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

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.