  We address the problem of chaotic temperature dependence in disordered glassy
systems at equilibrium by following states of a random-energy random-entropy
model in temperature; of particular interest are the crossings of the
free-energies of these states. We find that this model exhibits strong, weak or
no temperature chaos depending on the value of an exponent. This allows us to
write a general criterion for temperature chaos in disordered systems,
predicting the presence of temperature chaos in the Sherrington-Kirkpatrick and
Edwards-Anderson spin glass models, albeit when the number of spins is large
enough. The absence of chaos for smaller systems may justify why it is
difficult to observe chaos with current simulations. We also illustrate our
findings by studying temperature chaos in the naive mean field equations for
the Edwards-Anderson spin glass.
