  We study the response of a trapped Bose-Einstein condensate to a sudden
turn-on of a rotating drive by solving the two-dimensional Gross-Pitaevskii
equation. A weakly anisotropic rotating potential excites a quadrupole shape
oscillation and its time evolution is analyzed by the quasiparticle projection
method. A simple recurrence oscillation of surface mode populations is broken
in the quadrupole resonance region that depends on the trap anisotropy, causing
stochastization of the dynamics. In the presence of the phenomenological
dissipation, an initially irrotational condensate is found to undergo damped
elliptic deformation followed by unstable surface ripple excitations, some of
which develop into quantized vortices that eventually form a lattice. Recent
experimental results on the vortex nucleation should be explained not only by
the dynamical instability but also by the Landau instability; the latter is
necessary for the vortices to penetrate into the condensate.
