Finsler geometry and Jacobi stability analysis of higher-order ODEs using Kosambi–Cartan–Chern theory

In this paper, we use the geometric invariants of the KCC theory to characterize the time evolution of systems of second and higher-order ordinary differential equations (ODEs). In particular, we analyze some fourth-order ODEs, admitting a Lagrangian representation, which may be expressed as systems of second-order ODEs from the point of view of Jacobi stability of their equilibrium points. Jacobi stability describes the robustness of a trajectory in a dynamical system to small perturbations, indicating whether nearby paths converge or diverge. The study demonstrates the utility of the Kosambi–Cartan–Chern (KCC) theory in providing a geometric framework for stability analysis, highlighting its ability to complement traditional Lyapunov stability assessments.