Attached is an image of the problem I am dealing with. I am extremely lost on ho

Question

Attached is an image of the problem I am dealing with. I am extremely lost on how to write the coding in MATLAB to generate the required graphs for the following ODE's. Please provide a detailed coding for MATLAB!

A classical ODE system can be described by three odes. The equations are Here sigma, r, b are all positive parameters. Consider the scenario that sigma = 10, b = 8/3, r = 28. Try to solve this ode with the initial condition to be [0,1,0) with t to be long enough. Then Ploty vs t. Look at the figure and try to describe what you observed. Hint: do you see a pattern? Hint: think again, is it really a pattern? Maybe try to run it longer? Plot z vs x. Look at the figure and try to describe what you observed. Hint: Do you see a pattern? Hint: think again. Plot x vs y vs z in 3D. Describe what you see.

Explanation / Answer

%run this code by copying i a matlab file and for plotting in 3-d use plot tools

% dx/dt = 10*(y-x); dy/dt = 28x-y-xz; dz/dt = xy-(8/3)z;

%using forward difference euler method for numerical solving.i.e

%dx/dt=(x(i+1)-x(i))/dt;

%dy/dt=(y(i+1)-y(i))/dt;

%dz/dt=(z(i+1)-z(i))/dt;

clc

clear all

dt=0.01;

t(1)=0;

x(1)=0;

y(1)=1;

z(1)=0;

for i=1:1000

t(i+1)=t(i)+dt;

x(i+1)=x(i)+10*(y(i)-x(i))*dt;

y(i+1)=y(i)+(28*x(i)-y(i)-x(i)*z(i))*dt;

z(i+1)=z(i)+(x(i)*y(i)-(8/3)*z(i))*dt;

fprintf('x(%d)=%f -> y(%d)=%f ->z(%d)=%f ',i,x(i),i,y(i),i,z(i));

end;

p=plot(x,t);

hold all;

q=plot(y,t);

hold all;

o=plot(z,t);

hold all;

title('plots in 2-d');

xlabel('t-axis');

ylabel('Function Value of x,y,z');

set(p,'Color','red','LineWidth',1)

set(q,'Color','green','LineWidth',1)

set(o,'Color','blue','LineWidth',1)

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