# Draw the vector field and eigenvectors in the phase portrait for Van der Pol ODE

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F.O on 30 Mar 2019
Answered: Agnish Dutta on 8 Apr 2019
I have the follwing system which represent the Van der Pol oscillator with the inital condition and parameters are given. I draw the phase porrait using plot and ode45 but dont know how to draw the vector field and the eigenvectors with direction on them.
%function to solve the system with the time dependent term zero
function [dxdt] = vdp1(t,x,lambda,gamma,omega)
dxdt=zeros(2,1);
dxdt(1)=x(2);
dxdt(2)=lambda.*(1-x(1)^2)*x(2)-x(1)+gamma.*sin(omega*t);
end
%function to solve the system with the time dependent not zero
function [dxdt] = myode(t,x,gt,g,lambda,gamma,omega)
g=interp1(gt,g,t);
dxdt=zeros(2,1);
dxdt(1)=x(2);
dxdt(2)=lambda.*(1-x(1)^2)*x(2)-x(1)+g;
end
%script
lambda=[0.01 0.1 1 10 100] ;
gamma=[0 0.25];
omega=[0 1.04 1.1];
x0=[1 0];
x01=[3 0];
tspan=[0 500];
tspan1=[0 100];
%Numerical solution for the first initial value
[t,x]=ode45(@(t,x) vdp1(t,x,lambda(1),gamma(1),omega(1)),tspan,x0);
%Numerical solution for the second initial value
[t1,x1]=ode45(@(t,x) vdp1(t,x,lambda(1),gamma(1),omega(1)),tspan,x01);
%plotting x1,x2 aginst t
figure(3)
plot(x(:,1),x(:,2),'g-.')
hold on;
plot(x1(:,1),x1(:,2),'r-.')
xlabel('x1');
ylabel('x2')
legend('Solution first initial condition','Solution with the second initial condition')
title('phase portrait with t=[0 500] ,gamma=0,omega=0,lambda=0.01')
[tt1,xx1]=ode45(@(t,x) vdp1(t,x,lambda(2),gamma(1),omega(1)),tspan1,x0);
[tt2,xx2]=ode45(@(t,x) vdp1(t,x,lambda(3),gamma(1),omega(1)),tspan1,x0);
[tt3,xx3]=ode45(@(t,x) vdp1(t,x,lambda(4),gamma(1),omega(1)),tspan1,x0);
[tt4,xx4]=ode45(@(t,x) vdp1(t,x,lambda(5),gamma(1),omega(1)),tspan1,x0);
[tt11,xx11]=ode45(@(t,x) vdp1(t,x,lambda(2),gamma(1),omega(1)),tspan1,x01);
[tt22,xx22]=ode45(@(t,x) vdp1(t,x,lambda(3),gamma(1),omega(1)),tspan1,x01);
[tt33,xx33]=ode45(@(t,x) vdp1(t,x,lambda(4),gamma(1),omega(1)),tspan1,x01);
[tt44,xx44]=ode45(@(t,x) vdp1(t,x,lambda(5),gamma(1),omega(1)),tspan1,x01);
figure(6)
plot(xx1(:,1),xx1(:,2),'g-.')
hold on;
plot(xx11(:,1),xx11(:,2),'r-.')
xlabel('x1');
ylabel('x2')
legend('Solution first initial condition','Solution with the second initial condition')
title('phase portrait with t=[0 100] ,gamma=0,omega=0,lambda=0.1')
figure(8)
plot(xx2(:,1),xx2(:,2),'g-.')
hold on;
plot(xx22(:,1),xx22(:,2),'r-.')
xlabel('x1');
ylabel('x2')
legend('Solution first initial condition','Solution with the second initial condition')
title('phase portrait with t=[0 100] ,gamma=0,omega=0,lambda=1')
figure(10)
plot(xx3(:,1),xx3(:,2),'g-.')
hold on;
plot(xx33(:,1),xx33(:,2),'r-.')
xlabel('x1');
ylabel('x2')
legend('Solution first initial condition','Solution with the second initial condition')
title('phase portrait with t=[0 100] ,gamma=0,omega=0,lambda=10')
figure(12)
plot(xx4(:,1),xx4(:,2),'g-.')
hold on;
plot(xx44(:,1),xx44(:,2),'r-.')
xlabel('x1');
ylabel('x2')
legend('Solution first initial condition','Solution with the second initial condition')
title('phase portrait with t=[0 100] ,gamma=0,omega=0,lambda=100')
gt=[0 500];
g=gamma(2).*sin(omega(2).*gt);
g1=gamma(2).*sin(omega(3).*gt);
opts = odeset('RelTol',1e-2,'AbsTol',1e-4);
[t2,x2]=ode45(@(t,x) myode(t,x,gt,g,lambda(1),gamma(2),omega(2)),tspan,x0,opts);
[t22,x22]=ode45(@(t,x) myode(t,x,gt,g,lambda(1),gamma(2),omega(2)),tspan,x01,opts);
[t3,x3]=ode45(@(t,x) myode(t,x,gt,g,lambda(1),gamma(2),omega(3)),tspan,x0,opts);
[t33,x33]=ode45(@ (t,x) myode(t,x,gt,g1,lambda(1),gamma(2),omega(3)),tspan,x01,opts);
figure(14)
plot(x2(:,1),x2(:,2),'g-.')
hold on;
plot(x22(:,1),x22(:,2),'r-.')
xlabel('x1');
ylabel('x2')
legend('Solution first initial condition','Solution with the second initial condition')
title('phase portrait with t=[0 500] ,gamma=0.25,omega=1.04,lambda=0.01')
figure(16)
plot(x3(:,1),x3(:,2),'g-.')
hold on;
plot(x33(:,1),x33(:,2),'r-.')
xlabel('x1');
ylabel('x2')
legend('Solution first initial condition','Solution with the second initial condition')
title('phase portrait with t=[0 500] ,gamma=0.25,omega=1.1,lambda=0.01')
F.O on 2 Apr 2019
@Jan They restore my previous question ? which I replaced it by junk because no one would delet it.
Are you happy now?
Anyway I learned something here.

Agnish Dutta on 8 Apr 2019
If you can calculate the vector field values at every point, then the resulting data can be plotted using the “quiver” function, the details of which are in the following document:
I also found a few useful resources on the internet pertinent to what you are trying to do. Refer to the “computing the vector field” section of the following website:

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