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<p>This is a full study of the dynamics of polynomial planar vector fields whose linear part is a multiple of the identity and whose nonlinear part is a homogeneous polynomial of arbitrary degree <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(n&gt;1\)</EquationSource> </InlineEquation>. It extends previous work by other authors that was mainly concerned with the existence and number of limit cycles. The general results are also applied to two classes of examples where the nonlinearities have degrees 2 and 3, for which we provide a set of phase portraits.</p>

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Qualitative Planar Dynamics with Star Nodes and Homogeneous Nonlinearities: Beyond Hilbert’s 16th Problem

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年7月22日

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  • Begoña Alarcón,
  • Sofia B. S. D. Castro,
  • Isabel S. Labouriau

摘要

This is a full study of the dynamics of polynomial planar vector fields whose linear part is a multiple of the identity and whose nonlinear part is a homogeneous polynomial of arbitrary degree \(n>1\) . It extends previous work by other authors that was mainly concerned with the existence and number of limit cycles. The general results are also applied to two classes of examples where the nonlinearities have degrees 2 and 3, for which we provide a set of phase portraits.