# Asynchronous discrete dynamical systems

## TL;DR

This paper investigates the effects of asynchronous updates in coupled discrete-time systems, constructing complex models to analyze stability and demonstrate how asynchronicity influences dynamical behavior.

## Contribution

It introduces a novel framework for modeling asynchronous coupled systems using complex-valued and higher-dimensional systems, providing new insights into stability analysis.

## Key findings

- Asynchronicity significantly impacts system stability.
- Constructed complex models effectively analyze asynchronous dynamics.
- Examples show both stability and instability scenarios.

## Abstract

We study two coupled discrete-time equations with different (asynchronous) periodic time scales. The coupling is of the type sample and hold, i.e., the state of each equation is sampled at its update times and held until it is read as an input at the next update time for the other equation. We construct an interpolating two-dimensional complex-valued system on the union of the two time scales and an extrapolating four-dimensional system on the intersection of the two time scales. We discuss stability by several results, examples and counterexamples in various frameworks to show that the asynchronicity can have a significant impact on the dynamical properties.