  We study the chaotic behavior of the Gibbs state of spin-glasses under the
application of an external magnetic field, in the crossover region where the
field intensity scales proportional to $1/\sqrt{N}$, being $N$ the system size.
We show that Replica Symmetry Breaking (RSB) theory provides universal
predictions for chaotic behavior: they depend only on the zero-field overlap
probability function $P(q)$ and are independent of other features of the
system. Using solely $P(q)$ as input we can analytically predict quantitatively
the statistics of the states in a small field. In the infinite volume limit,
each spin-glass sample is characterized by an infinite number of states that
have a tree-like structure. We generate the corresponding probability
distribution through efficient sampling using a representation based on the
Bolthausen-Sznitman coalescent. In this way, we can compute quantitatively
properties in the presence of a magnetic field in the crossover region, the
overlap probability distribution in the presence of a small field and the
degree of decorrelation as the field is increased. To test our computations, we
have simulated the Bethe lattice spin glass and the 4D Edwards-Anderson model,
finding in both cases excellent agreement with the universal predictions.
