Dynamical Systems: Finding Directions Around Equilibria? I'm struggling to find the best way to determine directions about equilibria in order to draw a phase plot of this non linear dynamical system:
$x' = x − xy,$
$y' = \dfrac{4}{5} − x^2 + x − y.$
By setting both equations = 0 nullclines can then be drawn on x,y plane giving 3 equilibria.
How do I then determine direction around the equilibria?
As far as I can see there are 2 viable methods:
1. Testing points about the equilibria
2. Forming the generic Jacobian and taking eigenvalues at each equilibrium
Both methods to me seem to be extremely tedious (especially as these phase plots are only worth 1 mark in my exam coming up), is there a quicker way to do it with visual inspection? And if not which method is the best for multiple equilibria?
Thanks for your time!
 A: I'm afraid the only generally applicable method is to find the eigenvalues and eigenvectors of the Jacobian at each equilibrium. I agree with you that it can be quite tedious. Testing points is often not conclusive, and can take quite a lot of time if you test a lot of points.
However, for two-dimensional systems, it's easy to find the eigenvectors once you've found the eigenvalues. Given a matrix
\begin{equation}
A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}
\end{equation}
with eigenvalues $\lambda_{1,2}$, the associated eigenvectors are given by
\begin{equation}
 \xi_{1,2} = \begin{pmatrix} \lambda_{1,2} - a_{22} \\ a_{21} \end{pmatrix}.
\end{equation}
The eigenvalues themselves are the solution of the characteristic equation of $A$, as you undoubtedly know. In the two-dimensional case, this characteristic equation takes the convenient form
\begin{equation}
 \lambda^2 - \text{tr }A\,\lambda + \text{det }A = 0.
\end{equation}
This should help you get going. I agree with you that 1 mark for a phase space plot might not seem worth the effort, but it often helps to guide your thinking in the follow-up questions, so I would advice to put some effort and time into sketching a good phase space plot.
