linear stability analysis of ODE

Hello there!

I have a question regarding linear stability analysis in MatLab.

given is the ODE

x' = x - x^3

of which I have calculated the fixed points x1 = 0 x2 = 1 x3 = -1

Now these points have to be checked for stability, both graphically and by means of linear stability analysis. I started doing that, by doing a linearization of the given differential equation and trying to set up a Jacobian.

But: since I've only got one variable and one equation, the Jacobian is reduced to a skalar, or am I seeing this wrong?

I'd need the Jacobian to get the eigenvalues and further the stability, so I can plot it in MatLab, but I don't see how I could do it with this equation.

So I would really be grateful for some help. Greetings, Tanja :-)

[Sagar Doshi]

You are correct about the Jacobian for single equation with one variable is a scalar. This can also be seen in the Wikipedia link for liner stability here .