da Silva Vinícius Barros, Vieira João Peres, Leonel Edson Denis
Department of Physics, Universidade Estadual Paulista "Júlio de Mesquita Filho", Campus de Rio Claro, São Paulo 13506-900, Brazil.
Department of Mathematics, Universidade Estadual Paulista "Júlio de Mesquita Filho", Campus de Rio Claro, São Paulo 13506-900, Brazil.
Entropy (Basel). 2024 Aug 30;26(9):745. doi: 10.3390/e26090745.
The detection of limit cycles of differential equations poses a challenge due to the type of the nonlinear system, the regime of interest, and the broader context of applicable models. Consequently, attempts to solve Hilbert's sixteenth problem on the maximum number of limit cycles of polynomial differential equations have been uniformly unsuccessful due to failing results and their lack of consistency. Here, the answer to this problem is finally obtained through information geometry, in which the Riemannian metrical structure of the parameter space of differential equations is investigated with the aid of the Fisher information metric and its scalar curvature R. We find that the total number of divergences of |R| to infinity provides the maximum number of limit cycles of differential equations. Additionally, we demonstrate that real polynomial systems of degree n≥2 have the maximum number of 2(n-1)(4(n-1)-2) limit cycles. The research findings highlight the effectiveness of geometric methods in analyzing complex systems and offer valuable insights across information theory, applied mathematics, and nonlinear dynamics. These insights may pave the way for advancements in differential equations, presenting exciting opportunities for future developments.
由于非线性系统的类型、感兴趣的区域以及适用模型的更广泛背景，微分方程极限环的检测面临挑战。因此，由于结果失败及其缺乏一致性，试图解决希尔伯特关于多项式微分方程极限环最大数量的第十六问题一直未成功。在此，通过信息几何最终获得了该问题的答案，其中借助费希尔信息度量及其标量曲率(R)研究了微分方程参数空间的黎曼度量结构。我们发现(|R|)趋于无穷大的发散总数给出了微分方程极限环的最大数量。此外，我们证明了次数(n\geq2)的实多项式系统具有最多(2(n - 1)(4(n - 1) - 2))个极限环。研究结果突出了几何方法在分析复杂系统中的有效性，并在信息论、应用数学和非线性动力学方面提供了有价值的见解。这些见解可能为微分方程的进展铺平道路，为未来发展带来令人兴奋的机会。
