Criticality and Chaos in Systems of Communities

TL;DR

This paper investigates how different dynamics models of community interactions can lead to stable, oscillatory, or chaotic behaviors, highlighting the emergence of chaos through local-time updates in community systems.

Contribution

It introduces and compares four dynamic models for community interactions, revealing how local-time updates can induce chaos in systems with bilinear couplings.

Findings

Flow dynamics always converge to equilibrium.

Map dynamics with unfriendly couplings can produce period-two oscillations.

Local-time update map can generate true or marginal chaos.

Abstract

We consider a simple model of communities interacting via bilinear terms. After analyzing the thermal equilibrium case, which can be described by an Hamiltonian, we introduce the dynamics that, for Ising-like variables, reduces to a Glauber-like dynamics. We analyze and compare four different versions of the dynamics: flow (differential equations), map (discrete-time dynamics), local-time update flow, and local-time update map. The presence of only bilinear interactions prevent the flow cases to develop any dynamical instability, the system converging always to the thermal equilibrium. The situation is different for the map when unfriendly couplings are involved, where period-two oscillations arise. In the case of the map with local-time updates, oscillations of any period and chaos can arise as a consequence of the reciprocal "tension" accumulated among the communities during their sleeping time interval. The resulting chaos can be of two kinds: true chaos characterized by positive Lyapunov exponent and bifurcation cascades, or marginal chaos characterized by zero Lyapunov exponent and critical continuous regions.