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Phase Plot and Phase Portrait

 Thu 23 July 2020 by Pranabendra Prasad Chandra

Phase Plot and Phase Portrait

Using Simulink® and MATLAB®

Contents

  • ODE
  • Simulink Model
  • Defining the state equation
  • Phase plane
  • Calculating gradient and plotting the vector field
  • Check for different initial conditions

ODE

imshow('VDP.png')

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Simulink Model

close
% Assuming initial condition = (1,0)
open_system('van_der_pol_model')
fprintf('\n')
plot(simout_x1.data, simout_x2.data)
xlim([-3 3])
xlabel('x_1')
ylabel('x_2')

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Defining the state equation

f = @(t,X) [X(2); X(2)*(1 - X(1)^2) - X(1)];

Phase plane

x1 = linspace(-5,5,20);
x2 = linspace(-5,5,20);
[x,y] = meshgrid(x1,x2);

Calculating gradient and plotting the vector field

u = zeros(size(x));
v = zeros(size(x));
t=0;

for i = 1:numel(x)
    Xdot = f(t,[x(i); y(i)]);
    u(i) = Xdot(1);
    v(i) = Xdot(2);
end

quiver(x,y,u,v,'r'); figure(gcf)
xlabel('x_1')
ylabel('x_2')
axis tight equal;
hold on;

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Check for different initial conditions

for y20 = -4:2:4
    [ts,ys] = ode45(f,[0,20],[0;y20]);
    plot(ys(:,1),ys(:,2))
    plot(ys(1,1),ys(1,2),'x')
end

for y10 = -4:2:4
    [ts,ys] = ode45(f,[0,20],[y10;0]);
    plot(ys(:,1),ys(:,2))
    plot(ys(1,1),ys(1,2),'x')
end
hold off

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