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.

翻译：Kosambi-Cartan-Chern (KCC) 经典理论在微分几何中发展而来，为分析动力系统行为提供了强大方法。在KCC理论中，动力系统性质通过五个几何不变量描述，其中第二个不变量对应于系统的所谓雅可比稳定性。与文献中广泛研究的李雅普诺夫稳定性不同，雅可比稳定性分析近期才借助几何概念与工具展开研究。现有雅可比稳定性分析工作仍停留在理论层面，其算法化与符号化处理问题尚未得到解决。本文针对一类任意维度的常微分方程系统率先开展研究，提出两种基于符号计算的算法方案，用于检验非线性动力系统是否具有雅可比稳定性。第一种方案基于特征多项式复根结构的构造和量词消去方法，能够检测给定动力系统雅可比稳定性的存在性。第二种算法方案利用半代数系统求解方法，可确定给定动力系统具有预设数量雅可比稳定不动点的参数条件。通过多个算例验证了所提算法方案的有效性。