Homework Answers
In MATLAB:
function yp = programa (t,y)
y1=15;
y2=15;
y3=36
yp1=10*(y2-y1);
yp2=28*y1-y2-y1*y3;
yp3=y1*y2-(8/3)*y3;
yp=[yp1;yp2;yp3];
create new script with the following:
[t,y]= ode45(@programa, [0,100], [1,4]); #(change the initial values)
plot (t,y (:,1), t,y(:,2)); #(change for (y1,y3) and (y2,y3))
grid on
title (exercise);
xlabel(´t´);
xlabel(´y´);
Add Answer to:
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t0 = 0; tf = 20; y0 = [10;60]; a = .8; b = .01; c...
t0 = 0; tf = 20; y0 = [10;60];
a = .8; b = .01; c = .6; d = .1;
[t,y] = ode45(@f,[t0,tf],y0,[],a,b,c,d);
u1 = y(:,1); u2 = y(:,2); % y in output has 2 columns
corresponding to u1 and u2
figure(1);
subplot(2,1,1); plot(t,u1,'b-+'); ylabel('u1'); subplot(2,1,2);
plot(t,u2,'ro-'); ylabel('u2');
figure(2) plot(u1,u2); axis square; xlabel('u_1');
ylabel('u_2'); % plot the phase plot
%----------------------------------------------------------------------
function dydt = f(t,y,a,b,c,d)
u1 = y(1); u2 = y(2);
dydt = [ a*u1-b*u1*u2 ; -c*u2+d*u1*u2 ];
end
Only...
Q5 (30 pts). A well-studied first order system of ODEs is given by which are called the Lorenz equations, and they were derived by the Meteorologist Edward Lorenz in the early 1960's as a simple model of weather (more precisely as a model of a pair of coupled convection cells in the atmosphere). In the model o, β, ρ > 0 are positive real parameters. Fix the classical parameter values to be σ = 10, β-8/3 and ρ 28. Fix...
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t0 = 0; tf = 20; y0 = [10;60];
a = .8; b = .01; c = .6; d = .1;
[t,y] = ode45(@f,[t0,tf],y0,[],a,b,c,d);
u1 = y(:,1); u2 = y(:,2); % y in output has 2 columns
corresponding to u1 and u2
figure(1);
subplot(2,1,1); plot(t,u1,'b-+'); ylabel('u1'); subplot(2,1,2);
plot(t,u2,'ro-'); ylabel('u2');
figure(2) plot(u1,u2); axis square; xlabel('u_1');
ylabel('u_2'); % plot the phase plot
%----------------------------------------------------------------------
function dydt = f(t,y,a,b,c,d)
u1 = y(1); u2 = y(2);
dydt = [ a*u1-b*u1*u2 ; -c*u2+d*u1*u2 ];
end
Only...
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