From discrete to continuous dynamics and back: How large is 1?

TL;DR

This paper explores the relationship between discrete and continuous dynamical systems, highlighting how the size of the time step affects system behavior, especially in the context of the Verhulst logistic equation.

Contribution

It provides a detailed analysis of how the size of the time step influences the dynamics when transitioning between discrete and continuous models, with insights rooted in numerical analysis.

Findings

Discrete systems can exhibit chaos not present in continuous counterparts

The size of the time step critically affects the dynamics of the logistic equation

Caveats are identified in the common assumptions about the equivalence of discrete and continuous models

Abstract

Discrete autonomous dynamical systems in dimension 1 can exhibit chaotic behavior, whereas the corresponding continuous evolution equations rule it out, and cannot even possess a nontrivial periodic solution. Therefore the passage from discrete to continuous equations (and conversely) is all but harmless. We address this issue and evidence some caveats on the paradigmatic Verhulst logistic equation, investigating in particular the status and influence of the actual size of the unit time step in discrete modelings, rooted in well-known numerical analysis.