# Chaos in a continuous-time Boolean network

## TL;DR

This paper investigates chaos in continuous-time Boolean networks, demonstrating that even simple 4-dimensional systems with Boolean interactions can exhibit complex, chaotic dynamics similar to Smale horseshoes.

## Contribution

It provides a concrete example of chaos in a continuous-time Boolean network, linking discrete map properties to continuous dynamics and revealing complex behaviors in low-dimensional systems.

## Key findings

- Presence of chaotic dynamics in a 4D Boolean network
- Identification of a Smale horseshoe-like structure
- Chaotic behavior in systems with four or more variables

## Abstract

Continuous-time systems with switch-like behaviour occur in chemical kinetics, gene regulatory networks and neural networks. Networks with hard switching, as a limiting case of smooth sigmoidal switching, retain the richest possible range of behaviors but are mathematically more tractable. The form of an underlying discrete (fractional-linear) map encodes information on existence, stability and exact periods of periodic orbits. In richly connected structures with four or more variables, aperiodic behaviour can occur. We investigate a simple 4-dimensional example with Boolean interaction terms in which a Smale horseshoe-like object reveals chaotic dynamics.