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On discrete-time polynomial dynamical systems on hypergraphs

Shaoxuan Cui, Guofeng Zhang, Hildeberto Jardón-Kojakhmetov, Ming Cao

TL;DR

This work addresses stability analysis for discrete-time polynomial dynamical systems on hypergraphs, leveraging the tensor Perron–Frobenius theory and $Z$-eigenvalues to characterize the origin's stability. The authors develop results for both uniform (homogeneous) and non-uniform (non-homogeneous) hypergraphs, deriving conservative domains of attraction from system parameters and proposing a feedback control that shifts critical eigenvalue thresholds. Key contributions include explicit domain-of-attraction formulas for uniform and non-uniform cases, relaxation of common eigenvector requirements, and a practical control design to modulate stability regions, all validated by numerical examples. The findings enable open-loop stability assessment and targeted domain manipulation for higher-order network dynamics with potential applications in epidemiology and ecology.

Abstract

This paper studies the stability of discrete-time polynomial dynamical systems on hypergraphs by utilizing the Perron-Frobenius theorem for nonnegative tensors with respect to the tensors Z-eigenvalues and Z-eigenvectors. Firstly, for a multilinear polynomial system on a uniform hypergraph, we study the stability of the origin of the corresponding systems. Next, we extend our results to non-homogeneous polynomial systems on non-uniform hypergraphs. We confirm that the local stability of any discrete-time polynomial system is in general dominated by pairwise terms. Assuming that the origin is locally stable, we construct a conservative (but explicit) region of attraction from the system parameters. Finally, we validate our results via some numerical examples.

On discrete-time polynomial dynamical systems on hypergraphs

TL;DR

This work addresses stability analysis for discrete-time polynomial dynamical systems on hypergraphs, leveraging the tensor Perron–Frobenius theory and -eigenvalues to characterize the origin's stability. The authors develop results for both uniform (homogeneous) and non-uniform (non-homogeneous) hypergraphs, deriving conservative domains of attraction from system parameters and proposing a feedback control that shifts critical eigenvalue thresholds. Key contributions include explicit domain-of-attraction formulas for uniform and non-uniform cases, relaxation of common eigenvector requirements, and a practical control design to modulate stability regions, all validated by numerical examples. The findings enable open-loop stability assessment and targeted domain manipulation for higher-order network dynamics with potential applications in epidemiology and ecology.

Abstract

This paper studies the stability of discrete-time polynomial dynamical systems on hypergraphs by utilizing the Perron-Frobenius theorem for nonnegative tensors with respect to the tensors Z-eigenvalues and Z-eigenvectors. Firstly, for a multilinear polynomial system on a uniform hypergraph, we study the stability of the origin of the corresponding systems. Next, we extend our results to non-homogeneous polynomial systems on non-uniform hypergraphs. We confirm that the local stability of any discrete-time polynomial system is in general dominated by pairwise terms. Assuming that the origin is locally stable, we construct a conservative (but explicit) region of attraction from the system parameters. Finally, we validate our results via some numerical examples.
Paper Structure (8 sections, 6 theorems, 18 equations, 2 figures)

This paper contains 8 sections, 6 theorems, 18 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries on tensors and hypergraphs
  3. Discrete-time polynomial dynamical systems on hypergraphs
  4. Discrete-time polynomial systems on uniform hypergraphs
  5. Discrete-time polynomial systems on non-uniform hypergraphs
  6. Feedback control strategies
  7. Numerical examples and further discussions
  8. Conclusions

Key Result

Lemma 1

If $A \in \mathbb{R}_{++}^{[{{k}}, n]}$, then there exists a Z-eigenvalue $\lambda_0 \geq 0$ and a nonnegative Z-eigenvector $x_0 \neq \mathbf{0}$ of $A$ such that $A x_0^{{{k}}-1}=\lambda_0 x_0$. We call $\lambda_0, x_0$ the Perron-Z-eigenvalue and -eigenvector respectively, and refer to $(\lambda_

Figures (2)

  • Figure 1: Illustration of the infection process. The infection rate of $\beta_1$ with normal edges provides classical pairwise interactions, while the infection rate of $\beta_2$ with hyperedges of three elements provides higher-order group-wise interactions.
  • Figure 2: Points with 'o' are within the domain of attraction of the origin while points with 'x' are out of the domain. The square area (without the boundary) denotes the conservative region of attraction from Theorem \ref{['thm:sysnon2']} and Corollary \ref{['cor:qua']}.

Theorems & Definitions (12)

  • Lemma 1: Theorems 4.5 and 4.6 chang2013survey
  • Theorem 1
  • proof
  • Remark 1
  • Theorem 2
  • proof
  • Remark 2
  • Theorem 3
  • proof
  • Remark 3
  • ...and 2 more