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

Johannes Hagel

TL;DR

This work develops a systematic method to classify integrable cases of the time-periodic nonlinear oscillator $z'' + \omega^2 z + g(t) z^m = 0$ ($m>1$) by seeking first integrals quadratic in $p=z'$. Using a Courant--Snyder reduction from Hill equations to the normal form, the authors derive precise conditions on $g(t)$ that guarantee such invariants, obtaining $g(t) = [A + B \cos(2\omega t) + C \sin(2\omega t)]^{-(m+3)/2}$ for $m>2$, and a three-parameter/trigonometric family for $m=2$. For the $m=2$ case they construct an explicit invariant $I(z,p,t)$ and verify its conservation numerically; they also derive an exact analytical stability boundary in terms of the invariant and initial data, which matches extensive numerical experiments. These results unify the integrable classes of these nonlinear oscillators and provide a foundation for extending to generalized invariants and higher nonlinearities with rigorous stability criteria.

Abstract

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.

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

TL;DR

This work develops a systematic method to classify integrable cases of the time-periodic nonlinear oscillator () by seeking first integrals quadratic in . Using a Courant--Snyder reduction from Hill equations to the normal form, the authors derive precise conditions on that guarantee such invariants, obtaining for , and a three-parameter/trigonometric family for . For the case they construct an explicit invariant and verify its conservation numerically; they also derive an exact analytical stability boundary in terms of the invariant and initial data, which matches extensive numerical experiments. These results unify the integrable classes of these nonlinear oscillators and provide a foundation for extending to generalized invariants and higher nonlinearities with rigorous stability criteria.

Abstract

Starting from the nonlinear ODE with , we show that after a suitable normal-form reduction of any Hill equation one may, without loss of generality, fix the linear part as (with constant). For the class with , our goal is to compile a catalogue of all possible integrable cases. We restrict attention to integrals that are polynomial in the variables and . The Hamiltonian does not provide such an integral because it is explicitly time dependent. Instead, we search for invariants that are quadratic in . We show that such invariants exist precisely when satisfies the linear third-order ODE . This yields the three-parameter solution . For this reproduces the trigonometric structure with exponent 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.

Paper Structure

This paper contains 12 sections, 33 equations, 4 figures.

Table of Contents

  1. Introduction
  2. General formulation, integrability ansatz and example
  3. Problem class and objectives
  4. (i) $m=2$:
  5. (ii) $m>2$:
  6. The case $f(t) = \omega^2$
  7. (i) $m=2$:
  8. (ii) $m>2$:
  9. Integral for $m=2$ and demonstration example
  10. Phase curve at the Poincaré section $t_k = k\pi/\omega$
  11. Stability analysis by means of the invariant curve
  12. Conclusions

Figures (4)

  • Figure 1: Constancy of the invariant: plot of $I(t)/I(0)-1$ over $0<t<600$ (parameters as in text). The range shown is $[-10^{-5},+10^{-5}]$.
  • Figure 2: Poincaré phase section at $t_k = k\pi/\omega$: stroboscopic points $(z(t_k),p(t_k))$ together with the analytic curve given by (2.38).
  • Figure 3: Analytical stability boundary $z_{\mathrm{crit}}(\omega)$ from (3.4) for the example parameters $A=1.3$, $B=0.9$, $C=0$. Black dots: numerical results.
  • Figure 4: Time evolution $z(t)$ for $\omega=1.4$. Solid line: $z(0)=1.2$ (below the stability threshold, bounded). Dashed line: $z(0)=1.4$ (above the threshold, unbounded). The vertical axis is scaled to $[-5,5]$.