arXiv subjects

Johannes Hagel

Publications and source records attributed to Johannes Hagel.

4 recordsLinked to original sources

arXiv2026

Algebraic Detection of Tube Rupture via a Cubic Discriminant Criterion

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.

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arXiv2025

Tube Rupture in Aperiodic Nonlinear Oscillators: Theory and Simulation

We study the long-term behaviour of the nonlinear, aperiodically and parametrically forced oscillator z'' + z + g(tau) z^2 = 0, g(tau) = y(tau)^(-5/2), where y(tau) is the strictly positive solution of a weakly forced third-order equation. Building on the algebraic invariant constructed in our previous work, we show that the motion of z(tau) is confined to a two-dimensional invariant tube in the extended phase space (z, p, tau) as long as the corresponding invariant level set remains closed. The main result of this paper is an explicit analytical rupture criterion that predicts the precise time at which the invariant tube loses regularity. After transforming the invariant into polar coordinates and analysing the discriminant of the resulting cubic equation for the radial coordinate, we obtain a compact Cardano-type expression for the rupture time. Direct numerical integrations of the z-equation confirm the analytical prediction to within a few percent over a wide parameter range. The results establish the rupture time as a robust and quantitatively accurate indicator for the onset of unbounded behaviour in aperiodically and parametrically forced nonlinear oscillators. The method is based purely on algebraic properties of the invariant and remains valid throughout the asymptotic domain of the perturbative expansion for y(tau). Keywords: nonlinear oscillators; aperiodic forcing; invariant surfaces; tube integrability; rupture time; Cardano discriminant; secular perturbation; nonlinear dynamics.

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arXiv2025

Tube Integrability in a Time-Dependent Nonlinear Oscillator

We study the nonlinear oscillator z'' + omega^2 z + g(t) z^2 = 0 with a time-dependent coefficient g(t). We show that this equation admits an exact quadratic invariant I(z,p,t) provided that g(t) = alpha2(t)^(-5/2) and that alpha2(t) satisfies a nonlinear third-order differential equation. The resulting invariant constrains the dynamics to a smooth two-dimensional surface in the extended phase space (z,p,t). If alpha2(t) is periodic, this surface forms a compact invariant torus. However, we show that periodic solutions of alpha2(t) are generically obstructed by a resonance mechanism, leading instead to an aperiodic but non-chaotic evolution. In this regime the invariant surface is non-compact and extends along the time direction, forming a tube rather than a torus. We therefore propose the term "tube integrability" for integrable systems whose invariant manifolds are non-compact in time. A perturbation expansion for alpha2(t) up to third order is derived and compared with numerical integration, clarifying the parameter regimes in which the truncated series provides quantitatively accurate approximations. The breakdown of the series for small y0 reflects the asymptotic nature of the expansion rather than a loss of integrability.

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arXiv2025

Integrable nonlinear oscillators with polynomial invariants: construction, Poincare geometry, and an analytic stability boundary

Starting from the nonlinear ODE $z'' + f(t)\,z + g(t)\, z^{m}=0$ with $m>1$, we show that after a suitable normal-form reduction of any Hill equation one may, without loss of generality, fix the linear part as $f(t)\equiv ω^{2}$ (with $ω>0$ constant). For the class $z''+ω^{2}z+g(t)\, z^{m}=0$ with $m>1$, our goal is to compile a catalogue of all possible integrable cases. We restrict attention to integrals that are polynomial in the variables $z$ and $p=z'$. The Hamiltonian does not provide such an integral because it is explicitly time dependent. Instead, we search for invariants that are quadratic in $p=z'$. We show that such invariants exist precisely when $α_2(t):=g(t)^{-2/(m+3)}$ satisfies the linear third-order ODE $α_2''' + 4ω^2 α_2'=0$. This yields the three-parameter solution $g(t)=[a_0+a_1\cos(2ωt)+a_2\sin(2ωt)]^{-(m+3)/2}$. For $m=2$ this reproduces the trigonometric structure with exponent $-5/2$ found in Hagel--Bouquet (1992). In addition we present a detailed stability analysis based on the invariant using Poincaré sections and find full agreement with numerical simulations.

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