  We discuss non-linear excitation and amplitude saturation of $g$-modes,
$r$-modes and overstable convective (OsC) modes in early type main sequence
stars, taking account of the effects of three-mode couplings on amplitude
evolutions. OsC modes are rotationally stabilized convective modes in the
convective core and they resonantly excite low frequency $g$-modes to obtain
large amplitudes in the envelope when the rotation rate of the core is larger
than critical rates. We use, for a network of three-mode couplings, amplitude
equations governing the time evolution of the mode amplitudes where each of
three-mode couplings is assumed to occur between two stable modes and one
unstable mode. Assuming that the unstable modes in the couplings are OsC modes
in the core and the stable modes are $g$- and $r$-modes in the envelope, we
integrate the amplitude equations to see how the $g$- and $r$-modes are
non-linearly excited by the OsC modes and whether or not the amplitude
evolutions tend toward a state of finite amplitudes. We find that the
non-linear three-mode couplings do excite low frequency $g$- and $r$-modes but
they are not necessarily effective to achieve amplitude saturation since the
three-mode couplings between the OsC modes with large growth rates and $g$- and
$r$-modes with small damping rates tend to destabilize amplitude evolutions.
