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Tube Integrability in a Time-Dependent Nonlinear Oscillator

Johannes Hagel

TL;DR

Let the nonlinear oscillator $z''+\omega^2 z+ g(t) z^2=0$ be studied; an exact quadratic invariant $I(z,p,t)$ exists when $g(t)=\alpha_2(t)^{-5/2}$ with $\alpha_2$ obeying a nonlinear third-order equation. When $\alpha_2(t)$ is periodic the dynamics lie on invariant tori in the extended space; generically, a second-order resonance obstructs exact periodicity, producing a non-compact invariant tube that winds along the time axis. The authors develop a third-order perturbative expansion for $\alpha_2(t)$, compare with numerical integration, and show the expansion is asymptotic; breakdown reflects nearby complex singularities rather than loss of integrability. The results introduce tube integrability, extending Liouville concepts to non-compact invariant geometries and clarifying the limits of perturbative approximations.

Abstract

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.

Tube Integrability in a Time-Dependent Nonlinear Oscillator

TL;DR

Let the nonlinear oscillator be studied; an exact quadratic invariant exists when with obeying a nonlinear third-order equation. When is periodic the dynamics lie on invariant tori in the extended space; generically, a second-order resonance obstructs exact periodicity, producing a non-compact invariant tube that winds along the time axis. The authors develop a third-order perturbative expansion for , compare with numerical integration, and show the expansion is asymptotic; breakdown reflects nearby complex singularities rather than loss of integrability. The results introduce tube integrability, extending Liouville concepts to non-compact invariant geometries and clarifying the limits of perturbative approximations.

Abstract

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.

Paper Structure

This paper contains 23 sections, 1 theorem, 63 equations, 7 figures.

Table of Contents

  1. Introduction
  2. General formulation and derivation of the invariant
  3. Tube integrability
  4. Example: tube integrability via the Ermakov–Pinney construction
  5. Tube (vs. torus) behavior.
  6. Chaotic forcing example (logistic driver).
  7. Properties and computation of the function $\alpha_2(t)$
  8. Transformation of the independent variable
  9. Positivity of $y(\tau)$
  10. Reduction of the non autonomous term to a single cosine function
  11. Concerning the existence of a periodic solution for $y(\tau)$
  12. Fourier expansion and resonance mechanism
  13. First order
  14. Second order
  15. Third order
  16. ...and 8 more sections

Key Result

Proposition 4.1

Let $m>0$ and suppose that $y(\tau)=\alpha_2(t)$ is a $2\pi$--periodic solution of eq:alpha2-eq with mean value Then for $\varepsilon \neq 0$ and $C_1\neq 0$ no such solution bifurcates from the constant state $y\equiv m$.

Figures (7)

  • Figure 1: Schematic illustration of tube integrability. The $t$–axis (right) runs inside the tube, while the cross–section $(z,p)$ varies aperiodically so that no invariant torus arises. Instead, the solution winds around the $t$–axis on a non-periodic tube surface.
  • Figure 2: Illustration of tube integrability in the case of a nonperiodic $\alpha_2(t)$. The function $\alpha_2(t)$ shows dominant frequencies $\omega$ and $2\omega$ together with a slowly modulated envelope. The coefficient $g(t)$ injects energy, leading to irregular but bounded oscillations of $z(t)$. Nevertheless the quadratic invariant $I(t)$ remains conserved up to numerical accuracy, confirming motion on an invariant tube.
  • Figure 3: Numeric versus analytic solution of (4.6) for $y_0=1$, $\varepsilon=0.1$.
  • Figure 4: Numeric versus analytic solution when $y_0=0.7$, $\varepsilon=0.1$ and $0<t<300$
  • Figure 5: Numeric versus analytic solution when $y_0=0.7$, $\varepsilon=0.1$ and $300<t<500$
  • ...and 2 more figures

Theorems & Definitions (6)

  • Definition 3.1: Tube integrability
  • Remark 3.1
  • proof : Sketch
  • Proposition 4.1: Non-existence of small $2\pi$--periodic solutions
  • proof : Sketch of proof
  • Remark 4.1