Fast symplectic mapping, based on a canonical generator of the full-turn map in polar coordinates (I, Φ), is a powerful tool to study long-term stability in large hadron storage rings. Accurate maps for realistic lattices can be constructed in a few hours on a workstation computer, and can be iterated to follow orbits for 10⁷ turns in less than 4 hours. Orbits of the map can also be used in a non-perturbative construction of quasi-invariant actions. By bounding the small changes in quasi-invariants along many short orbits, one can derive conservative estimates of survival time for long orbits, for any initial condition in a region of phase space. A first quasi-invariant vector, J, arises from a canonical transformation (1, Φ) → (J, Ψ), based on interpolation of invariant tori surrounding the origin. The variation of J is relatively large near a broad resonance. In such a region a second canonical transformation, associated with pendulum-like motion in appropriate variables, is required to produce a good quasi-invariant. This quasi-invariant is used to set a long-term bound on motion near a broad, large-amplitude resonance in a realistic model of the Large Hadron Collider (LHC). Interesting general properties of the pseudo-pendulum motion are studied.