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<p>It is proved that there are many (positive Lebesgue measure) Kolmogorov-Arnold-Moser (KAM for short) tori at infinity and thus all solutions are bounded for the Duffing equations <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11401_2025_20_Article_IEq1.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="72" Type="Linedraw" Width="208" /> </InlineMediaObject> <EquationSource Format="TEX">\({\ddot x}+x^{2n+1}+\sum_{j=0}^{2n}p_{i}(t)x^{j}=0\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mrow> <mover> <mi>x</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mo>+</mo> <msup> <mi>x</mi> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <munderover> <mo>∑</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </munderover> <msub> <mi>p</mi> <mrow> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mi>x</mi> <mrow> <mi>j</mi> </mrow> </msup> <mo>=</mo> <mn>0</mn> </math></EquationSource> </InlineEquation> with <i>p</i><sub><i>j</i></sub>(<i>t</i>)’s being time-quasi-periodic smooth functions.</p>

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Lagrange Stability and KAM Tori for Duffing Equations with Quasi-periodic Coefficients

  • 期刊论文

  • 发表日期：

    2025年5月1日

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  • Huining Xue,
  • Xiaoping Yuan

摘要

It is proved that there are many (positive Lebesgue measure) Kolmogorov-Arnold-Moser (KAM for short) tori at infinity and thus all solutions are bounded for the Duffing equations \({\ddot x}+x^{2n+1}+\sum_{j=0}^{2n}p_{i}(t)x^{j}=0\) x ¨ + x 2 n + 1 + ∑ j = 0 2 n p i ( t ) x j = 0 with pj(t)’s being time-quasi-periodic smooth functions.