  The nonlinear collisional dynamics of coupled driven plasma waves in the
presence of background dissipation is studied analytically within kinetic
theory. Sufficiently near marginal stability, phase space correlations are
poorly preserved and time delays become unimportant. The system is then shown
to be governed by two first-order coupled autonomous differential equations of
cubic order for the wave amplitudes and two complementary first-order equations
for the evolution of their phases. That system of equations can be decoupled
and further simplified to a single second-order differential equation of
Li\'enard's type for each amplitude. Numerical solutions for this equation are
obtained in the general case while analytic solutions are obtained for special
cases in terms of parameters related to the spacing of the resonances of the
two waves in frequency space, e.g., wave lengths and oscillation frequencies.
These parameters are further analyzed to find classes of quasi-steady
saturation and pulsating scenarios. To classify equilibrium points, local
stability analysis is applied, and bifurcation conditions are determined. When
the two waves saturate at similar amplitude levels, their combined signal is
shown to invariably exhibit amplitude beating and phase jumps of nearly $\pi$.
The obtained analytical results can be used to benchmark simulations and to
interpret eigenmode amplitude measurements in fusion experiments.
