Use Matlab make a bifurcation diagram like Figure 2.16. The code uses parameter b=-0.3 and a ranging from 0 to 2 by increments of da = 0.001. For each value of the parameter a, the code chooses the initial point (0, 2) and calculates its 1 orbit under the Henon map to step 104. It then plots the last N iterates (x-coordinates of the orbit) where N is small enough to allow the orbit to settle down to the attracting set (and not crush the graphics card in your computer). 2.5 -2.50 1.25 Figure 2.16 Bifurcation diagram for the Hnon map, b = 0.4. Each vertical slice shows the projection onto the x-axis of an attractor for the map for a fixed value of the parameter a. 1 (a) Is the picture different if the y-coordinate is plotted instead? (b) Make an additional diagram varying the parameter a from 1.925 to 1.975 by increments of da = 0.0001. Are there any periodic windows? For which values of a? (c) Generate an estimate of Feigenbaum's constant for the period-doubling cascade using a sequence of bifurcations up to period-64. You may need to make da smaller in order to resolve when a bifurcation takes place. Why doesn't your estimate approach 4.669...? %% define parameters amin = 0; amax = 2; b = = -0.3; %N = 10^3; N = 10^4; da = 10^-3; IC = [0;2]; x = zeros (N,1); y = zeros (N, 1); | %% make empty figure figure; %% iterate over parameter values a %for a = linspace(amin, amax, 25) for a = amin:da: amax disp(a) x(1) = IC(1); y(1) = IC(2); %% iterate the map for this a value for i = 1:N x(i+1) = a-x(i)^2+b*y(i); y(i+1) = x(i); end S; %% determine if there is a periodic orbit of period less than 2 %% we do this to avoid plotting thousands of points on the same S = 10; % length of longest per p = % initialize length of the orbit we will plot tol = 10^(-15); % tolerance within whic inside=0; % trigger telling me if for i = 0 :S % loop looking for orbi difference(i+1) = norm(x(end)-x(end-2^i)); % compu if (difference(i+1)