  We show how various mathematical formalisms, specifically the catastrophe
formalism and group theory, aid in the study of relevant systems in quantum
optics. We describe the phase transition of the Dicke model for a finite number
N of atoms, via 3 different methods, which lead to universal parametric curves
for the expectation value of the first quadrature of the electromagnetic field
and the expectation value of the number operator, as functions of the atomic
relative population. These are valid for all values of the matter-field
coupling parameter, and valid for both the ground and first-excited states.
Using these mathematical tools, the critical value of the atom-field coupling
parameter is found as a function of the number of atoms, from which its
critical exponent is derived.
