  In this paper we give a complete analysis of the phase transitions in the
mean-field Blume-Emery-Griffiths lattice-spin model with respect to the
canonical ensemble, showing both a second-order, continuous phase transition
and a first-order, discontinuous phase transition for appropriate values of the
thermodynamic parameters that define the model. These phase transitions are
analyzed both in terms of the empirical measure and the spin per site by
studying bifurcation phenomena of the corresponding sets of canonical
equilibrium macrostates, which are defined via large deviation principles.
Analogous phase transitions with respect to the microcanonical ensemble are
also studied via a combination of rigorous analysis and numerical calculations.
Finally, probabilistic limit theorems for appropriately scaled values of the
total spin are proved with respect to the canonical ensemble. These limit
theorems include both central-limit-type theorems, when the thermodynamic
parameters are not equal to critical values, and noncentral-limit-type
theorems, when these parameters equal critical values.
