  We consider the finite temperature metal-insulator transition in the half
filled paramagnetic Hubbard model on the infinite dimensional Bethe lattice. A
new method for calculating the Dynamical Mean Field Theory fixpoint surface in
the phase diagram is presented and shown to be free from the convergence
problems of standard forward recursion. The fixpoint equation is then analyzed
using dynamical systems methods. On the fixpoint surface the eigenspectra of
its Jacobian is used to characterize the hysteresis boundaries of the first
order transition line and its second order critical end point. The critical
point is shown to be a cusp catastrophe in the parameter space, opening a
pitchfork bifurcation along the first order transition line, while the
hysteresis boundaries are shown to be saddle-node bifurcations of two merging
fixpoints. Using Landau theory the properties of the critical end point is
determined and related to the critical eigenmode of the Jacobian. Our findings
provide new insights into basic properties of this intensively studied
transition.
