New insights into a chaotic system with only a Lyapunov stable equilibrium

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.

Economic system entanglement on intra-firm trade portfolios: the impact of counterparty credit ratings on business-to-business credit dynamics

“…The system expressed by (10) shows a point of equilibrium, generically denoted by , which is stable for each ̂> , according to Lyapunov [7]. For every perturbation ̂> , the vicinity with radius that exists around the point of equilibrium delimits the bounds of the system trajectory: for these values, an -limit cycle is formed.…”

Qualitative geometric analysis of traveling wave solutions of the modified equal width Burgers equation

“…In recent years, many scholars concentrated on Jacobi stability of three-dimensional systems and even high-dimensional systems. [41][42][43][44][45][46][47] A common and effective method to explore the complex dynamical behavior of a system is to add a small disturbance to the system, in which the periodic disturbance has a great influence on the system. In their work, 48,49 Saha et al applied a periodic disturbance to the planar autonomous system; the obtaining results explain that the planar system with disturbance presents periodic, quasi-periodic, and chaotic complex behaviors.…”

“…In literature, 40 Jacobi stability of one‐dimensional typical bifurcations (saddle‐node, transcritical, and pitchfork bifurcations) was researched, including the stability of equilibrium points and nonequilibrium region. In recent years, many scholars concentrated on Jacobi stability of three‐dimensional systems and even high‐dimensional systems 41–47 …”

Jagged-shape chaotic attractors of a megastable oscillator with spatially square-wave damping

“…After them, many chaotic oscillators with various equilibrium points were proposed [6]. Chaotic oscillators with a line of equilibria [7], the curve of equilibria [8], and one Lyapunov stable equilibrium [9] are some of them. Investigating the dynamics of chaotic oscillators has been a hot topic [10].…”