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<p>In this paper we are concerned with the characterization of globally asymptotic stable planar polynomial differential systems. Such systems are very important in the qualitative theory of differential equations because all their trajectories are defined for all positive time and the <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1326_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(\omega \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ω</mi> </math></EquationSource> </InlineEquation>-limit of any point is a unique equilibrium point. The characterization of quadratic planar polynomial differential systems that are globally asymptotic stable is known. Here, we characterize all planar cubic polynomial differential systems with a nilpotent point at the origin, without quadratic terms and with an invariant straight line through the origin, that are globally asymptotically stable.</p>

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Global Asymptotic Stability in Cubic Systems with a Nilpotent Point at the Origin and with An Invariant Straight Line

  • 期刊论文

  • 发表日期：

    2025年7月7日

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  • Érika Diz-Pita,
  • M. Victoria Otero-Espinar,
  • Claudia Valls

摘要

In this paper we are concerned with the characterization of globally asymptotic stable planar polynomial differential systems. Such systems are very important in the qualitative theory of differential equations because all their trajectories are defined for all positive time and the \(\omega \) ω -limit of any point is a unique equilibrium point. The characterization of quadratic planar polynomial differential systems that are globally asymptotic stable is known. Here, we characterize all planar cubic polynomial differential systems with a nilpotent point at the origin, without quadratic terms and with an invariant straight line through the origin, that are globally asymptotically stable.