Completely degenerate responsive tori in Hamiltonian systems <sup>*</sup>
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Abstract
Abstract We consider the existence of responsive tori for the completely degenerate Hamiltonian system with the following Hamiltonian <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>H</mml:mi> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi>θ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>I</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ϵ</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mrow> <mml:mo stretchy="false">⟨</mml:mo> <mml:mrow> <mml:mi>ω</mml:mi> <mml:mo>,</mml:mo> <mml:mi>I</mml:mi> </mml:mrow> <mml:mo stretchy="false">⟩</mml:mo> </mml:mrow> <mml:mo>+</mml:mo> <mml:mi>λ</mml:mi> <mml:mfrac> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:mfrac> <mml:mo>+</mml:mo> <mml:mfrac> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>y</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>m</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:mrow> <mml:mi>m</mml:mi> </mml:mrow> </mml:mfrac> <mml:mo>+</mml:mo> <mml:mi>ϵ</mml:mi> <mml:mi>P</mml:mi> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi>θ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ϵ</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> <mml:mspace class="nbsp" width="0.3333em"/> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi>θ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>I</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mo>∈</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">T</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>d</mml:mi> </mml:mrow> </mml:msup> <mml:mo>×</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> </mml:math> which is associated with the standard symplectic structure, where λ = ±1 and n > 2, n ⩾ m ⩾ 2 are integers. With P satisfying certain non-degenerate conditions, we obtain the following results: (1) For λ = −1 and ϵ sufficiently small, responsive tori exist for each ω satisfying a weak non-resonant condition; (2) For λ = 1 and ϵ * sufficiently small, there exists a Cantor set <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi mathvariant="script">E</mml:mi> <mml:mo>⊂</mml:mo> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:msub> <mml:mrow> <mml:mi>ϵ</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>*</mml:mo> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> with almost full Lebesgue measure such that responsive tori exist for each <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>ϵ</mml:mi> <mml:mo>∈</mml:mo> <mml:mi mathvariant="script">E</mml:mi> </mml:math> if ω satisfies a Diophantine condition. Non-existence of responsive tori are also discussed when P fails to satisfy the non-degenerate condition. Our results are directly applicable to the existence problem of quasi-periodic responsive solutions of degenerate harmonic oscillators.
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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