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Consider a dynamical system with a differential equation as \(\dot{x}_{1} \equiv \frac{{dx_{1} }}{dt} = P(x_{1} ,x_{2} ), \, \dot{x}_{2} \equiv \frac{{dx_{2} }}{dt} = Q(x_{1} ,x_{2} )\) where \(P(x_{1} ,x_{2} )\) and \(Q(x_{1} ,x_{2} )\) are real polynomials of degree \(n\) . The second part of Hilbert's 16th problem is to decide an upper bound for the number of limit cycles in polynomial vector fields of degree \(n\) and, similar to the first part, investigate their relative positions. The original problem can be found.

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Introduction

  • 章节

  • 发表日期：

    2025年

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  • Albert C. J. Luo

摘要

Consider a dynamical system with a differential equation as \(\dot{x}_{1} \equiv \frac{{dx_{1} }}{dt} = P(x_{1} ,x_{2} ), \, \dot{x}_{2} \equiv \frac{{dx_{2} }}{dt} = Q(x_{1} ,x_{2} )\) where \(P(x_{1} ,x_{2} )\) and \(Q(x_{1} ,x_{2} )\) are real polynomials of degree \(n\) . The second part of Hilbert's 16th problem is to decide an upper bound for the number of limit cycles in polynomial vector fields of degree \(n\) and, similar to the first part, investigate their relative positions. The original problem can be found.