期刊
INTERNATIONAL JOURNAL OF GEOMETRIC METHODS IN MODERN PHYSICS
卷 17, 期 4, 页码 -出版社
WORLD SCIENTIFIC PUBL CO PTE LTD
DOI: 10.1142/S0219887820500620
关键词
Chaotic system; equilibrium point; periodic orbit; invariant; Jacobi stability
资金
- National Natural Science Foundation of China [11961074, 11971414]
- Natural Science Foundation of Guangxi Province [2018GXNSFDA281028, 2017GXNSFAA198234]
- Youth Project of Hunan Provincial Education Department [18B518, 18B082]
- High Level Innovation Team Program from Guangxi Higher Education Institutions of China [[2018] 35]
- Science Technology Program of Yulin Normal University [2017YJKY28]
- Postgraduate Innovation Program of Guangxi University for Nationalities [GXUN-CHXZS2018042]
Little seems to be known about the study of the chaotic system with only Lyapunov stable equilibria from the perspective of differential geometry. Therefore, this paper presents Jacobi analysis of an unusual three-dimensional (3D) autonomous chaotic system. Under certain parameter conditions, this system has positive Lyapunov exponents and only two linear stable equilibrium points, which means that chaotic attractor and Lyapunov stable equilibria coexist. The dynamical behavior of the deviation vector near the whole trajectories (including all equilibrium points) is analyzed in detail. The results show that the value of the deviation curvature tensor at. equilibrium points is only related to parameters; the two equilibrium points of the system are Jacobi stable if the parameters satisfy certain conditions. Particularly, for a specific set of parameters, the linear stable equilibrium points of the system are always Jacobi unstable. A periodic orbit that is Lyapunov stable is also proven to be always Jacobi unstable. Next, Jacobi-stable regions of the Lorenz system, the Chen system and the system under study are compared for specific parameters. It can be found that although these three chaotic systems are very similar, their regions of Jacobi stable parameters are much different. Finally, by comparing Jacobi stability with Lyapunov stability, the obtained results demonstrate that the Jacobi stable parameter region is basically symmetric with the Lyapunov stable parameter region.
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