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<p>The interest in studying the uniform isochronous centers goes back to C. Huygens in the XVII century. Since then, many papers have been published on this subject. In particular, the phase portraits of the polynomial uniform isochronous center up to degree four have been classified. In this paper, we classify the topological phase portraits of polynomial differential systems with a uniform isochronous center, whose nonlinear part is a homogeneous polynomial of degree 5. We prove that there are three distinct topological phase portraits in the Poincaré disc for such polynomial differential systems.</p>

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The Uniform Isochronous Centers with Homogeneous Nonlinearities of Degree 5

  • 期刊论文

  • 发表日期：

    2025年4月14日

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  • Guangfeng Dong,
  • Jaume Llibre

摘要

The interest in studying the uniform isochronous centers goes back to C. Huygens in the XVII century. Since then, many papers have been published on this subject. In particular, the phase portraits of the polynomial uniform isochronous center up to degree four have been classified. In this paper, we classify the topological phase portraits of polynomial differential systems with a uniform isochronous center, whose nonlinear part is a homogeneous polynomial of degree 5. We prove that there are three distinct topological phase portraits in the Poincaré disc for such polynomial differential systems.