Question

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 the initial condition to be y(0) (1,1,1)T E R3. Using power series, fix a step size h 01 and Taylor order N 20 to compute 5000 steps of integration. Plot the resulting solution in space, that is plot the resulting orbit [(V1(t),v2(t),()): t [0,50 CR3.

Answer 1

%Matlab code for Lorenz attractor clear all close all %all parameters for Lorenz attractor sigma= 10 ; roh-28; beta-8/3; %function for differential equation f1-4 (t,y1,y2,y3) sigma (y2-yl); f2-e ( t , y 1 , y2 , y 3 ) yl*(ron-y 3 )-y2; f3-@ ( t , y 1 , y2 , у 3 ) ylty2-betar y3; step size h-o.01; t end-50 %initial guess y1 0-1; y2_0-1 y3_0-1; %A11 initial guess yl euler (1)-yl_0; y2_euler (1)-y2_0: y3_euler (1)-y3_0; t_euler(1)-0; n-t_end/h; %iteration for y1, y2 and y3 using Euler method for i-l:n t euler (i+1)-t euler (i)+h; +1)-yl_euler (i)+h*fl (t_euler (i),yl_euler(i),y2_euler (i),y3_euler (i)); +1)-y2_ euler (i)+h*f2 (t_euler (i),yl_euler(i),y2_euler(i),y3_euler (i)); +1)-y3_euler (i)+h*f3 (t_euler (i),yl_euler(i),y2_euler(i),y3_euler (i)); end y1_euler(i y2_euler (i y3-euler ( i %plotting the resulting orbit plot3 (yl_euler,y2_euler, y3 euler) xlabel 'yl' ylabel('y2") zlabel(y3') title 'Plot for Lorenz Attractor') 8888868888888888888888 End of Code 웅88888888888888888888움

60 Plot for Lorenz Attractor 50 40 20 10 0 20 0 y2 2020-15-10 -505 10 15 20 25 y 1

%Matlab code for Lorenz attractor
clear all
close all
%all parameters for Lorenz attractor
sigma=10; roh=28; beta=8/3;
%function for differential equation
f1=@(t,y1,y2,y3) sigma*(y2-y1);
f2=@(t,y1,y2,y3) y1*(roh-y3)-y2;
f3=@(t,y1,y2,y3) y1*y2-beta*y3;

%step size
h=0.01;
t_end=50;
%Initial guess
y1_0=1; y2_0=1; y3_0=1;
%All initial guess
y1_euler(1)=y1_0; y2_euler(1)=y2_0; y3_euler(1)=y3_0;
t_euler(1)=0; n=t_end/h;
%iteration for y1, y2 and y3 using Euler method
for i=1:n
  
  
    t_euler(i+1)=t_euler(i)+h;
    y1_euler(i+1)=y1_euler(i)+h*f1(t_euler(i),y1_euler(i),y2_euler(i),y3_euler(i));
    y2_euler(i+1)=y2_euler(i)+h*f2(t_euler(i),y1_euler(i),y2_euler(i),y3_euler(i));
    y3_euler(i+1)=y3_euler(i)+h*f3(t_euler(i),y1_euler(i),y2_euler(i),y3_euler(i));
  
end

%plotting the resulting orbit
plot3(y1_euler,y2_euler,y3_euler)
xlabel('y1')
ylabel('y2')
zlabel('y3')

title('Plot for Lorenz Attractor')

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