Positive circuits and maximal number of fixed points in discrete dynamical systems

TL;DR

This paper establishes an upper bound on the number of fixed points in discrete dynamical systems based on the topology of positive circuits in the interaction graph, generalizing previous theorems and results.

Contribution

It introduces a new upper bound on fixed points depending on the interaction graph's positive circuits, extending prior theorems to broader classes of systems.

Findings

Upper bound depends on the interaction graph topology

Generalizes Richard and Comet's theorem for discrete systems

Extends bounds for Boolean networks

Abstract

We consider the Cartesian product X of n finite intervals of integers and a map F from X to itself. As main result, we establish an upper bound on the number of fixed points for F which only depends on X and on the topology of the positive circuits of the interaction graph associated with F. The proof uses and strongly generalizes a theorem of Richard and Comet which corresponds to a discrete version of the Thomas' conjecture: if the interaction graph associated with F has no positive circuit, then F has at most one fixed point. The obtained upper bound on the number of fixed points also strongly generalizes the one established by Aracena et al for a particular class of Boolean networks.