用代数曲线分离的不连续分段线性微分系统的第16 Hilbert问题 \(y=x^{n}\)

IF 0.9 3区数学 Q3 MATHEMATICS, APPLIED Mathematical Physics, Analysis and Geometry Pub Date : 2023-09-19 DOI:10.1007/s11040-023-09467-4

Jaume Llibre, Claudia Valls

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引用次数: 0

摘要

我们考虑由线性中心或线性哈密顿鞍构成的平面分段不连续微分系统，它们由代数曲线\(y=x^n\)与\(n \ge 2\)分隔。我们用一种很简单的方法给出了这些微分系统关于n的极限环个数的上界，在这种情况下证明了扩展的16阶希尔伯特问题。特别地，我们证明了对于\(n=2\)这个边界是可以达到的。

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The 16th Hilbert Problem for Discontinuous Piecewise Linear Differential Systems Separated by the Algebraic Curve \(y=x^{n}\)

We consider planar piecewise discontinuous differential systems formed by either linear centers or linear Hamiltonian saddles and separated by the algebraic curve \(y=x^n\) with \(n \ge 2\). We provide in a very short way an upper bound of the number of limit cycles that these differential systems can have in terms of n, proving the extended 16th Hilbert problem in this case. In particular, we show that for \(n=2\) this bound can be reached.

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来源期刊

Mathematical Physics, Analysis and Geometry 数学-物理：数学物理

CiteScore

2.10

自引率

0.00%

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审稿时长

>12 weeks

期刊介绍： MPAG is a peer-reviewed journal organized in sections. Each section is editorially independent and provides a high forum for research articles in the respective areas. The entire editorial board commits itself to combine the requirements of an accurate and fast refereeing process. The section on Probability and Statistical Physics focuses on probabilistic models and spatial stochastic processes arising in statistical physics. Examples include: interacting particle systems, non-equilibrium statistical mechanics, integrable probability, random graphs and percolation, critical phenomena and conformal theories. Applications of probability theory and statistical physics to other areas of mathematics, such as analysis (stochastic pde''s), random geometry, combinatorial aspects are also addressed. The section on Quantum Theory publishes research papers on developments in geometry, probability and analysis that are relevant to quantum theory. Topics that are covered in this section include: classical and algebraic quantum field theories, deformation and geometric quantisation, index theory, Lie algebras and Hopf algebras, non-commutative geometry, spectral theory for quantum systems, disordered quantum systems (Anderson localization, quantum diffusion), many-body quantum physics with applications to condensed matter theory, partial differential equations emerging from quantum theory, quantum lattice systems, topological phases of matter, equilibrium and non-equilibrium quantum statistical mechanics, multiscale analysis, rigorous renormalisation group.

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