  We examine the problem of damage spreading in the off-equilibrium mode
coupling equations. The study is done for the spherical $p$-spin model
introduced by Crisanti, Horner and Sommers. For $p>2$ we show the existence of
a temperature transition $T_0$ well above any relevant thermodynamic transition
temperature. Above $T_0$ the asymptotic damage decays to zero while below $T_0$
it decays to a finite value independent of the initial damage. This transition
is stable in the presence of asymmetry in the interactions. We discuss the
physical origin of this peculiar phase transition which occurs as a consequence
of the non-linear coupling between the damage and the two-time correlation
functions.
