Algebraic Detection of Tube Rupture via a Cubic Discriminant Criterion

Abstract: We investigate the rupture of invariant tubes in a class of nonautonomous dynamical systems arising from time-dependent Ermakov-type equations. Starting from an exactly tube-integrable reference system, we analyze a time-dependent invariant obtained from a positivity-preserving second-order perturbative construction, which provides a near-integrable geometric description of the dynamics. While this approximation does not preserve exact invariance, its algebraic structure remains sufficiently robust to allow a precise characterization of tube opening and loss of confinement. For fixed time, the discriminant of the approximate invariant with respect to the momentum variable defines a cubic polynomial in the configuration variable. We show that the invariant tube admits an unbounded bridge if and only if the associated cubic possesses exactly one real root. This yields a purely algebraic rupture criterion based on the cubic discriminant and reduces the full geometric problem to the evaluation of a single scalar function of time. Applying this criterion reveals a sequence of isolated bridge windows whose temporal organization undergoes a transition from one opening per 2*pi cycle to two openings per cycle, corresponding to a period-halving in time. These windows can be represented compactly by a one-dimensional box-plot visualization, which faithfully captures the underlying geometry and highlights the progressive densification and widening of escape-enabling intervals. The results demonstrate that algebraic diagnostics derived from time-dependent invariants can retain sharp predictive power for rupture phenomena even when exact tube integrability is weakly perturbed.