Phase plane plot second order ODE

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Elinor Ginzburg
Elinor Ginzburg 2024-4-24
评论： Elinor Ginzburg 2024-4-24
Hi,
I don't have enough experience with Matlab, so forgive me for the question, it's probably pretty basic. have an ode of the following form where are know constants. How can I plot the phase plane?
this is the code I've used with and
% Define the system of first-order ODEs
function dydt = second_order_ode(t, y)
% y(1) = x, y(2) = dx/dt
dydt = [y(2); -y(1) - 0.5*y(2)];
end
% Define time span
tspan = [0 20];
% Define initial conditions
y0 = [0.5; 0]; % initial position and velocity
% Solve the ODE system
[t, y] = ode45(@second_order_ode, tspan, y0);
% Plot the phase plane
figure;
plot(y(:,1), y(:,2));
xlabel('x');
ylabel('dx/dt');
title('Phase Plane Plot for x'' + x'' + 0.5x = 0');
grid on;
But I think there's something wrong with it because the plot generated is not what I expected (straight lines since )
Thank for your help

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Sam Chak
Sam Chak 2024-4-24
编辑：Sam Chak 2024-4-24
Ran in:
Looks correct! No issue because I checked with the built-in @odephas2 function to generate the 2-D phase plane plot.
Edit: Upon rechecking your original code, I found that there were no errors during execution. However, I discovered that the code within the second_order_ode() function was incorrect. Please refer to the comment within the code for the necessary corrections.
ODE:
State-space:
%% Define the system of first-order ODEs
function dydt = second_order_ode(t, y, param)
% parameters
alpha = param.alpha;
beta = param.beta;
gamma = param.gamma;
% ODEs
dydt = [y(2);
gamma - alpha*y(2) - beta*y(1)]; % <-- Corrections in this line
end
%% set parameters
param.alpha = 1;
param.beta = 0.5;
param.gamma = 0;
%% Define time span
tspan = [0 20];
% Define initial conditions
y0 = [0.5; 0]; % initial position and velocity
%% Solve the ODE system
opts = odeset('OutputFcn',@odephas2, 'Stats', 'on');
[t, y] = ode45(@(t, y) second_order_ode(t, y, param), tspan, y0, opts);
22 successful steps 0 failed attempts 133 function evaluations
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Sam Chak
Sam Chak 2024-4-24
Ran in:
Hi @Elinor Ginzburg
The short answer is because the solution is a decaying sinusoidal function.
A phase plane plot, also known as a phase portrait or phase diagram, is a graphical representation used in the field of dynamical systems to visualize the behavior of a system of differential equations. It provides insight into the qualitative behavior and stability of the system.
In a phase plane plot, the state variables of the system are typically represented on the x and y axes. Each point in the plot corresponds to a specific combination of values for the state variables at a given time. By plotting many points over time, the evolution of the system can be observed.
By the way, I have edited the code in my Answer above as the dynamics was incorrect in the second_order_ode() function.
syms y(t)
alpha = 1;
beta = 0.5;
gamma = 0;
dy = diff(y,t);
ddy = diff(y,t,2);
eqn = ddy + alpha*dy + beta*y == gamma;
cond = [y(0)==0.5, dy(0)==0];
ySol(t) = dsolve(eqn, cond)
ySol(t) = 
dySol(t) = diff(ySol)
dySol(t) = 
Elinor Ginzburg
Elinor Ginzburg 2024-4-24
I think I got it. Thank you for your help! I appreciate it very much.

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