数学物理学报 ›› 2021, Vol. 41 ›› Issue (3): 783-796.
Rabinovich系统的Jacobi分析
刘永建1,*(
),黄秋健2
- 1 玉林师范学院 广西高校复杂系统优化与大数据处理重点实验室 广西玉林 537000
2 广西民族大学理学院 & 广西大学附属中学 南宁 530006
-
收稿日期:2020-03-08出版日期:2021-06-26发布日期:2021-06-09 -
通讯作者:刘永建 E-mail:liuyongjianmaths@126.com -
基金资助:国家自然科学基金(11961074);广西自然科学基金重点项目(2018GXNSFDA281028);广西高校高水平创新团队项目([2018]35);玉林师范学院高层次人才启动项目(G2019ZK51)
Jacobi Analysis of the Rabinovich System
Yongjian Liu1,*(),Qiujian Huang2
- 1 Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing, Yulin Normal University, Guangxi Yulin 537000
2 College of Sciences, Guangxi University for Nationalities & High School Affiliated to Guangxi University, Nanning 530006
-
Received:2020-03-08Online:2021-06-26Published:2021-06-09 -
Contact:Yongjian Liu E-mail:liuyongjianmaths@126.com -
Supported by:the NSFC(11961074);the NSF of Guangxi Province(2018GXNSFDA281028);the High Level Innovation Team Program from Guangxi Higher Education Institutions of China([2018]35);the Senior Talent Research Foundation of Yulin Normal University(G2019ZK51)
摘要：
该文运用微分几何技术开展三维微分系统的复杂性研究.基于Kosambi-Cartan-Chern（KCC）理论，从系统轨线的任意点出发，分析三维Rabinovich系统的Jacobi稳定性态，并给出系统所有平衡点的Jacobi稳定的条件；在获得系统平衡点附近偏离向量及其分量的时间演化的基础上，通过引入不稳定性指数和曲率，同时结合数值仿真对系统的混沌机理进行探讨性分析，数值结果有力地验证了已有的理论分析结果.
中图分类号:
- O175
引用本文
刘永建,黄秋健. Rabinovich系统的Jacobi分析[J]. 数学物理学报, 2021, 41(3): 783-796.
Yongjian Liu,Qiujian Huang. Jacobi Analysis of the Rabinovich System[J]. Acta mathematica scientia,Series A, 2021, 41(3): 783-796.
图 1
Rabinovich系统混沌吸引子的相图($ v_{1} = 1 $, $ v_{2} = 4 $, $ v_{3} = 1 $, $ h = 6.75 $)"
表 1
Rabinovich系统平衡点的稳定性($v_1 = v_3 = 1$, $v_2 = 4$)"
| 参数取值 | 平衡点 | 线性稳定性态 | Jacobi稳定性态 |
| | 不稳定 | 不稳定 | |
| | 不稳定 | 不稳定 | |
| | | 不稳定 | 不稳定 |
| | 不稳定 | 不稳定 | |
| | | 不稳定 | 不稳定 |
| | 不稳定 | 不稳定 | |
| | 不稳定 | 不稳定 | |
| | 不稳定 | 不稳定 |
图 2
Rabinovich系统混沌吸引子的相图($v_{1} = 1$, $v_{2} = 4$, $v_{3} = 1$, $h = 5.5$(左上); $h = 6.75$(右上); $h = 10$(左下); $h = 15$(右下))."
图 3
偏离向量$ \xi $及其分量$ \xi_{i} $在平衡点$ S_{0} $附近的时间演化, $ i = 1, 2, 3 $. $ v_{1} = 1, v_{2} = 4, $ $ v_{3} = 1 $, $ h = 5.5 $(蓝色曲线); $ h = 6.75 $(红色曲线); $ h = 10 $(黑色曲线); $ h = 15 $(紫色曲线). 初始条件$ \xi_{1}(0) = \xi_{2}(0) = \xi_{3}(0) = 0 $, $ \dot{\xi}_{1}(0) = \dot{\xi}_{2}(0) = \dot{\xi}_{3}(0) = 10^{-13} $."
