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：在微分几何中发展起来的Kosambi-Cartan-Chern (KCC) 经典理论为分析动力系统的行为提供了强有力的方法。在KCC理论中，动力系统的性质通过五个几何不变量来描述，其中第二个不变量对应于系统的所谓雅可比稳定性。与文献中已被广泛研究的李雅普诺夫稳定性不同，雅可比稳定性的分析是较近期才开始利用几何概念和工具进行的。现有关于雅可比稳定性分析的工作仍停留在理论层面，而算法化与符号化处理雅可比稳定性分析的问题尚未得到解决。本文针对一类任意维度的常微分方程组系统，首次开展该问题的研究，并提出两种利用符号计算来检验非线性动力系统是否可能呈现雅可比稳定性的算法方案。第一种方案基于特征多项式复根结构的构造以及量词消去方法，能够检测给定动力系统是否存在雅可比稳定性。第二种算法方案利用半代数系统求解方法，能够确定使给定动力系统具有指定数量雅可比稳定不动点的参数条件。通过若干实例验证了所提算法方案的有效性。