Algebraic Detection of Tube Rupture via a Cubic Discriminant Criterion
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866917206102638592 |
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| author | Hagel, Johannes |
| author_facet | Hagel, Johannes |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_10736 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Algebraic Detection of Tube Rupture via a Cubic Discriminant Criterion Hagel, Johannes Dynamical Systems 37J25 37C10 34C15 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. |
| title | Algebraic Detection of Tube Rupture via a Cubic Discriminant Criterion |
| topic | Dynamical Systems 37J25 37C10 34C15 |
| url | https://arxiv.org/abs/2601.10736 |