Jacobi stability analysis for systems of ODEs with symbolic computation

The classical theory of Kosambi–Cartan–Chern (KCC) developed in differential geometry provides a powerful method for analyzing the behaviors of dynamical systems. In the KCC theory, the properties of a dynamical system are described in terms of five geometrical invariants, of which the second corres...

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Bibliographic Details
Published in:Journal of symbolic computation Vol. 132; p. 102478
Main Authors: Huang, Bo, Wang, Dongming, Wang, Xinyu, Yang, Jing
Format: Journal Article
Language:English
Published: Elsevier Ltd 01.01.2026
Subjects:
Algorithmic approach
Jacobi stability
KCC theory
Quantifier elimination
Semi-algebraic system
Symbolic computation
68W30
Quantifier elimination
KCC theory
Semi-algebraic system
Jacobi stability
Algorithmic approach
Symbolic computation
34C07
ISSN:0747-7171
Online Access:Get full text
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Summary:The classical theory of Kosambi–Cartan–Chern (KCC) developed in differential geometry provides a powerful method for analyzing the behaviors of dynamical systems. In the KCC theory, the properties of a dynamical system are described in terms of five geometrical invariants, of which the second corresponds to the so-called Jacobi stability of the system. Different from that of the Lyapunov stability that has been studied extensively in the literature, the analysis of the Jacobi stability has been investigated more recently using geometrical concepts and tools. It turns out that the existing work on the Jacobi stability analysis remains theoretical and the problem of algorithmic and symbolic treatment of Jacobi stability analysis has yet to be addressed. In this paper, we initiate our study on the problem for a class of ODE systems of arbitrary dimension and propose two algorithmic schemes using symbolic computation to check whether a nonlinear dynamical system may exhibit Jacobi stability. The first scheme, based on the construction of the complex root structure of a characteristic polynomial and on the method of quantifier elimination, is capable of detecting the existence of the Jacobi stability of the given dynamical system. The second algorithmic scheme exploits the method of semi-algebraic system solving and allows one to determine conditions on the parameters for a given dynamical system to have a prescribed number of Jacobi stable fixed points. Several examples are presented to demonstrate the effectiveness of the proposed algorithmic schemes. The computational results on Jacobi stability of these examples are further verified by numerical simulations.
ISSN:0747-7171
DOI:10.1016/j.jsc.2025.102478