Skip to main content
SpringerLink
Log in

Invariant Synchrony and Anti-synchrony Subspaces of Weighted Networks

Journal of Nonlinear Science volume 33, Article number: 63 (2023) Cite this article

Abstract

The internal state of a cell in a coupled cell network is often described by an element of a vector space. Synchrony or anti-synchrony occurs when some of the cells are in the same or the opposite state. Subspaces of the state space containing cells in synchrony or anti-synchrony are called polydiagonal subspaces. We study the properties of several types of polydiagonal subspaces of weighted coupled cell networks. In particular, we count the number of such subspaces and study when they are dynamically invariant. Of special interest are the evenly tagged anti-synchrony subspaces in which the number of cells in a certain state is equal to the number of cells in the opposite state. Our main theorem shows that the dynamically invariant polydiagonal subspaces determined by certain types of couplings are either synchrony subspaces or evenly tagged anti-synchrony subspaces. A special case of this result confirms a conjecture about difference-coupled graph network systems.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Data Availability

This paper’s data are computer-generated and available at Nijholt et al. (2022).

References

  • Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions with formulas, graphs, and mathematical tables. Reprint of the 1972 ed. A Wiley-Interscience Publication. Selected Government Publications. New York: John Wiley & Sons, Inc; Washington, D.C.: National Bureau of Standards. xiv, 1046 pp (1984)

  • Adler, V.E.: Set partitions and integrable hierarchies. Theor. Math. Phys. 187(3), 842–870 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  • Aguiar, M., Dias, A.: Synchrony and antisynchrony in weighted networks. SIAM J. Appl. Dyn. Syst. 20(3), 1382–1420 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  • Aguiar, M.A.D., Dias, A.P.S.: The lattice of synchrony subspaces of a coupled cell network: characterization and computation algorithm. J. Nonlinear Sci. 24(6), 949–996 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Arenas, A., Díaz-Guilera, A., Kurths, J., Moreno, Y., Zhou, C.: Synchronization in complex networks. Phys. Rep. 469(3), 93–153 (2008)

    Article  MathSciNet  Google Scholar 

  • Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Sciences. Society for Industrial and Applied Mathematics (1994)

    Book  MATH  Google Scholar 

  • Bóna, M.: A walk through combinatorics. An introduction to enumeration and graph theory. With a foreword by Richard Stanley. Hackensack, NJ: World Scientific (2017)

  • da Conceição, A., Leite, M., Golubitsky, M.: Homogeneous three-cell networks. Nonlinearity 19(10), 2313–2363 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Diggans, C.T., Fish, J., AlMomani, A.A.R., Bollt, E.M.: The essential synchronization backbone problem. Chaos Interdiscip. J. Nonlinear Sci. 31(11), 113142 (2021)

    Article  Google Scholar 

  • Dionne, B., Golubitsky, M., Stewart, I.: Coupled cells with internal symmetry: I: wreath products. Nonlinearity 9(2), 559–574 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  • Dionne, B., Golubitsky, M., Stewart, I.: Coupled cells with internal symmetry: II: direct products. Nonlinearity 9(2), 575–599 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  • Fries, P.: A mechanism for cognitive dynamics: neuronal communication through neuronal coherence. Trends Cogn. Sci. 9(10), 474–480 (2005)

    Article  Google Scholar 

  • Golubitsky, M., Stewart, I., Schaeffer, D.G.: Singularities and groups in bifurcation theory. Vol. II, volume 69 of Applied Mathematical Sciences. Springer-Verlag, New York (1988)

  • Golubitsky, M., Stewart, I., Török, A.: Patterns of synchrony in coupled cell networks with multiple arrows. SIAM J. Appl. Dyn. Syst. 4(1), 78–100 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Hegselmann, R., Krause, U., et al.: Opinion dynamics and bounded confidence models, analysis, and simulation. J Arti Soc Soc Simul 5(3), 1 (2002)

    Google Scholar 

  • Hu, J., Zheng, W.X.: Bipartite consensus for multi-agent systems on directed signed networks. In: 52nd IEEE conference on decision and control, pp 3451–3456. IEEE (2013)

  • Huang, L., Chen, Q., Lai, Y.-C., Pecora, L.M.: Generic behavior of master-stability functions in coupled nonlinear dynamical systems. Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 80(3), 036204 (2009)

    Article  Google Scholar 

  • Kaplansky, I.: Linear Algebra and Geometry: A Second Course, p. 139. Allyn and Bacon, Inc, Boston (1969)

    MATH  Google Scholar 

  • Kim, C.-M., Rim, S., Kye, W.-H., Ryu, J.-W., Park, Y.-J.: Anti-synchronization of chaotic oscillators. Phys. Lett. A 320(1), 39–46 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • Liu, W., Qian, X., Yang, J., Xiao, J.: Antisynchronization in coupled chaotic oscillators. Phys. Lett. A 354(1–2), 119–125 (2006)

