方程的相位图 $$ddot{x}+ax\dot{x}+bx^{3}=0$$

IF 0.8 4区数学 Q3 MATHEMATICS, APPLIED

Regular and Chaotic Dynamics Pub Date : 2024-09-05 DOI:10.1134/s1560354724560053

Jaume Llibre, Claudia Valls

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引用次数: 0

摘要

二阶微分方程((a,bin\mathbb{R}\)(ddot{x}+ax\dot{x}+bx^{3}=0)已经被多位学者研究，这主要是由于它的应用。在这里，我们首次根据其参数 $a$ 和 $b$ 对其所有相位肖像进行了分类。这种分类是在庞加莱圆盘中进行的，目的是控制从无穷大逃逸或来自无穷大的轨道。我们证明，在与二阶微分方程相关的一阶微分系统的Poincaré圆盘中，正好有六个拓扑不同的相位图。此外，我们还证明了该系统始终是可积分的，并明确提供了其第一积分。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

$Phase Portraits of the Equation $$\ddot{x}+ax\dot{x}+bx^{3}=0$$$

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Phase Portraits of the Equation $$\ddot{x}+ax\dot{x}+bx^{3}=0$$

The second-order differential equation $\ddot{x}+ax\dot{x}+bx^{3}=0$ with $a,b\in\mathbb{R}$ has been studied by several authors mainly due to its applications. Here, for the first time, we classify all its phase portraits according to its parameters $a$ and $b$. This classification is done in the Poincaré disc in order to control the orbits that escape or come from infinity. We prove that there are exactly six topologically different phase portraits in the Poincaré disc of the first-order differential system associated to the second-order differential equation. Additionally, we show that this system is always integrable, providing explicitly its first integrals.

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来源期刊

Regular and Chaotic Dynamics 物理-力学

CiteScore

2.50

自引率

7.10%

发文量

审稿时长

>12 weeks

期刊介绍： Regular and Chaotic Dynamics (RCD) is an international journal publishing original research papers in dynamical systems theory and its applications. Rooted in the Moscow school of mathematics and mechanics, the journal successfully combines classical problems, modern mathematical techniques and breakthroughs in the field. Regular and Chaotic Dynamics welcomes papers that establish original results, characterized by rigorous mathematical settings and proofs, and that also address practical problems. In addition to research papers, the journal publishes review articles, historical and polemical essays, and translations of works by influential scientists of past centuries, previously unavailable in English. Along with regular issues, RCD also publishes special issues devoted to particular topics and events in the world of dynamical systems.