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

Abstract

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.

Jacobi Stability Analysis of Liu System: Detecting Chaos

“…To ensure the integrity of the paper, the basic concepts of the KCC geometric theory and Jacobi stability are briefly reviewed. For detailed discussions on the mathematical aspects of these topics, see [5,6,23,[26][27][28][29].…”

“…Obviously, ∥Y∥ 2 := ⟨⟨Y, Y⟩⟩ = 1. When t approaches 0 + , the trajectories of system (2) are [5,23,31] (1) bunching together if and only if the real part of the eigenvalues of P i j (0) are strictly negative. That is, bunching together if and only if ∥ξ(t)∥ < t 2 , t ≈ 0 + .…”

“…Practically, robustness is a measure of the model's sensitivity and responsiveness to changes in its internal parameters and surroundings. The Jacobi stability analysis of dynamical models can classify the roles of stable and unstable fixed points, and it has been widely used, such as in the Chen System [6], Lotka-Volterra system [18], Lorenz system [5], Navier-Stokes system [19], Rikitake system [20], modified Chua circuit system [21] and other systems [22][23][24][25].…”

KCC Theory of the Oregonator Model for Belousov-Zhabotinsky Reaction

“…Jacobi stability analysis for different systems like Lorenz system [12], Chua circuit system [13] and other systems [14][15][16][17][18][19][20][21] have been studied. According to the articles [22,23], one of the geometrical invariants that identifies the beginning of chaos is the deviation vector from the so-called Jacobi equation. Jacobi stability has been analyzed by a large number of authors in the past years as an effective method for predicting chaotic behaviour of the systems [24][25][26].…”

Kosambi–Cartan–Chern Stability in the Intermediate Nonequilibrium Region of the Brusselator Model

“…Because a dynamic system is often described by differential equations, KCC theory has been applied to the geometric aspects of various dynamic structures, including those of physical (e.g., [Kumar et al, 2019;Krylova et al, 2019;Alawadi et al, 2020;Liu et al, 2020;Klën, & Molina, 2020]), biological (e.g., [Antonelli et al, 1993;Yamasaki & Yajima, 2013;Antonelli et al, 2014;Antonalli et al, 2019;Kolebaja, & Popoola, 2019]) and general (e.g., [Gupta, & Yadav, 2017;Chen, & Yin, 2019;) systems. Moreover, it has been applied in mathematics to resolve the inverse problem of updating the general parameters of dynamic systems [Sulimov et al, 2018], and the geometric parameters of complex systems, including chaotic ones (e.g., [Oiwa, & Yajima, 2017;Huang et al, 2019;Chen et al, 2020;Feng et al, 2020;Liu et al, 2021]).…”