About the Jacobi Stability of a Generalized Hopf-Langford System through the Kosambi-Cartan-Chern Geometric Theory

In this work, we will consider an autonomous three-dimensional quadratic system of first-order ordinary differential equations, with five parameters and with symmetry relative to the z-axis, which generalize the Hopf-Langford system. By reformulating the system as a system of two second-order ordinary differential equations and using the Kosambi-Cartan-Chern (KCC) geometric theory, we will investigate this system from the perspective of Jacobi stability. We will compute the five invariants of KCC theory which determine the own geometrical properties of this system, especially the deviation curvature tensor. Additionally, we will search for necessary and sufficient conditions on the five parameters of the system in order to reach the Jacobi stability around each equilibrium point.