Asymptotic Behavior of Conjunctive Boolean Networks over Weakly Connected Digraphs

Xudong Chen, Zuguang Gao, Tamer Basar

Research output: Contribution to journalArticlepeer-review

Abstract

A conjunctive Boolean network (CBN) is a finite state dynamical system, whose variables take values from a binary set, and the value update rule for each variable is a Boolean function consisting only of logic and operations. We investigate the asymptotic behavior of CBNs by computing their periodic orbits. When the underlying digraph is strongly connected, the periodic orbits of the associated CBN have been completely understood, one-to-one corresponding to binary necklaces of a certain length given by the loop number of the graph. We characterize in the paper the periodic orbits of CBNs over arbitrary weakly connected digraphs. We establish, among other things, a new method to investigate their asymptotic behavior. Specifically, we introduce a graphical approach, termed system reduction, which turns the underlying digraph into a special weakly connected digraph, whose strongly connected components are all cycles. We show that the reduced system uniquely determines the asymptotic behavior of the original system. Moreover, we provide a constructive method for computing the periodic orbit of the reduced system, which the system will enter for a given, but arbitrary initial condition.

Original languageEnglish (US)
Article number8770125
Pages (from-to)2536-2549
Number of pages14
JournalIEEE Transactions on Automatic Control
Volume65
Issue number6
DOIs
StatePublished - Jun 2020

Keywords

  • Asymptotic behavior
  • Boolean network
  • graph theory
  • system reduction

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Computer Science Applications
  • Electrical and Electronic Engineering

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