# 2D Bifurcation Classification

Given the system with m as a varying parameter:

$\dot x = mx^2-y$

and

$\dot y = m+y - x$

Determine any bifurcations that occur

Attempt:

x nullcline

$y=mx^2$

y nullcline

$y=x-m$

Fixed points:

$mx^2=x-m$

solving for x:

x = ${1 \pm \sqrt{1-4m^2} \over 2m}$

y = $({1 \pm \sqrt{1-4m^2}})^2 \over {4m}$

This shows a bifurcation point (unstable) exists at (1,0.5) for an m = 0.5 and there are no fixed points if $1-4m^2 <0$

As you go past the bifurcation point, two fixed points appear but these two are unstable. This has properties of a saddle node bifurcation since two fixed points appear as m goes past the bifurcation point. However, I do not have one stable and one unstable fixed point appearing but two unstable fixed points appearing instead. Can this still be classified as a saddle node bifurcation?