  We study the Loschmidt echo and Berry phase in a nonlinear system transported
around a double and triple degeneracy. The nonlinearity of the system makes the
Berry phase different from that in linear systems. We propose a witness of
nonlinearity for the nonlinear system and show its dependence on the parameters
of the system, taking the standard Landau-Zener model as an example. We
calculate the Loschmidt echo(LE) or quantum fidelity in the quantum dynamics
under perturbations around the degeneracy, and establish a connection between
the LE and the witness of nonlinearity. These phases and Loschmidt echo can be
observed with current experimental technology in a nonlinear resonator.
