  This thesis investigates in the time domain a particular class of second
order perturbations of a perfect fluid non-rotating compact star: those arising
from the coupling between first order radial and non-radial perturbations. This
problem has been treated by developing a gauge invariant formalism based on the
2-parameter perturbation theory (Sopuerta, Bruni and Gualtieri, 2004) where the
radial and non-radial perturbations have been separately parameterized. The
non-linear perturbations obey inhomogeneous partial differential equations,
where the structure of the differential operator is given by the previous
perturbative orders and the source terms are quadratic in the first order
perturbations. In the exterior spacetime the sources vanish, thus the
gravitational wave properties are completely described by the second order
Zerilli and Regge-Wheeler functions.
  As main initial configuration we have considered a first order differentially
rotating and radially pulsating star. Although at first perturbative order this
configuration does not exhibit any gravitational radiation, we have found a new
interesting gravitational signal at non-linear order, in which the radial
normal modes are precisely mirrored. In addition, a resonance effect is present
when the frequencies of the radial pulsations are close to the first axial
w-mode. Finally, we have roughly estimated the damping times of the radial
pulsations due to the non-linear gravitational emission. The coupling near the
resonance results to be a very effective mechanism for extracting energy from
the radial oscillations.
