Tube Integrability in a Time-Dependent Nonlinear Oscillator

TL;DR

This paper introduces the concept of 'tube integrability' in a nonlinear oscillator with time-dependent parameters, demonstrating exact invariants, analyzing the nature of invariant manifolds, and exploring the effects of perturbations and resonances.

Contribution

It establishes the existence of exact quadratic invariants under specific conditions and defines 'tube integrability' for systems with non-compact invariant manifolds in time.

Findings

Invariant surfaces can form tori or tubes depending on periodicity.

Resonance mechanisms prevent periodic solutions of alpha2(t).

Perturbation series accurately approximate alpha2(t) within certain regimes.

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.