Asteroseismic studies of red giants generally assume that the oscillation modes can be treated as linear perturbations to the background star. However, observations by the Kepler mission show that the oscillation amplitudes increase dramatically as stars ascend the red giant branch. The importance of nonlinear effects should therefore be assessed. In previous work, we found that mixed modes in red giants are unstable to nonlinear three-wave interactions over a broad range of stellar mass and evolutionary state. Here we solve the amplitude equations that describe the mode dynamics for large networks of nonlinearly coupled modes. The networks consist of stochastically driven parent modes coupled to resonant secondary modes (daughters, granddaughters, etc.). We find that nonlinear interactions can lower the energy of gravity-dominated mixed modes by $gtrsim 80%$ compared to linear theory. However, they have only a mild influence on the energy of pressure-dominated mixed modes. Expressed in terms of the dipole mode visibility $V^2$, i.e., the summed amplitudes of dipole modes relative to radial modes, we find that $V^2$ can be suppressed by $50-80%$ relative to the linear value for highly-evolved red giants whose frequency of maximum power $ u_{rm max} lesssim 100,mutextrm{Hz}$. However, for less evolved red giants with $150lesssim u_{rm max} lesssim 200,mutextrm{Hz}$, $V^2$ is suppressed by only $10-20%$. We conclude that resonant mode coupling can have a potentially detectable effect on oscillations at $ u_{rm max} lesssim 100,mutextrm{Hz}$ but it cannot account for the population of red giants that exhibit dipole modes with unusually small amplitudes at high $ u_{rm max}$.
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