# Fixed point of non-linear system: infinite eigenvector

I've come across a $$2d$$ non-linear dynamical system (autonomous) the stability properties of which I would like to understand better. Computing the stability matrix, its eigenvalues and eigenvectors, I get finite eigenvalues at a fixed point which interests me, however, the eigenvector for one of the eigenvalues seems to blow up in one of it's two entries. I took the limits for the fixed point from different directions and the problem remains as it is. This seems to be related to the fact that one of the variables factorizes in the second equation. From the perspective of theory, what is known about such eigenvectors?

The system is given by $$(x(t),y(t))$$ with

\begin{align} \dot{x}&=-(128 x^3 + 8 x^2 (-75 + 16 y) + 8 (-15 + 8 y) y^2 + x (15 + 8 y (-45 + 8 y))),\\ \dot{y}&=y (-15 + 360 y - 8 x (-105 + 32 x + 32 y)). \end{align}

I refer to the fixed point at about $$(0.025,0)$$.

Any advice would be greatly appreciated.

• How can components of an eigenvector blow up? You just care about the direction of it surely. Are you artificially fixing that eg the first component is $1$? – jacob1729 Apr 19 at 8:41
• Well, computing the eigenvectors gives in one entry a $1/y$ which does not cancel with what is written in the numerator. For the given fixed point it will blow up since $y^{\ast}=0$. – Hamurabi Apr 19 at 10:10
• It would help if you posted the eigenvectors you are getting. However, assuming you have something like $v=(1/y,a)$ for constant $a$ then notice that if you normalise this to the unit vector $\hat{v}=(1/y,a)/(y^{-2}+a^2)^{-1/2}$ that it is well behaved as $y\to 0$. – jacob1729 Apr 19 at 10:39
• Thanks @jacob1729. The mathematica output is pretty lengthy I must say. Also when running a Normalize[] over it, the problem persits. – Hamurabi Apr 19 at 12:43
• It shouldn't persist. Mathematica Normalize is probably not doing what you think it is. Normalize[matrix] I think just devides by the operator norm, not normalize each column. – jacob1729 Apr 19 at 13:23