Tube Rupture in Aperiodic Nonlinear Oscillators: Theory and Simulation

TL;DR

The work addresses long-time dynamics of a nonlinear, aperiodically forced oscillator by leveraging an algebraic invariant whose level sets form tubular surfaces in the extended phase space. By transforming to polar coordinates, the authors reduce the problem to a cubic in the radial coordinate and derive a purely algebraic rupture criterion: rupture occurs when the cubic discriminant vanishes, yielding a closed-form rupture time τ_rupt that scales as ε^{-2} and depends on initial conditions via a Cardano-type expression. They justify sampling the invariant at τ=nπ, show no frequency shift to the required order, and validate the rupture time against direct numerics with small 3–15% discrepancies, thereby establishing rupture time as a robust predictor of the onset of unbounded behavior in aperiodically forced nonlinear oscillators. The results rely on tube integrability and offer potential extensions to chaotic forcing, highlighting the geometric nature of rupture over purely dynamic blow-up times.

Abstract

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.

Figures (5)

• Figure 1: Validity region of the closed rupture-time formula in the $(y_0,z_0)$–plane, defined by the condition $y_0(y_0 - C^{1/3}) < 1$. The red point marks the parameter pair $(y_0,z_0)=(1,0.25)$ used in the numerical experiments, which lies well inside the domain of validity.
• Figure 2: Analytic rupture index $n_{\mathrm{crit}}(\varepsilon,z_0)$ computed from the closed formula. (Data source: fig_rupture_time.pdf.)
• Figure 3: Contour plot of the analytic rupture index. (Data source: fig_rupture_time_contour.pdf.)
• Figure 4: Rupture of the invariant tube in the $(z,p)$–section. The figure shows the deformation of the sampled invariant $I(z,p,n\pi)$ for increasing sampling index $n$. For small $n$ the level sets form perfectly closed tube cross sections. As $n$ approaches the analytically predicted rupture index $n_{\mathrm{crit}}$ from Eq. (33), the tube develops a distinct horseshoe-shaped opening, corresponding to the vanishing of the cubic discriminant. This opening marks the exact geometric rupture point of the invariant tube.
• Figure 5: Time series of $z(\tau)$ corresponding to the rupture shown in Fig. \ref{['fig:tuberupture']}. After thousands of oscillations with irregular, chaotic–looking amplitudes, the trajectory escapes through the newly formed opening of the invariant tube, leading to a rapid divergence of $z(\tau)$. The onset of unbounded growth occurs shortly after the analytically predicted rupture time $\tau_{\mathrm{rupt}}=\pi n_{\mathrm{crit}}$.

Theorems & Definitions (1)

• Remark 1: Closed-form nature of the rupture condition