<p>This paper studies the global dynamics of the rolling motion of ships in random beam seas model described by the equation <Equation ID="Equ17"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1228_Article_Equ17.gif" Format="GIF" Height="36" Rendition="HTML" Resolution="72" Type="Linedraw" Width="252" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} \ddot{x}+ (2\mu + \delta x^2){\dot{x}} + \omega ^2 x-\frac{1}{a^2} x^3=0, \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mover accent="true"> <mi>x</mi> <mo>¨</mo> </mover> <mo>+</mo> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>μ</mi> <mo>+</mo> <mi>δ</mi> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> <mover accent="true"> <mi>x</mi> <mo>˙</mo> </mover> <mo>+</mo> <msup> <mi>ω</mi> <mn>2</mn> </msup> <mi>x</mi> <mo>-</mo> <mfrac> <mn>1</mn> <msup> <mi>a</mi> <mn>2</mn> </msup> </mfrac> <msup> <mi>x</mi> <mn>3</mn> </msup> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1228_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="77" /> </InlineMediaObject> <EquationSource Format="TEX">$0 < \mu \ll 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>μ</mi> <mo>≪</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1228_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="71" /> </InlineMediaObject> <EquationSource Format="TEX">$0< \delta < 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>δ</mi> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1228_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$\omega =1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ω</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1228_Article_IEq4.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="72" /> </InlineMediaObject> <EquationSource Format="TEX">$0< a < 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>a</mi> <mo><</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. We fully explain its dynamics in the Poincaré compactification of <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1228_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">${{\mathbb {R}}}^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>.</p>

Global Phase Portrait of a Rolling Motion of Ships Equation

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2025年1月30日

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Martha Álvarez–Ramírez,
Jaume Llibre

摘要

This paper studies the global dynamics of the rolling motion of ships in random beam seas model described by the equation $\begin{aligned} \ddot{x}+ (2\mu + \delta x^2){\dot{x}} + \omega ^2 x-\frac{1}{a^2} x^3=0, \end{aligned}$ $\begin{matrix} \ddot{x} + (2 μ + δ x^{2}) \dot{x} + ω^{2} x - \frac{1}{a^{2}} x^{3} = 0, \end{matrix}$ where $0 < \mu \ll 1$ $0 < μ ≪ 1$ , $0< \delta < 1$ $0 < δ < 1$ , $\omega =1$ $ω = 1$ and $0< a < 3$ $0 < a < 3$ . We fully explain its dynamics in the Poincaré compactification of ${{\mathbb {R}}}^2$ $R^{2}$ .