The system with linearisation matrix given by
has eigenvalues which are the solution to
These eigenvalues are
The eigenvectors corresponding to these eigenvectors will be complex when solved using
We can however transform coordinates. When we transform to polar coordinates, the system becomes much simpler. If the eigenvalues are
then define
Similarly
and differentiation gives
Hence
We obtain the simple equations
and
We can solve these to obtain
and
Elimination of
between these two equations gives
If
and
or
and
then
increases exponentially as
increases. If
and
have the same sign thendecreases exponentially asincreases.