Jacobi Stability for T-System

Abstract

In this paper will be considered a three-dimensional autonomous quadratic polynomial system of first-order differential equations with three real parameters, the so-called T-system. This system is symmetric relative to the Oz-axis and represents a special type of the generalized Lorenz system. The approach of this work will consist of the study of the nonlinear dynamics of this system through the Kosambi–Cartan–Chern (KCC) geometric theory. More exactly, we will focus on the associated system of second-order differential equations (SODE) from the point of view of Jacobi stability by determining the five invariants of the KCC theory. These invariants determine the internal geometrical characteristics of the system, and particularly, the deviation curvature tensor is decisive for Jacobi stability. Furthermore, we will look for necessary and sufficient conditions that the system parameters must satisfy in order to have Jacobi stability for every equilibrium point.

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].…”

“…In simpler terms, a system is Lyapunov stable if small disturbances to its state result in the system returning to its original state (or a nearby state) over time. For more detailed information about Lyapunov stability, please refer to [5,6,23,26,28,29].…”