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Article Contents

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

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.

Mathematics Subject Classification: Primary: 34D20, 34D45; Secondary: 34K18.

 Citation:

• Figure 1.  Attractors of Yang-Chen system with (a) $a = 10$, $b = 8/3$, and $c = 16$; (b) $a = 35$, $b = 3$, and $c = 35$

Figure 2.  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

Figure 3.  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

Figure 4.  Phase portrait of the system (10), which corresponds to the phase portrait of the Yang-Chen system at infinity in the local charts $U_2$

Figure 5.  Phase portrait of the system (12), which corresponds to the phase portrait of the Yang-Chen system at infinity in the local charts $U_3$

Figure 6.  Phase portrait of system (1) at infinity

Figure 7.  Time variation of the deviation vector and its curvature near $E_{1}$, for $a = 35$, $b = 3$

Figure 8.  Time variation of instability exponent $\delta(E_{1})$ for $a = 35$, $b = 3$, and different values of $c$

Figure 9.  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 $\xi_{1}(0) = \xi_{2}(0) = 0$, $\dot{\xi}_{1}(0) = \dot{\xi}_{2}(0) = 10^{-6}$

Figure 10.  Time variation of curvature $\kappa_{0}$ of deviation vector near equilibrium points $E_{1}$ with $a = 35, b = 3$

Figure 11.  Time variation of curvature $\kappa_{0}$ of deviation vector near equilibrium points $E_{2, 3}$ with $a = 35, b = 3$

Figure 12.  A large version of Fig. 11 at time $0.25$ to $0.55$

Figures(12)