3D Phase portrait for a set of differential equations

I have a set of three differential equations and I want to make a phase portrait of them. I have some idea of using quiver or plot3 to get a phase portrait of a set of 3 differential equations. I am unable to do for this case.

Here is my attempt:

timerange= 0:0.5:350;
IC= [0.1,0,0];
IC1= [0.1,0.2,0];
[t,y] =ode45(@(t,y) fn(t,y),timerange, IC);
[t1,y1] =ode45(@(t,y) fn(t,y),timerange, IC1);

plot3(y(:,1),y(:,2),y(:,3)); 
hold on
plot3(y1(:,1),y1(:,2),y1(:,3)); 

grid 
xlabel('T')
ylabel('A')
zlabel('I')

function rk1 =fn(t,y)

r=0.00173;K=0.03166;A0=0.4;gammaA=0.04;eps = 0.00055;rho= 0.025;alpha1= 1.30187;c1=0.63433;

I0= 0.3;gammaI=0.0208;
alpha= 0.0002802;

n= y(1);
A= y(2);
I= y(3);
    

rk1(1)= r*n*(1- n/K)+ alpha*A*n -eps*n*I;
rk1(2) = A0 - gammaA*A;
rk1(3) = I0 + (rho*I*n)/(alpha1+n) -c1*I*n - gammaI*I;

rk1=rk1(:);
endI

What I am doing is merely changing the initial conditions The actual result should be something like this.

Any help would be greately helpful. This is what I am getting from above code.

Thank You

[darova]

Phase portrait it's a graph and y. HOw to plot 3D phase portrait? What those axes represent

Her is the link from wikipedia: LINK

[Vira Roy]

Thanks for reply darova. I was trying to do as Lorenz attractor done by James adams in this code. I have three differential equations and wanted to get a plot similar to above one i have shown.

What I wanted to ask is that, Can we get that by just choosing some initial conditions and plot the three solutions against each other. If not If have to get a phase portrait in 3 dimensions how should I proceed.

Those axes represent the three variables in the differential equation. I, n and A. Similar to x ,y ,z in Lorentz attractor.

Thank you anyways Darova.

Answer 1

[darova]

I tried to plot T A and I separately. X Y Z axis represents T A and I respectively. Color represents derivative of T A or I

I used surface to plot color lines (plotted curves as surfaces, made them nx2 size)

derivative just difference

[Vira Roy]

Thanks for this Darova. I am accepting the answer as it was helpful. Sorry for being naive but can you please explain a bit.

1.The graph I get has different colour scheme than shown in your diagram. Is that system specific or am i missing something.

2. The commented part of the code, what does this represent. I removed the comment and ran but found no difference in the plot.

dy = diff(y);%./repmat(diff(t),[1 3]);

3. I can only see one graph, unlike the two you have pictured above, dy/dt and dy. Do i have make some changes in the code to get that.

I am sorry if i am demanding too much. even a link to documentation will be fine . Either way I am accepting the answer.

edits made as I found some of the answers i asked.

[darova]

• The graph I get has different colour scheme than shown in your diagram. Is that system specific or am i missing something.

It's because im from Poland. Poles like more red colors

colormap jet        % try to change colormap

• The commented part of the code, what does this represent. I removed the comment and ran but found no difference in the plot.

Impossible! Maybe color axis (range) didn't after you uncommented. Try to add clf at the beginning of the script

clf

• I can only see one graph, unlike the two you have pictured above, dy/dt and dy. Do i have make some changes in the code to get that.

I just run the script, saved the picture. Then commented line and run again (save second picture)

• I am accepting the answer as it was helpful.

But i don't see any

[Vira Roy]

The colors were somehow the same but the values are different. Saw that after adding the color bar.

Thanks for your help.