# Fixed points of Van Der Pol equation

1. Nov 26, 2011

### Koeneuze

1. The problem statement, all variables and given/known data
"Discuss the fixed points of the Van Der Pol equation depending on the perturbation parameter µ when µ >= 0"

2. Relevant equations
$$x'' - \mu(1-x^2)x' + x = 0$$ is the Van Der Pol Equation.

3. The attempt at a solution
Well, this question is really easy for most values of µ. The only fixed point is (0,0), and after linearising you get 4 situations:
1) µ = 0
This gives two complex eigenvalues that don't have a real part, so we either have a focus/center, or a spiral point. Graphing clearly shows it's a center.
2) 0 < µ < 2
Spiral point.
3) µ > 2
Unstable node.
4) µ = 2. Now, this is one of the situations where linearisation can't tell much, except that you either have a node or a spiral point. I haven't been able to find a general solution for this, everything I've found only relates to the linear system, and even those aren't too clear.

So, my questions are:
1. Am i correct that, for the linearised system, this is an unstable improper node?
2. Is there any way to derive what kind of fixed point the original (non-linearised) system has? Or is there a way to deduct that from a graph?

I've searched around a bit, and I seem to find both unstable star/proper node and unstable improper node for the fixed point of the original system... Star seems to be the one with the more trustworthy sources, but at the same time, how do you define a star node for a non-linear system? For linear it just meant that all solutions were straight lines...