Direction field and phase potrait

Hello, I need to draw a direction field and sketch phase potrait for this differential equation:

dL/dA = (-0.5L+0.0001AL)/2A(1-0.0001A)-0.01AL

How would I do it?

thank you for helping!

Answer 1

[Sam Chak]

Hi @Tuân Nguyen

You can basically plot the direction field like this:

[A, L] = meshgrid(0.1:10/14:10.1, -5:10/14:5);

M = (- 0.5*L + 0.0001*A.*L)./(2*A.*(1 - 0.0001*A) - 0.01*A.*L);

N = sqrt(1 + M.^2);

U = 1./N;

V = M./N;

quiver(A, L, U, V, 0.5)

axis square

hold on

% differential equation

f = @(A, L) (- 0.5*L + 0.0001*A*L)/(2*A*(1 - 0.0001*A) - 0.01*A*L);

tspan = 0.1:0.01:10.1; % simulation time

init = 4; % initial condition L(0.1) = 4

[A, L] = ode45(f, tspan, init);

plot(A, L, 'r', 'linewidth', 1.5)

hold off

Result:

For more info, please visit the documentation:

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

[Tuân Nguyen]

hello, thank for your help but I made a big mistake, it should be

dL/dA = (-0.5L+0.0001AL)/(2A(1-0.0001A)-0.01AL)

not dL/dA = (-0.5L+0.0001AL)/2A(1-0.0001A)-0.01AL

It is still right if I changed the second line to the right DE?

[Sam Chak]

Alright, I've fixed the the ODE as requested.

Note that the simulation cannot be run from A = 0, else it will cause singularity issue (division-by-zero).

In this example, I run A from 0.1 to 10.1, and select the initial condition, L(0.1) = 4.

[Tuân Nguyen]

Understood!

Thank you for helping!

[Sam Chak]

Cảm ơn for your acceptance!