Nonsingularity of Grain-like cascade FSRs via semi-tensor product

  • Jianquan LuEmail author
  • Meilin Li
  • Yang Liu
  • Daniel W.C. Ho
  • Jürgen. Kurths
Research Paper Special Focus on Analysis and Control of Finite-Valued Network Systems


In this paper, Grain-like cascade feedback shift registers (FSRs) are regarded as two Boolean networks (BNs), and the semi-tensor product (STP) of the matrices is used to convert the Grain-like cascade FSRs into an equivalent linear equation. Based on the STP, a novel method is proposed herein to investigate the nonsingularity of Grain-like cascade FSRs. First, we investigate the property of the state transition matrix of Grain-like cascade FSRs. We then propose their sufficient and necessary nonsingularity condition. Next, we regard the Grain-like cascade FSRs as Boolean control networks (BCNs) and further provide a sufficient condition of their nonsingularity. Finally, two examples are provided to illustrate the results obtained in this paper.


Grain-like cascade FSRs Boolean control networks Boolean networks semi-tensor product nonsingularity 



This work was supported by National Natural Science Foundation of China (Grant Nos. 61573102, 11671361), Natural Science Foundation of Jiangsu Province of China (Grant No. BK20170019), Jiangsu Provincial Key Laboratory of Networked Collective Intelligence (Grant No. BM2017002), China Postdoctoral Science Foundation (Grant Nos. 2014M560377, 2015T80483), Jiangsu Province Six Talent Peaks Project (Grant No. 2015-ZNDW-002), and Fundamental Research Funds for the Central Universities.


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Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  • Jianquan Lu
    • 1
    • 2
    Email author
  • Meilin Li
    • 1
  • Yang Liu
    • 2
  • Daniel W.C. Ho
    • 3
  • Jürgen. Kurths
    • 4
  1. 1.School of MathematicsSoutheast UniversityNanjingChina
  2. 2.College of Mathematics, Physics and Information EngineeringZhejiang Normal UniversityJinhuaChina
  3. 3.Department of MathematicsHong KongChina
  4. 4.Potsdam Institute for Climate Impact ResearchPotsdamGermany

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