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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2510.24080 |
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| _version_ | 1866915582316642304 |
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| author | Hagel, Johannes |
| author_facet | Hagel, Johannes |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_24080 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Integrable nonlinear oscillators with polynomial invariants: construction, Poincare geometry, and an analytic stability boundary Hagel, Johannes Dynamical Systems 34A05, 34C14, 37J30 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. |
| title | Integrable nonlinear oscillators with polynomial invariants: construction, Poincare geometry, and an analytic stability boundary |
| topic | Dynamical Systems 34A05, 34C14, 37J30 |
| url | https://arxiv.org/abs/2510.24080 |