Algorithmic analysis of Jacobi stability for systems of second order ordinary differential equations

This paper aims to provide an algorithmic method for analyzing Jacobi stability of systems of second order ordinary differential equations (ODEs) using Kosambi–Cartan–Chern (KCC) theory. We develop an efficient symbolic program using Maple for computing the second KCC invariant for systems of second order ODEs in arbitrary dimension. The program allows us to systematically analyze Jacobi stability of a system of second order ODEs by means of real solving and solution classification using symbolic computation. Moreover, we present an alternative method for detecting Jacobi stability for systems of second order ODEs by making use of parametric discriminants for multiplicities of univariate polynomials. This alternative method is based on the expression of the deviation equation, which appears to be useful for detecting Jacobi stability of non-parametric models. The effectiveness of the proposed approach is illustrated by a model of wound strings, a two-dimensional airfoil model with cubic nonlinearity in supersonic flow and a 3-DOF tractor seat-operator model. The computational results on Jacobi stability of these models are further verified by numerical simulations. Our algorithmic approach allows us to detect hand-guided computation errors in published papers. In addition, we conduct a comparative analysis of the Jacobi stability and the linear stability of these models.