Emergence of chaos in interacting communities

TL;DR

This paper presents a simple model of two interacting communities showing how cooperative interactions lead to equilibrium, while anti-cooperative interactions can cause oscillations and chaos, with potential implications for stock market dynamics.

Contribution

The paper introduces a minimal dynamical model demonstrating how different types of community interactions lead to equilibrium or chaotic behavior, highlighting the role of intra- and inter-community couplings.

Findings

Cooperative couplings lead to equilibrium states.

Anti-cooperative couplings can cause oscillations and chaos.

Moderate consensus in one community can suppress chaos.

Abstract

We introduce a simple dynamical model of two interacting communities whose elements are subject to stochastic discrete-time updates governed by only bilinear interactions. When the intra- and inter-couplings are cooperative, the two communities reach asymptotically an equilibrium state. However, when the intra- or inter-couplings are anti-cooperative, the system may remain in perpetual oscillations and, when the coupling values belong to certain intervals, two possible scenarios arise, characterized either by erratic aperiodic trajectories and high sensitiveness to small changes of the couplings, or by chaotic trajectories and bifurcation cascades. Quite interestingly, we find out that even a moderate consensus in one single community can remove the chaos. Connections of the model with interacting stock markets are discussed.