I am trying to classify the type of bifurcation for the dynamical system given by:
$\dot x = x^2+y^2-2my$
$\dot y= mx-y$
with m as a varying parameter
The fixed points are at (0,0) and ($2m^2 \over m^2+1$ $2m^3 \over m^2+1$)
This has features of a transcritical bifurcation as the fixed point (0,0) exists for all m. The problem I am having is at the bifurcation point which I deem to be m=0. As m goes from negative values to positive values, the stability does not change for this fixed point as a transcritical bifurcation should. The other fixed point should also change stability but linear analysis classifies the point as a saddle for all m (i.e. the determinant of the Jacobian < 0). What other bifurcation could it be?