at every grid point, the MATLAB Grader may time out. You can accelerate your code by using a while loop instead of a for loop.)
Script E 1 r 28; sigma 18; beta-8/3; 2 x1=0; y1@; 21-0; 3 X2-sqrt(beta"(r-1)); y2 sqrt(beta*(n-1)); 22-r-1; 4 X-sqrt(beta" (n-1)); y3u-sqrt(beta(r-1)); 23-r-1; 5 nx=500; nz 500; 6 xmin.40; xmax 40; zmin=-40; max-40; 7 X_grid-linspace(xmin, xmax, nx); z grid-linspace(zmin, zmax, nz); 8 [X,Z]-meshgrid(x_grid, z_grid); 9 10 % Write Newton's method using every grid point as the initial condition 11 % Perform enough iterations that every initial condition converges to a root 12 % Save the x-values of the converged roots in the matrix x 13 % to pass the assessment, every pixel in the figure must be correctly colored 14 15 %!!!!!!!!! Set initial value y=3*sqrt(2) for all values (x,z) on the grid 16 17 18 19 20 21 22 eps-1.e-23; 23 X1 = abs(x-x1) < eps; X2= abs(X-X2) < eps; X3 = abs(X-X3) < eps 24 X4 = (x1+X2+X3); 25 figure; 26 map = [1 0 0; 0 1 0; 081; 8 0]; colormap(map); [red; green;blue;black] 27 X=(x1+2*X2+3*X3+4*X4); 28 image([xmin xmax], [zmin zmax), x); set(gca, voir', 'normal'); 29 xlabel('$X$', 'Interpreter', 'latex', 'Fontsize', 14); 30 ylabel('$2$', 'Interpreter', 'latex', 'Font size, 14); 31 title('Fractal from the Lorenz Equations', 'Interpreter', 'latex', 'FontSize', 16) 32