I am referring to this article (pp. 437) about FitzHugh-Nagumo.
Consider the ODE system $$ u'=v,~~~~~v'=-cv-f(u)+w,~~~~~w'=(-(\epsilon / c)(u-\gamma w), $$ where $f(u)=u(u-a)(1-u),~a<\frac{1}{2}$ and constants $\gamma, \epsilon$ are positive.
Let $S_{\epsilon}=(u_{\epsilon},v_{\epsilon},w_{\epsilon})$ be a solution.
Consider the variational equations $$ \delta u'=\delta v,~~~~~\delta v'=-c\delta v-f'(u_{\epsilon})\delta u+\delta w,~~~~~\delta w'=-(\epsilon / c)(\delta u-\gamma\delta w)~~~~~~~(*) $$
and the system linearised at the point $U_1=(u_1,v_1,w_1)$ with $\epsilon=0$: $$ \delta u'=\delta v,~~~~~\delta v'=-c\delta v-f'(u_1)\delta u + \delta w,~~~~~\delta w'=0.~~~~~~~(**) $$
Now, there is a part (p. 437 below) that I do not understand, I cite it:
Because they are linear, both $(*)$ and $(**)$ induce flows on $S^2$ by equating two vectors in $\mathbb{R}^3\setminus\left\{0\right\}$ if one is a positive multiple of the other. The flow of $(**)$ is qualitatively the same as the linearisation at rest. It has one unstable subspace and two stable ones. Let these be spans of the eigenvectors $X_1$ (unstable), $X_2$ and $X_3$.
(1.) Could you please explain me in which way $(*)$ and $(**)$ induce flows on $S^2$?
(2.) Why is the flow induced by $(**)$ qualitative the same as the linearisation at rest (what does that mean)?
(3.) Which subspaces/ eigenvectors are meant?
Would be thankful for any help. Especially understanding how the flow is meant would be essential, I think.