I'm puzzled by a question of how dimension reduction influences the bifurcation diagram of a dynamical system.
Say we have a 3-D system with variables $x_1,x_2,x_3$ and we reduce it to 2-D by looking at the difference between the two first variables and the third $y : = x_1 - x_2$, $x_3$. If we in this 2-D system find a parameter value for which a saddle node bifurcation appears, what happens in the 3-D system? $\dot{y} = 0$ does not imply $\dot{x}_1 = 0 \wedge\dot{x}_2 = 0$, which would be necessary for a fix point to appear in the 3-D system. Are fixed points in the 2-D system not generally speaking periodic orbits, and so isn't the bifurcation diagram of the 3-D system fundamentally different?