How to find the equilibrium points of dynamics system?

Hello! I have this scalar dynamic system.

How to find the equilibrium points in order to check the stability with two methods Lyapunov?

Thanks in advance!

What i do? I solve in matlab this f(x) = 0 and find x=1 x=2 and x=2

Answer 1

[Sam Chak]

The system has two equilibrium points ( and ). It is possible to evaluate the system's stability through a graphical method. From the vector field, we can observe how the system's state will evolve near these equilibrium points.

[T, X] = meshgrid(-0:5/15:5, 0:3/21:3);
S      = - (X - 1).*(X - 2).^2;
L      = sqrt(1 + S.^2);
U      = 1./L;
V      = S./L;
quiver(T, X, U, V, 0.5)
axis tight
xlabel('t'), ylabel('x(t)')

For , it takes an eternity to converge to the equilibrium. For , the trajectory converges in finite time.

fcn = @(t, x) - (x - 1).*(x - 2).^2;
[t, x] = ode45(@(t, x) fcn(t, x), [0 2000], 2.01);
figure(2)
plot(t, x), grid on, xlabel('t'), ylabel('x(t)')

[t, x] = ode45(@(t, x) fcn(t, x), [0 200], 1.99);
figure(3)
plot(t, x), grid on, xlabel('t'), ylabel('x(t)')