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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?

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This would still classify as a transcritical bifurcation, as far as I'm concerned, although a degenerate one. That is, the intersection between one family of equilibria $(\frac{2m^2}{m^2+1},\frac{2m^3}{m^2+1})$ and the other family $(0,0)$ is not transversal. This degeneracy has the effect of a 'double stability switch': at the bifurcation point, there is a double exchange of stability, which is equivalent to no exchange of stability at all.

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