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.
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