# American Institute of Mathematical Sciences

## Dynamics at infinity and Jacobi stability of trajectories for the Yang-Chen system

 1 Guangxi Colleges and Universities Key Laboratory of Complex System Optimization, and Big Data Processing, Yulin Normal University, Yulin 537000, China 2 College of Science, Guangxi University for Nationalities, Guangxi, 530006, China 3 School of Mathematics and Physics, China University of Geosciences (Wuhan), Wuhan, Hubei 430074, China 4 Zhejiang Institute, China University of Geosciences, Hangzhou, Zhejiang 311305, China

* Corresponding author: weizhouchao@163.com

Received  January 2020 Revised  May 2020 Published  August 2020

Fund Project: The first author is supported by National Natural Science Foundation of China (Grant No. 11961074), Natural Science Foundation of Guangxi Province (Grant Nos. 2018GXNSFDA281028, 2017GXNSFAA198234), the High Level Innovation Team Program from Guangxi Higher Education Institutions of China (Document No. [2018] 35), and the Science Technology Program of Yulin Normal University (Grant No. 2017YJKY28). The second author is supported by the Postgraduate Innovation Program of Guangxi University for Nationalities (Grant No. GXUN-CHXZS2018042). The third author is supported by National Natural Science Foundation of China (Grant No. 11772306), Zhejiang Provincial Natural Science Foundation of China under Grant (No.LY20A020001), and the Fundamental Research Funds for the Central Universities, China University of Geosciences (CUGGC05)

The present work is devoted to giving new insights into a chaotic system with two stable node-foci, which is named Yang-Chen system. Firstly, based on the global view of the influence of equilibrium point on the complexity of the system, the dynamic behavior of the system at infinity is analyzed. Secondly, the Jacobi stability of the trajectories for the system is discussed from the viewpoint of Kosambi-Cartan-Chern theory (KCC-theory). The dynamical behavior of the deviation vector near the whole trajectories (including all equilibrium points) is analyzed in detail. The obtained results show that in the sense of Jacobi stability, all equilibrium points of the system, including those of the two linear stable node-foci, are Jacobi unstable. These studies show that one might witness chaotic behavior of the system trajectories before they enter in a neighborhood of equilibrium point or periodic orbit. There exists a sort of stability artifact that cannot be found without using the powerful method of Jacobi stability analysis.

Citation: Yongjian Liu, Qiujian Huang, Zhouchao Wei. Dynamics at infinity and Jacobi stability of trajectories for the Yang-Chen system. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020235
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Attractors of Yang-Chen system with (a) $a = 10$, $b = 8/3$, and $c = 16$; (b) $a = 35$, $b = 3$, and $c = 35$
Dynamics of the Yang-Chen system near the sphere at infinity in the local charts $U_1$ for (blue) $(a, b, c) = (0.5, 1.01, 1)$ with initial conditions $(z_1(0), z_2(0), z_(0)) = (0.03, 0.03, -0.03)$, (red) $(a, b, c) = (1, 1.01, 1)$ with initial conditions $(z_1(0), z_2(0), z_(0)) = (0.03, 0.03, -0.03)$, (black) $(a, b, c) = (0.1, 1.01, 1)$ with initial conditions $(z_1(0), z_2(0), z_(0)) = (0.03, 0.03, -0.01)$, respectively
Dynamics of the Yang-Chen system near the sphere at infinity in the local charts $V_1$ (blue) $(a, b, c) = (0.5, 1.01, 1)$ with initial conditions $(z_1(0), z_2(0), z_(0)) = (0.03, 0.03, 0.03)$, (red) $(a, b, c) = (1, 1.01, 1)$ with initial conditions $(z_1(0), z_2(0), z_(0)) = (0.03, 0.03, 0.03)$, (black) $(a, b, c) = (0.1, 1.01, 1)$ with initial conditions $(z_1(0), z_2(0), z_(0)) = (0.03, 0.03, 0.01)$, respectively
Phase portrait of the system (10), which corresponds to the phase portrait of the Yang-Chen system at infinity in the local charts $U_2$
Phase portrait of the system (12), which corresponds to the phase portrait of the Yang-Chen system at infinity in the local charts $U_3$
Phase portrait of system (1) at infinity
Time variation of the deviation vector and its curvature near $E_{1}$, for $a = 35$, $b = 3$
Time variation of instability exponent $\delta(E_{1})$ for $a = 35$, $b = 3$, and different values of $c$
Time variation of the deviation vector and its curvature near $E_{2, 3}$ with $a = 35, b = 3$. Initial conditions used to integrate deviation equations are $\xi_{1}(0) = \xi_{2}(0) = 0$, $\dot{\xi}_{1}(0) = \dot{\xi}_{2}(0) = 10^{-6}$
Time variation of curvature $\kappa_{0}$ of deviation vector near equilibrium points $E_{1}$ with $a = 35, b = 3$
Time variation of curvature $\kappa_{0}$ of deviation vector near equilibrium points $E_{2, 3}$ with $a = 35, b = 3$
A large version of Fig. 11 at time $0.25$ to $0.55$
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