  We theoretically investigate quadrupolar collective excitations in
two-component Bose-Einstein condensates and their nonlinear dynamics associated
with harmonic generation and mode coupling. Under the Thomas-Fermi
approximation and the quadratic polynomial ansatz for density fluctuations, the
linear analysis of the superfluid hydrodynamic equations predicts excitation
frequencies of three normal modes constituted from monopole and quadrupole
oscillations, and those of three scissors modes. We obtain analytically the
resonance conditions for the second harmonic generation in terms of the trap
aspect ratio and the strength of intercomponent interaction. The numerical
simulation of the coupled Gross-Pitaevskii equations vindicates the validity of
the analytical results and reveals the dynamics of the second harmonic
generation and nonlinear mode coupling that lead to nonlinear oscillations of
the condensate with damping and recurrence reminiscent of the Fermi-Pasta-Ulam
problem.
