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Jacobi Stability Analysis for Systems of ODEs Using Symbolic Computation

Bo Huang, Dongming Wang, Jing Yang

TL;DR

This paper introduces algorithmic, symbolic methods to analyze Jacobi stability within the Kosambi–Cartan–Chern (KCC) framework for general first-order ODE systems $\dot{\boldsymbol{x}}=\boldsymbol{f}(\boldsymbol{x},\boldsymbol{\mu})$. It presents two schemes: a QE-based approach (Jacobi Test I) to decide the existence of Jacobi-stable fixed points, and a semi-algebraic approach (Jacobi Test II) to determine parameter regions yielding a prescribed number of Jacobi-stable fixed points. Theoretical results connect Jacobi stability to the spectrum of $J^2$, with practical reductions to algebraic conditions and real solution classifications, enabling application to models such as the Brusselator, Cdc2-Cyclin B/Wee1, and Lorenz–Stenflo. Experiments demonstrate the feasibility and limitations of the methods, highlighting that the symbolic approach is most tractable for small to moderate systems and may benefit from model reduction for larger ones. Overall, the work provides a principled, algorithmic pathway to quantify Jacobi stability in nonlinear dynamical systems.

Abstract

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.

Jacobi Stability Analysis for Systems of ODEs Using Symbolic Computation

TL;DR

This paper introduces algorithmic, symbolic methods to analyze Jacobi stability within the Kosambi–Cartan–Chern (KCC) framework for general first-order ODE systems . It presents two schemes: a QE-based approach (Jacobi Test I) to decide the existence of Jacobi-stable fixed points, and a semi-algebraic approach (Jacobi Test II) to determine parameter regions yielding a prescribed number of Jacobi-stable fixed points. Theoretical results connect Jacobi stability to the spectrum of , with practical reductions to algebraic conditions and real solution classifications, enabling application to models such as the Brusselator, Cdc2-Cyclin B/Wee1, and Lorenz–Stenflo. Experiments demonstrate the feasibility and limitations of the methods, highlighting that the symbolic approach is most tractable for small to moderate systems and may benefit from model reduction for larger ones. Overall, the work provides a principled, algorithmic pathway to quantify Jacobi stability in nonlinear dynamical systems.

Abstract

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.
Paper Structure (15 sections, 8 theorems, 57 equations, 1 figure, 1 table)

This paper contains 15 sections, 8 theorems, 57 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. KCC Theory and Jacobi Stability of Dynamical Systems
  3. Theoretical Results on the Jacobi Stability
  4. Algorithmic Analysis of the Jacobi Stability
  5. Jacobi Test I
  6. Jacobi Test II
  7. Examples and Experiments
  8. Illustrative Example: The Brusselator
  9. Other Models and Remarks
  10. The Cdc2-Cyclin B/Wee1 System
  11. The Lorenz--Stenflo System
  12. Conclusion and Future Work
  13. Proof of Lemma \ref{['lem-main-1']}
  14. List of Expressions
  15. The Lorenz--Stenflo System

Key Result

Lemma 1

Let $\bar{\boldsymbol{x}}=(\bar{x}_1,\bar{x}_2,\ldots,\bar{x}_n)$ be an isolated fixed point of system eq1-0. Denoted by $J(\bar{\boldsymbol{x}},\boldsymbol{\mu})$ the Jacobian matrix of system eq1-0 evaluated at the fixed point $\bar{\boldsymbol{x}}$. Then $\bar{\boldsymbol{x}}$ is Jacobi stable if

Figures (1)

  • Figure 1: In the dark-gray and light-gray regions, system \ref{['kcc-ex-2']} has one and zero Jacobi stable fixed point, respectively. The blue curve is defined by $R_0=0$.

Theorems & Definitions (18)

  • Definition 1
  • Lemma 1
  • Theorem 1
  • proof
  • Remark 1
  • Proposition 1
  • proof
  • Remark 2
  • Theorem 2
  • proof
  • ...and 8 more