In this paper, the research of the Jacobi stability of the Chen system is performed by using the KCC-theory. By associating a nonlinear connection and a Berwald connection, five geometrical invariants of the Chen system are obtained. The Jacobi stability of the Chen system at equilibrium points and a periodic orbit is investigated in terms of the eigenvalues of the deviation curvature tensor. The obtained results show that the origin is always Jacobi unstable, while the Jacobi stability of the other two nonzero equilibrium points depends on the values of the parameters. And a periodic orbit of the Chen system is proved to be also Jacobi unstable. Furthermore, Jacobi stability regions of the Chen system and the Lorenz system are compared. Finally, the dynamical behavior of the components of the deviation vector near the equilibrium points is also discussed.
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.
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.
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