First of all, this is my fourth question about dynamical systems in a week, sorry for that.
Considering a linear bidimensional dynamical (autonomous) system, the orbits can be plotted in the phase state as so:
Let me specify that I already managed to solve the dynamical equations in every case (regarding eigenvalues), and plot similar orbits.
My question is on the third image: there, there is one double (positive) eigenvalue. The dimension of the unstable manifold is therefore $2$. It is spanned by an eigenvector (here, $(1,0)$ for e.g.) and a generalized eigenvector.
My question is, is there a link between the shape of the trajectories and a generalized eigenvector? For the eigenvector the link is quite obvious (also in the first and last cases in the figure), but I am not able to show a generalized eigenvector from the orbits.
edit: I wrote the dynamical equations for complex conjugate eigenvalues, and: - the (un)stable manifold is of obviously bidimensional (stable/unstable manifold theorem); - but nothing prevents me from trying to find a 1D-submanifold of the (un)stable manifold, by seeking a parametric curve of the form $(x,h(x))$ and writing that it is invariant ($h$ is written as its Taylor expansion up to a finite degree). The interesting thing is that there is no such curve!
As I have not read anything about this, I'd be willing to have your opinions: in some cases (1 and 4 in the figure), the (un)stable manifold is of dimension 2, and two submanifolds of dimension 1 can always be calculated. If the eigenvalues are complex (not real) (case 2 in above figure), the (un)stable manifold is of dimension 2, but this manifold cannot be "decomposed" in two (un)stable submanifolds. I'm not sure what I'm taking about when writing about decomposition in submanifolds... Highlight welcome!