Consider the linear 3-dim. ODE system $$ \dot{\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}}=\underbrace{\begin{pmatrix}0 & 1 & 0\\-f'(u_1) & -c & 1\\0 & 0 & 0\end{pmatrix}}_{=:A}\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}, $$ where $f'(u_1)<0, c>0$ and $U_1=(u_1,v_1,w_1)$ is some point we linearized at.
Then, $A$ has the following three eigenvalues:
$$ \lambda_1=-\frac{c}{2}+\sqrt{\frac{c^2}{4}-f'(u_1)},~~\lambda_2=0,~~~\lambda_3=-\frac{c}{2}-\sqrt{\frac{c^2}{4}-f'(u_1)}. $$
$\lambda_1$ is positive, $\lambda_3$ is negative.
Hence, in terms of stability theory, the ODE has two stable subspaces (namely the stable subset that is spanned by the eigenvector $X_3$ belonging to $\lambda_3$ and the centre subspace which is spanned by the eigenvector $X_2$ belonging to $\lambda_2$) and one unstable subspace (spanned by the eigenvector $X_1$ belonging to $\lambda_1$).
When I now project the ODE onto the sphere $S^2$, I get a new ODE which acts on the sphere (more precisely: a new ODE on the projective space by identifying opposite points on the sphere). And it follows that the unstable subspace of the original ODE becomes two attracting points for the ODE on the sphere (the red point in the picture below and its opposite point) and the stable subspace of the original ODE becomes two repelling points (the blue point in the picture and its opposite point) for the ODE on the sphere.
The green point is a saddle point (and also its opposite point) which arises from the centre subspace of the original ODE.
My questions now are the following ones:
(1.) Where on the picture do we see $\text{span}\left\{X_1\right\}\cap S^2$? I guess this is nothing else but the red point (and its opposite point).
(2.) Where on the picture do we see $C:=\text{span}\left\{X_2,X_3\right\}\cap S^2$?
(3.) Somebody told me that "inside $C$" we have that $\text{span}\left\{X_1\right\}\cap S^2$ is attracting. But, frankly speaking, I have no idea what may be meant by inside $C$. What may be meant by inside $C$? Where is inside $C$ on the picture?
Maybe you can help me. I would be very thankful for any help. Sorry for too much text but I had to describe my problem a bit more detailed.
Kind regards!
