  In many mean-field glassy systems, the low-temperature Gibbs measure is
dominated by exponentially many metastable states. We analyze the evolution of
the metastable states as temperature changes adiabatically in the solvable case
of the spherical $s+p$-spin glass model, extending the work of Barrat, Franz
and Parisi J. Phys. A 30, 5593 (1997). We confirm the presence of level
crossings, bifurcations, and temperature chaos. For the states that are at
equilibrium close to the so-called dynamical temperature $T_d$, we find,
however, that the following state method (and the dynamical solution of the
model as well) is intrinsically limited by the vanishing of solutions with
non-zero overlap at low temperature.
