Algebraic Detection of Tube Rupture via a Cubic Discriminant Criterion

TL;DR

The paper addresses rupture of time-dependent invariant tubes in nonautonomous Ermakov-type systems by constructing a positivity-preserving perturbative invariant, yielding a discriminant $Disc(z,t)$ that is cubic in $z$. A purely algebraic bridge criterion uses the real-root structure of $Disc(z,t)=0$, reduced further to the sign of the cubic discriminant $ riangle_z[Disc(z,t)]$, enabling time-local rupture detection. The results reveal a sequence of isolated bridge windows with a period-doubling organization (from one opening per $2oldsymbol{ m\pi}$ cycle to two openings per cycle) and a compact box-plot representation that captures the evolving geometry. This work demonstrates that algebraic diagnostics derived from time-dependent invariants can robustly predict rupture phenomena even when exact tube integrability is weakly perturbed, offering a scalable scalar diagnostic for near-integrable, time-dependent dynamics.

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.

Figures (6)

• Figure 1: Comparison of the numerical solution of the $y$-equation (top) with the exponential perturbative approximation (bottom) for $\varepsilon=0.08$ and $y_0=1$.
• Figure 2: Straightforward perturbative expansion of the $y$-equation up to second order in $\varepsilon$. While accurate at early times, this approximation deteriorates at large times and does not preserve positivity.
• Figure 3: Geometry of the approximate invariant surface $p=p(z,t)$ at early and late times for $y_0=1$, $\varepsilon=0.08$ , $z_0=0.25$. While the surface remains single-valued and tube-like at early times, neighboring branches begin to overlap at later times, marking the geometric onset of the loss of confinement.
• Figure 4: Onset of the bridge phenomenon in the discriminant surface $\mathop{\mathrm{Disc}}\nolimits(z,t)$. For $t\approx 328\pi$ no bridge is present and the invariant tube remains closed. Around $t\approx 330\pi$ the discriminant tongues nearly touch, indicating the imminent opening of a bridge. At $t\approx 332\pi$ a clear local opening is visible, corresponding to the first time interval in which the cubic equation $\mathop{\mathrm{Disc}}\nolimits(z,t)=0$ possesses exactly one real root.
• Figure 5: Emergence of period--doubling in the bridge geometry of the discriminant surface $\mathop{\mathrm{Disc}}\nolimits(z,t)$. Panel (a) shows the last regime with a single bridge per $2\pi$ cycle, while panel (b) demonstrates the fully developed period--halved structure. Careful inspection reveals the formation of secondary bulges in the central channel, indicating early geometric signatures of a subsequent doubling, although no further transition is claimed at this stage.

Theorems & Definitions (6)

• Remark 2.1: Historical note on the Bouquet--Hagel system
• Definition 2.2: Tube integrability
• Remark 2.3
• Remark 2.4: No boundedness requirement
• Remark 2.5: Integrability versus tube integrability
• Remark 3.1: Explicit form of the cubic discriminant