Mean field model for synchronization of coupled two-state units and the effect of memory

Daniel Elias Escaff, K. Lindenberg

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8 Scopus citations

Abstract

A prototypical model for a mean field second order transition is presented, which is based on an ensemble of coupled two-states units. This system is used as a basic model to study the effect of memory. To wit, we distinguish two types of memories: weak and strong, depending on the feasibility of linearizing the generalized mean field master equation. For weak memory we find static solutions that behave much like those of the memoryless (Markovian) system. The latter exhibits a pitchfork bifurcation as the control parameter is increased, with two stable and one unstable solution. The former exhibits an imperfect pitchfork bifurcation to states with the same behaviors. In both cases, the stability of the static solutions is analyzed via the usual linearization around the equilibrium solution. For strong memories we again find an imperfect pitchfork bifurcation, with two stable and one unstable branch. However, it is no longer possible to analyze these behaviors via the usual linearization, which is local in time, because a strong memory requires knowledge of the system for its entire past. Finally, we are pleased to dedicate this publication to Helmut Brand on the occasion of his 60th birthday.

Original languageEnglish
Pages (from-to)155-166
Number of pages12
JournalEuropean Physical Journal: Special Topics
Volume223
Issue number1
DOIs
StatePublished - Jan 2014

Bibliographical note

Funding Information:
D. Escaff acknowledge the financial support of FAI-2013-Puente (Universidad de los Andes).

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