I'm confused about a question in dynamical systems, and whether it can be done at all.
Say we consider the lorenz system
$ \dot{x} = \sigma(y-x)\\ \dot{y} = x(\rho-z)-y\\ \dot{z} = xy-\beta z $
And we seek to do a bifurcation analysis, but we are for other reasons only interested in the bifurcation of the subsystem $\lbrace\dot{x},\dot{y}\rbrace$. How can we analyse this? Do we then consider $z$ to be a fourth parameter, which is time dependent?
Ideas are welcome
It really doesn't work like that: either $z$ "matters" or not, and in the first case it isn't possible to do what you want.
When one linearizes the system, the third equation becomes $z'=-\beta z$. Since $\beta$ is usually assumed to be positive, this doesn't contribute to any bifurcation.
On the other hand, the linearization of the first two equation doesn't include the variable $z$ and you can check that there is a bifurcation at $\rho=1$ passing from a saddle to a sink.
This bifurcation is also (necessarily) present in the full linearization and so also in the original equation (because it goes from a hyperbolic equilibrium to a hyperbolic equilibrium).