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<p>In the framework of the spatial circular Hill three-body problem, we illustrate the application of symplectic invariants to analyze the network structure of symmetric periodic orbits families. The extensive collection of families within this problem constitutes a complex network, fundamentally comprising the so-called basic families of periodic solutions, including the orbits of the satellite <i>g</i>, <i>f</i>, the libration (Lyapunov) <i>a</i>,&#xa0;<i>c</i>, and collision <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10569_2025_10241_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathscr {B}}_0\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">B</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> families. Since the Conley–Zehnder index leads to a grading on the local Floer homology and its Euler characteristics, a bifurcation invariant, the computation of those indices facilitates the construction of well-organized bifurcation graphs depicting the interconnectedness among families of periodic solutions. The critical importance of the symmetries of periodic solutions in comprehending the interaction among these families is demonstrated.</p>

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Studying network of symmetric periodic orbit families of the Hill problem via symplectic invariants

  • 期刊论文

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  • 发表日期：

    2025年3月13日

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  • Cengiz Aydin,
  • Alexander Batkhin

摘要

In the framework of the spatial circular Hill three-body problem, we illustrate the application of symplectic invariants to analyze the network structure of symmetric periodic orbits families. The extensive collection of families within this problem constitutes a complex network, fundamentally comprising the so-called basic families of periodic solutions, including the orbits of the satellite g, f, the libration (Lyapunov) a, c, and collision \({\mathscr {B}}_0\) B 0 families. Since the Conley–Zehnder index leads to a grading on the local Floer homology and its Euler characteristics, a bifurcation invariant, the computation of those indices facilitates the construction of well-organized bifurcation graphs depicting the interconnectedness among families of periodic solutions. The critical importance of the symmetries of periodic solutions in comprehending the interaction among these families is demonstrated.