Pattern selection and super-patterns in the bounded confidence model
1 Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory Los Alamos, NM 87545 USA
2 School of Mathematics, University of Minnesota - Minneapolis, MN 55455, USA
Received: 3 July 2015
Accepted: 7 October 2015
We study pattern formation in the bounded confidence model of opinion dynamics. In this random process, opinion is quantified by a single variable. Two agents may interact and reach a fair compromise, but only if their difference of opinion falls below a fixed threshold. Starting from a uniform distribution of opinions with compact support, a traveling wave forms and it propagates from the domain boundary into the unstable uniform state. Consequently, the system reaches a steady state with isolated clusters that are separated by distance larger than the interaction range. These clusters form a quasi-periodic pattern where the sizes of the clusters and the separations between them are nearly constant. We obtain analytically the average separation between clusters L. Interestingly, there are also very small quasi-periodic modulations in the size of the clusters. The spatial periods of these modulations are a series of integers that follow from the continued-fraction representation of the irrational average separation L.
PACS: 89.75.Kd – Patterns / 82.40.Ck – Pattern formation in reactions with diffusion, flow and heat transfer / 05.45.-a – Nonlinear dynamics and chaos
© EPLA, 2015