In hadron colliders, such as the Large Hadron Collider (LHC) to be built at CERN, the long-term stability of the single-particle motion is mostly determined by the field-shape quality of the superconducting magnets. The mechanism of particle loss may be largely enhanced by modulation of betatron tunes, induced either by synchro-betatron coupling (via the residual uncorrected chromaticity), or by unavoidable power supply ripple. This harmful effect is investigated in a simple dynamical system model, the Henon map with modulated linear frequencies. Then, a realistic accelerator model describing the injection optics of the LHC lattice is analyzed. Orbital data obtained with long-term tracking simulations ($10^5$-$10^7$ turns) are post-processed to obtain the dynamic aperture. It turns out that the dynamic aperture can be interpolated using a simple mpirical formula, and it decays proportionally to a power of the inverse logarithm of the number of turns. Furthermore, the extrapolation of tracking data at $10^5$ turns gives reliable estimates of the dynamic aperture for $10^7$ turns, which represent the expected duration of the LHC injection plateau.
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