\section{Symmetry Reduced Planar Equations}

This planar vector field comes from an analysis of how the reduction of
symmetry in a conservative mechanical system can lead to instability.
In a one degree of freedom Hamiltonian system with circular ($SO(2)$ or $S^1$)
symmetry, the origin is always Lyapunov stable. If the symmetry of a
system is broken to a $Z_2$ symmetry of rotation by $\pi$, then the 
stability of the system may change as the eigenvalues of the origin pass
through zero with a varying parameter. Generically, they do. This 
system describes the normal form for this problem. Included is a 
damping term with variable coefficient that shows the effect of damping
upon the phenomenon. In complex coordinates, the equations are
$$ \dot{A} = i(\lambda A - |A^2| A - \epsilon \bar{A}) - \delta A $$
The reference \cite{GuckenMahalov} applies the analysis of this simple 
system to explain a type of vortex instability in fluid flows.

