  Long-period variability in luminous red giants has several promising
applications, all of which require models able to accurately predict pulsation
periods. Linear pulsation models have proven successful in reproducing the
observed periods of overtone modes in evolved red giants, but they fail to
accurately predict their fundamental mode periods. Here, we use a 1D
hydrodynamic code to investigate the long-period variability of M-type
asymptotic giant branch stars in the nonlinear regime. We examine the period
and stability of low-order radial pulsation modes as a function of mass and
radius, and find overtone mode periods in complete agreement with predictions
from linear pulsation models. In contrast, nonlinear models predict an earlier
onset of dominant fundamental mode pulsation, and shorter periods at large
radii. Both features lead to a substantially better agreement with
observations, that we verify against OGLE and Gaia data for the Magellanic
Clouds. We provide simple analytic relations describing the nonlinear
fundamental mode period-mass-radius relation. Differences with respect to
linear predictions originate from the readjustment of the envelope structure
induced by large-amplitude pulsation. We investigate the impact of turbulent
viscosity on linear and nonlinear pulsation, and probe possible effects of
varying metallicity and carbon abundance.
