  We study the nonlinear evolution of the spherical symmetric black holes under
a small neutral scalar field perturbation in Einstein-Maxwell-dilaton theory
with coupling function $f(\phi)=e^{-b\phi}$ in asymptotic anti-de Sitter
spacetime. The non-minimal coupling between scalar and Maxwell fields allows
the transmission of the energy from the Maxwell field to the scalar field, but
also behaves as a repulsive force for the scalar. The scalar field oscillates
with damping amplitude and converges to a final value by a power law. The
irreducible mass of the black hole increases abruptly at initial times and then
saturates to the final value exponentially. The saturating rate is twice the
decaying rate of the dominant mode of the scalar. The effects of the black hole
charge, the cosmological constant and the coupling parameter on the evolution
are studied in detail. When the initial configuration is a naked singularity
spacetime with a large charge to mass ratio, a horizon will form soon and hide
the singularity.
