# Phase plane for a system of differential equations.

I have a system of 5 ODEs. A phase plane was drawn for two of the variables, to see how they interact together. This is the phase plane that I got.

How can I interpret this phase plane? If I have a system of 2 ODEs then I can say that this equilibrium point is asymptomatically stable. But here as there are 5 ODEs and I am only considering two of the variables can I still say that this is asymptomatically stable .

How to analyse systems with more than 2 ODEs?

In this can I say this spirals in, as I know it start with (1000,0) and when I use ode45 in Matlab and solve the system the equilibrium point found was in point A.

Code for direction fields

 %trajectory code
[tsol,usol]=ode15s(@rhs,[0 time],[1000,0,0, 10^8,0]);
A=usol(:,1)+usol(:,2);
B=usol(:,5);
figure
plot(A,B);
hold on;
%direction fields
Ngrid=100;
y1=linspace(1000,10^7,Ngrid);
y2=linspace(0,10^8,Ngrid);
[x,y]=meshgrid(y1,y2);

t=0;
for i=1:Ngrid
for j=1:Ngrid
Yprime=rhs(t,[x(i),0,0,10^8,y(j)]);
Yprime=Yprime/norm(Yprime);
u(i,j)=Yprime(2)+Yprime(1);
v(i,j)=Yprime(5);
sfactor=0.6;
end
end
quiver(x,y,u,v,sfactor,'r')


When I try to draw the direction fields it changes my trajectory plot as well. I used the code as in Plotting phase plane in Matlab for SIR model . I think the problem is the two variables I draw the phase plane of reaches large values such as $10^6$ and $10^7$. How can I adjust the code to draw the direction fields correctly? In the above code, why is t=0 chosen specifically? Is it because we should know the initial conditions? In the line Yprime=rhs(t,[x(i),0,0,10^8,y(j)]); I should give x,y coordnates to the variables I draw the phase plane and for the other variables the initial conditions right? rhs is the function with my 5 ODEs

The plot I get when I draw the direction fields is

• First, there should be a direction associated with that curve. Since you say that if those were the only variables this would be "asymptotically stable", I presume the direction is into the spiral. This tells you exactly that- that the evolution of these two variables, independent of the others, is asymptotically stable. Since you are ignoring the other three variables, this cannot tell you anything about them! – user247327 Jul 5 '17 at 13:23
• To get an understanding of a system with more than 2 degrees of freedom, you can always draw phase plots for all ${5 \choose 2} = 15$ "coordinate planes" in your system (rather than just the one you have) and examine the trajectories there. – Michael Seifert Jul 5 '17 at 13:44
• @user247327 can't I say the direction by looking at the initial point and the point at the equilibrium. I know it start with (1000,0) and when I use ode45 in Matlab and solve the system the equilibrium point found was in point A. So, shouldn't the direction be into the spiral? – sam_rox Jul 5 '17 at 13:48
• Your direction field has little sense unless there is some invariant 2d plane in the phase space. But you don't show the equations, so it's hard to say whether it makes sense or no. – Evgeny Jul 6 '17 at 9:00

• I tried to draw the direction fields and I have posted the code for it. But then it changes my trajectory plot as well. I used the code as in math.stackexchange.com/questions/2208608/… . I think the problem is the two variables I draw the phase plane of reaches large values such as $10^6$ and $10^7$. Would you please be able to tell what I should change in the code to get the direction fields correctly – sam_rox Jul 6 '17 at 0:42