How to draw a polar coordinate ODE system's phase portrait and trajectory?

For a linear system dotx=Ax ,with complex eigenvalues.

If it's written in polar coordinate,then it will be : dot r= \alpha *r ; dot \theta = \beta,

The initial condition will be like : r(t)=r_0 * exp ( \alpha * t) ; \theta(t)=\theta_0+\beta*t

How to draw it's phase portrait and trajectory? For there are 4 unknown segments in the system.

I tried to use quiver function to solve, but not successful.

Wish to receive a proper code and explainnation about it. Thanks everyone

Answer 1

[Yash Sharma]

Hi Zixuan,

I understand that you want to plot phase portrait and trajectory of a linear system of equations with complex eigen values

Here is an example code on how you can achieve the same.

% Define the parameters of the system 
alpha = 0.5;   % Parameter for r dynamics 
beta = 0.1;    % Parameter for theta dynamics 
% Define the time span for simulation 
tspan = 0:0.1:10; 
 
% Define the initial conditions 
r0 = 1;       % Initial r value 
theta0 = 0;   % Initial theta value 
 
% Define the ODE system in polar coordinates 
odefun = @(t, y) [alpha * y(1); beta]; 
 
% Solve the ODE system 
[t, y] = ode45(odefun, tspan, [r0; theta0]); 
 
% Extract the r and theta values 
r = y(:, 1); 
theta = y(:, 2); 
 
% Convert theta to radians 
theta_rad = deg2rad(theta); 
 
% Compute the x and y coordinates 
x = r .* cos(theta_rad); 
y = r .* sin(theta_rad); 
 
% Plot the phase portrait 
figure; 
quiver(x, y, cos(theta_rad), sin(theta_rad)); 
xlabel('x'); 
ylabel('y'); 
title('Phase Portrait'); 
axis equal; 
 
% Plot the trajectory 
figure; 
plot(x, y); 
xlabel('x'); 
ylabel('y'); 
title('Trajectory'); 
axis equal;

You can adjust the parameters and initial conditions according to your specific system, and further customize the plots as needed.

Please find links to below documentation which I believe will help you for further reference.

• ode45: https://www.mathworks.com/help/matlab/ref/ode45.html
• deg2rad: https://www.mathworks.com/help/matlab/ref/deg2rad.html
• quiver: https://www.mathworks.com/help/matlab/ref/quiver.html

[Sam Chak]

@Yash Sharma, are you sure that you understand that Zixuan wants to plot the phase portrait and trajectory, similar to your example? If not, please fix the mathematics in your example.

% Define the parameters of the system 
alpha = 0.5;   % Parameter for r dynamics 
beta = 0.1;    % Parameter for theta dynamics 
% Define the time span for simulation 
tspan = 0:0.1:10; 
 
% Define the initial conditions 
r0 = 1;       % Initial r value 
theta0 = 0;   % Initial theta value 
 
% Define the ODE system in polar coordinates 
odefun = @(t, y) [alpha * y(1); beta]; 
 
% Solve the ODE system 
[t, y] = ode45(odefun, tspan, [r0; theta0]); 
 
% Extract the r and theta values 
r = y(:, 1); 
theta = y(:, 2); 
 
% Convert theta to radians 
theta_rad = deg2rad(theta); 
 
% Compute the x and y coordinates 
x = r .* cos(theta_rad); 
y = r .* sin(theta_rad); 
 
% Plot the phase portrait 
figure; 
quiver(x, y, cos(theta_rad), sin(theta_rad)); 
xlabel('x'); 
ylabel('y'); 
title('Phase Portrait'); 
axis equal; 

 
% Plot the trajectory 
figure; 
plot(x, y); 
xlabel('x'); 
ylabel('y'); 
title('Trajectory'); 
axis equal;