# Stable Desynchronization for Wireless Sensor Networks: (III) Stability Analysis

## TL;DR

This paper analyzes the stability of desynchronization algorithms in wireless sensor networks using dynamical systems, eigenvalue bounds, and stability theorems, providing conditions for system stability at equilibrium.

## Contribution

It introduces a formal stability analysis framework for desynchronization algorithms, extending previous work with eigenvalue bounds and stability criteria.

## Key findings

- Eigenvalues determine stability conditions.
- Stability depends on the number of nodes within certain bounds.
- The analysis confirms stability at equilibrium under specified conditions.

## Abstract

In this paper, we use dynamical systems to analyze stability of desynchronization algorithms at equilibrium. We start by illustrating the equilibrium of a dynamic systems and formalizing force components and time phases. Then, we use Linear Approximation to obtain Jaconian (J) matrixes which are used to find the eigenvalues. Next, we employ the Hirst and Macey theorem and Gershgorins theorem to find the bounds of those eigenvalues. Finally, if the number of nodes (n) is within such bounds, the systems are stable at equilibrium. (This paper is the last part of the series Stable Desynchronization for Wireless Sensor Networks - (I) Concepts and Algorithms (II) Performance Evaluation (III) Stability Analysis)