    Article  Google Scholar 

  • Meng, J., Wang, X.Y.: Robust anti-synchronization of a class of delayed chaotic neural networks. Chaos Interdiscip. J. Nonlinear Sci. 17(2), 023113 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Miller, D.A., Kowalski, K.L., Lozowski, A.: Synchronization and anti-synchronization of chua’s oscillators via a piecewise linear coupling circuit. In 1999 IEEE international symposium on circuits and systems (ISCAS), volume 5, pages 458–462. IEEE (1999)

  • Morone, F., Leifer, I., Makse, H.A.: Fibration symmetries uncover the building blocks of biological networks. Proc. Natl. Acad. Sci. 117(15), 8306–8314 (2020)

    Article  Google Scholar 

  • Neuberger, J.M., Sieben, N., Swift, J.W.: Computing eigenfunctions on the Koch snowflake: a new grid and symmetry. J. Comput. Appl. Math. 191(1), 126–142 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Neuberger, J.M., Sieben, N., Swift, J.W.: Synchrony and Antisynchrony for Difference-Coupled Vector Fields on Graph Network Systems. SIAM J. Appl. Dyn. Syst. 18(2), 904–938 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  • Neuberger, J.M., Sieben, N., Swift, J.W.: Invariant synchrony subspaces of sets of matrices. SIAM J. Appl. Dyn. Syst. 19(2), 964–993 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  • Nijholt, E., Sieben, N., Swift, J.W.: Github repository. https://github.com/jwswift/Anti-Synchrony-Subspaces/ (2022)

  • Olfati-Saber, R.: Flocking for multi-agent dynamic systems: algorithms and theory. IEEE Trans. Autom. Control 51(3), 401–420 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Pecora, L.M., Carroll, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821–824 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  • Pirzada, S., Naikoo, T.A., Samee, U., Iványi, A.: Imbalances in directed multigraphs. Acta Univ. Sapientiae Math. 2(2), 137–145 (2010)

    MathSciNet  MATH  Google Scholar 

  • Sloane, N.J.A.: The OEIS Foundation Inc. The on-line encyclopedia of integer sequences, http://oeis.org, (2021)

  • Stewart, I., Golubitsky, M., Pivato, M.: Symmetry groupoids and patterns of synchrony in coupled cell networks. SIAM J. Appl. Dyn. Syst. 2(4), 609–646 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • Supina, M.: The Hopf monoid of orbit polytopes. J. Comb. 11(4), 575–601 (2020)

    MathSciNet  MATH  Google Scholar 

  • Watts, D.J., Strogatz, S.H.: Collective dynamics of ‘small-world’ networks. Nature 393(6684), 440–442 (1998)

    Article  MATH  Google Scholar 

  • Wedekind, I., Parlitz, U.: Experimental observation of synchronization and anti-synchronization of chaotic low-frequency-fluctuations in external cavity semiconductor lasers. Int. J. Bifurc. Chaos 11(04), 1141–1147 (2001)

    Article  Google Scholar 

  • Weisstein, E.W.: Modified bessel function of the first kind from MathWorld: a Wolfram Web Resource, https://mathworld.wolfram.com/ModifiedBesselFunctionoftheFirstKind.html. Last visited on 3/18/2022

  • West, D.B.: Introduction to Graph Theory, 2nd edn. Prentice-Hall of India, New Delhi (2005)

    MATH  Google Scholar 

  • Wilson, R.J.: Introduction to graph theory, 4th edn. Longman, Harlow (1996)

    MATH  Google Scholar 

Download references

Acknowledgements

E.N. acknowledges the support of the Center for Research in Mathematics Applied to Industry (FAPESP Cemeai grant 2013/07375-0) and the Serrapilheira Institute (Grant No. Serra-1709-16124)

Author information

Authors and Affiliations

  1. Department of Mathematics, Imperial College London, London, UK

    Eddie Nijholt

  2. Department of Mathematics and Statistics, Northern Arizona University, PO Box 5717, Flagstaff, AZ, 86011-5717, USA

    Nándor Sieben & James W. Swift

Authors
  1. Eddie Nijholt
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nándor Sieben
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. James W. Swift
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to James W. Swift.

Ethics declarations

Conflict of interest

The authors declare no financial conflict of interest.

Additional information

Communicated by Paul Newton.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nijholt, E., Sieben, N. & Swift, J.W. Invariant Synchrony and Anti-synchrony Subspaces of Weighted Networks. J Nonlinear Sci 33, 63 (2023). https://doi.org/10.1007/s00332-023-09924-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00332-023-09924-y

Keywords

Mathematics Subject Classification

search

Navigation