图 4
不稳定性指数$ \delta(S_{0}) $的时间演化结果. $ v_{1} = 1, v_{2} = 4, v_{3} = 1 $, $ h = 5.5 $(蓝色曲线); $ h = 6.75 $(红色曲线); $ h = 10 $(黑色曲线); $ h = 15 $(紫色曲线). 初始条件$ \xi_{1}(0) = \xi_{2}(0) = \xi_{3}(0) = 0 $, $ \dot{\xi}_{1}(0) = \dot{\xi}_{2}(0) = \dot{\xi}_{3}(0) = 10^{-13} $."
图 5
偏离向量$ \xi $及其分量$ \xi_{i} $在平衡点$ S_{1} $附近的时间演化, $ i = 1, 2, 3 $. $ v_{1} = 1, v_{2} = 4, $ $ v_{3} = 1 $, $ h = 5.5 $(蓝色曲线); $ h = 6.75 $(红色曲线); $ h = 10 $(黑色曲线); $ h = 15 $(紫色曲线). 初始条件$ \xi_{1}(0) = \xi_{2}(0) = \xi_{3}(0) = 0 $, $ \dot{\xi}_{1}(0) = \dot{\xi}_{2}(0) = \dot{\xi}_{3}(0) = 10^{-4} $."
图 6
不稳定性指数$ \delta(S_{1}) $的时间演化. $ v_{1} = 1, v_{2} = 4, v_{3} = 1 $, $ h = 5.5 $(蓝色曲线); $ h = 6.75 $(红色曲线); $ h = 10 $(黑色曲线); $ h = 15 $(紫色曲线). 初始条件$ \xi_{1}(0) = \xi_{2}(0) = \xi_{3}(0) = 0 $, $ \dot{\xi}_{1}(0) = \dot{\xi}_{2}(0) = \dot{\xi}_{3}(0) = 10^{-4} $."
图 7
偏离向量$ \xi $及其分量$ \xi_{i} $在平衡点$ S_{2} $附近的时间演化, $ i = 1, 2, 3 $. $ v_{1} = 1, v_{2} = 4, $ $ v_{3} = 1 $, $ h = 5.5 $(蓝色曲线); $ h = 6.75 $(红色曲线); $ h = 10 $(黑色曲线); $ h = 15 $(紫色曲线). 初始条件$ \xi_{1}(0) = \xi_{2}(0) = \xi_{3}(0) = 0 $, $ \dot{\xi}_{1}(0) = \dot{\xi}_{2}(0) = \dot{\xi}_{3}(0) = 10^{-4} $."
图 8
不稳定性指数$ \delta(S_{2}) $的时间演化. $ v_{1} = 1, v_{2} = 4, v_{3} = 1 $, $ h = 5.5 $(蓝色曲线); $ h = 6.75 $(红色曲线); $ h = 10 $(黑色曲线); $ h = 15 $(紫色曲线). 初始条件$ \xi_{1}(0) = \xi_{2}(0) = \xi_{3}(0) = 0 $, $ \dot{\xi}_{1}(0) = \dot{\xi}_{2}(0) = \dot{\xi}_{3}(0) = 10^{-4} $."
图 9
在平衡点$ S_{0} $(左上), $ S_{1} $(右上), $ S_{2} $(下方)处的曲率的时间演化. $ v_{1} = 1, v_{2} = 4, $ $ v_{3} = 1 $, $ h = 5.5 $(蓝色曲线); $ h = 6.75 $(红色曲线); $ h = 10 $(黑色曲线); $ h = 15 $(紫色曲线). 初始条件$ \xi_{1}(0) = \xi_{2}(0) = \xi_{3}(0) = 0 $, $ \dot{\xi}_{1}(0) = \dot{\xi}_{2}(0) = \dot{\xi}_{3}(0) = 10^{-4} $."
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