% here's a sample of how I used matlab to make those figures. I % created this file by first typing "diary for_class". % this is one of the fixed points. I found it using maple. >> a = .1030367663 ; b = .01045779210 ; % dpdt refers to a file called "dpdt.m" which contains the ODE I'm % solving. In your home directory you need a directory called "matlab" % and in that directory you need the file dpdt.m If your deptartment % machine doesn't have matlab, come see me. We have it on the math dept % pcs and I'll show you how to use it there. % numerically solve the equation with initial data that is near to (a,b): >> [t,y] = ode45('dpdt',[0,2000],[a+.1;b]); % make a plot of x(t) versus t: >> subplot(3,1,1) >> plot(t,y(:,1)); >> title('perturbation of steady state (.1030367663, .01045779210)'); >> axis([0,50,-.5,.5]) >> axis([0,50,-.5,.5]) >> subplot(3,1,2) >> plot(t,y(:,1)); >> axis([0,300,-.5,.5]) >> subplot(3,1,3) >> plot(t,y(:,1)); % save the figure, to be printed with lpr or ghostview: >> print -dps fig1.ps % now I want to make a figure like figure 7.3 So first I go to dpdt.m % and change k to 0. >> figure(2) % figure(2) pops open a new window. To get to the original window, % type "figure(1)" >> a = -.6734177811; b = 0; >> [t,y] = ode45('dpdt',[0,500],[a+.1;b]); >> plot(y(:,1),y(:,2),'-'); >> hold on % hold on makes it so that I can have more than one orbit per % figure. To stop holding on, type "hold off", or "clf" to clear % the figure-screen altogether. >> [t,y] = ode45('dpdt',[0,500],[a+.2;b]); >> plot(y(:,1),y(:,2),'-'); >> [t,y] = ode45('dpdt',[0,500],[a+.3;b]); >> plot(y(:,1),y(:,2),'-'); % here I noticed that my orbit didn't meet itself so I ran again with % a longer time interval: >> [t,y] = ode45('dpdt',[0,700],[a+.3;b]); >> plot(y(:,1),y(:,2),'-'); >> [t,y] = ode45('dpdt',[0,700],[a+.4;b]); >> plot(y(:,1),y(:,2),'-'); >> [t,y] = ode45('dpdt',[0,700],[a+.5;b]); >> plot(y(:,1),y(:,2),'-'); >> [t,y] = ode45('dpdt',[0,700],[a+.6;b]); >> plot(y(:,1),y(:,2),'-'); >> [t,y] = ode45('dpdt',[0,700],[a+.7;b]); >> plot(y(:,1),y(:,2),'-'); >> [t,y] = ode45('dpdt',[0,700],[a+.05;b]); >> plot(y(:,1),y(:,2),'-'); >> [t,y] = ode45('dpdt',[0,700],[a+.005;b]); >> plot(y(:,1),y(:,2),'-'); % here I decided that I didn't like the curve I just plotted. So I % painted over it with itself in black. (My figure is on a black % background.) If my figure was on a white background then I'd do 'w-' % in the options. To play around, try options '-', '--', '-.', and ':'. % These all have color versions 'b-', 'r-', 'y-', 'w-', etc. >> plot(y(:,1),y(:,2),'k-'); >> [t,y] = ode45('dpdt',[0,700],[a+.02;b]); >> plot(y(:,1),y(:,2),'-'); >> [t,y] = ode45('dpdt',[0,300],[a+1.5;b]); >> plot(y(:,1),y(:,2),'-'); >> [t,y] = ode45('dpdt',[0,300],[a+1.4;b]); >> plot(y(:,1),y(:,2),'-'); >> axis([-1.2,1.2,-1.2,1.2]) >> print -dps fig2.ps >> diary off >> quit 4691068 flops. % note that the above was a little sloppy --- I was always perturbing % off of the left-most fixed point. I really should have perturbed off % of all fixed points and I should have found the eigendirections for % any saddle points so that I could put initial data on those % eigendirections to try and capture the unstable and stable manifolds.