  We investigate a model of globally coupled conservative oscillators. Two
different algebraic potentials are considered that display in the canonical
ensemble either a second ($\phi^{4}$) or both a second and a first order phase
transition separated by tricritical points ($\phi^{6}$). The stability of
highly clustered states appearing in the low temperature/energy region is
studied both analytically and numerically for the $\phi^{4}$-model. Moreover,
long-lived out-of-equilibrium states appear close to the second order phase
transition when starting with "water-bag" initial conditions, in analogy with
what has been found for the Hamiltonian Mean Field (HMF) model. The
microcanonical simulations of the $\phi^{6}$-model show strong hysteretic
effects and metastability near the first-order phase transition and a narrow
region of negative specific heat.
