2D Phase portrait - Black hole?

[unscientific]

Homework Statement

Trajectories around a black hole can be described by $\frac{d^2u}{d\theta^2} + u = \alpha \epsilon u^2$, where $u = \frac{1}{r}$ and $\theta$ is azimuthal angle.

(a) By using $v = \frac{du}{d\theta}$, reduce system to 2D and find fixed points and their stability. Find direction of fastest perturbations.
(b) Sketch the phase portrait. Would stability of fixed points differ in the non-linearized version?

Homework Equations

The Attempt at a Solution

Part (a)
The equations now become $\delta v = \delta \dot u$ and $\delta \dot v + \delta u = 2\epsilon u \delta u$.

Fixed points are $\left( 0,0 \right)$ and $\left( \frac{1}{\epsilon}, 0 \right)$. At $(0,0)$, all eigenvalues are imaginary, so the fixed point is a center. At $(\frac{1}{\epsilon},0)$, eigenvalues are $\pm 1$ so fixed point is a saddle.
Eigenvalue of $J + J^T$ is $2\epsilon u$ and direction of fastest perturbation is $u=v$.

Part(b)
Eigenvalue in general is $\lambda^2 = (2\epsilon u - 1)$, so for $|u| > \frac{1}{2\epsilon}$, the particle doesn't get trapped by the black hole?

 

[unscientific]

bumpp

 

[unscientific]

bumpp