This is maybe an obvious question but I wanted to make sure.
Say that $\gamma$ is a curve encircling a fixed point $p$ in a two-dimensional dynamical system. I am aware that the Poincare index $I(\gamma)$ is equal to $1$ if $p$ is:
- a source $(\lambda_1,\lambda_2>0)$,
- a sink $(\lambda_1,\lambda_2<0)$,
- a spiral $(\lambda_1,\lambda_2 \text{ are complex and } \Re(\lambda)\neq0)$,
- a centre $(\lambda_1,\lambda_2 \text{ are complex and } \Re(\lambda)=0)$
and equal to $-1$ if $p$ is a saddle.
But what about the case when $\lambda_1=\lambda_2$ and there is only one eigenvalue $\vec{v_1}$ (not two, which would make it either a source or sink)? Looking at the phase plot for that kind of system, I would expect it to be equal to $1$, since the direction of the vector field along any $\gamma$ we choose is always outward/inward much like the source/sink case.
But my textbook doesn't seem to cover this case (and I have been taught it as distinct from the other cases I have mentioned) which makes me think there is a bit more to it?