New insights into a chaotic system with only a Lyapunov stable equilibrium. (18th June 2020)

• Record Type: Journal Article
• Authors: Chen, Biyu; Liu, Yongjian; Wei, Zhouchao; Feng, Chunsheng
• Is Part Of: Mathematical methods in the applied sciences. Volume 43:Number 15(2020)
• Page Start: 9262
• Page End: 9279
• Publication Date: 2020-06-18
• Subjects: chaotic system -- dynamics at infinity -- Jacobi analysis -- stable equilibrium; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.6619
• Languages: English
• ISSNs: 0170-4214

Abstract

Abstract : This paper gives some new insights into a chaotic system. The considered system has only one Lyapunov stable equilibrium and positive Lyapunov exponent in some certain parameter range, which means there coexists chaotic attractors and Lyapunov stable equibirium in system. First, from the perspective of analyzing the global structure of the system, based on the Poincaré compactification technique, the complete description of its dynamical behavior on the sphere at infinity is presented. The obtaining results show that its global dynamical behavior is very complex. Second, from a geometric perspective, the Jacobi stability of the system is investigated, including the unique equilibrium and a periodic orbit. Interestingly, we find that the unique Lyapunov stable equilibrium is always Jacobi unstable, a Lyapunov stable periodic orbit falls into both Jacobi stable and Jacobi unstable regions. It is shown that we might witness chaotic behavior of the system before the trajectories enter a neighborhood of the equilibrium point. It is hoped that the investigation of this paper can contribute to better understand and study the complex chaotic system with stable equilibria.