Skip to main content
SpringerLink
Log in

Discovering, Constructing, and Analyzing Synchronous Clusters of Oscillators in a Complex Network Using Symmetries

  • Chapter
  • First Online:
Advances in Dynamics, Patterns, Cognition

Part of the book series: Nonlinear Systems and Complexity ((NSCH,volume 20))

Abstract

Synchronization is a collective phenomenon that appears in many natural and man-made networks of oscillators or dynamical systems such as telecommunication, neuronal and biological networks. An interesting form of synchronization is cluster synchronization where the network becomes partitioned into groups of oscillator nodes which synchronize to each other, but not to other nodes in other groups or clusters. We present a technique and develop methods for the analysis of network dynamics that shows the connection between network symmetries and cluster formation. We also experimentally confirm these approaches in the context of real networks with heterogeneities and noise using an electro-optic network. We find an interesting scenario for the appearance of chimera synchronization in these cases of identical cluster synchronization. We also extend these methods to networks which have Laplacian coupling and show that we can analyze cases where there are synchronization clusters which do not directly arise from symmetries, but can be built from clusters found by a symmetry analysis.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Afraimovich, V.S., Verichev, N.N., Rabinovich, M.I.: Stochastic synchronization of oscillations in dissipative systems. Izvestiya-Vysshikh-Uchebnykh-Zavedenii,-Radiofizika 29, 1050–1060 (1986). Original Russian reference

    Google Scholar 

  2. Pikovsky, A.: On the interaction of strange attractors. Z. Phys. B 55, 149–154 (1984)

    Article  MathSciNet  Google Scholar 

  3. Fujisaka, H., Yamada, T.: Stability theory of synchronized motion in coupled-oscillator systems. Prog. Theor. Phys. 69, 32–47 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  4. Volkovskii, A.R., Rul’kov, N.F.: Experimental study of bifurcations at the threshold for stochastic locking. Sov. Tech. Phys. Lett. 15, 249–251 (1989)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Pecora, L.M., Carroll, T.L.: Master stability functions for synchronized coupled systems. Phys. Rev. Lett. 80, 2109–2112 (1998)

    Article  Google Scholar 

  7. Zhou, C., Kurths, J.: Hierarchical synchronization in complex networks with heterogeneous degrees. Chaos 16, 015104 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Do, A.-L., Höfener, J., Gross, T.: Engineering mesoscale structures with distinct dynamical implications. New J. Phys. 14, 115022 (2012)

    Article  Google Scholar 

  9. Dahms, T., Lehnert, J., Schöll, E.: Cluster and group synchronization in delay-coupled networks. Phys. Rev. E 86, 016202 (2012)

    Article  Google Scholar 

  10. Fu, C., Deng, Z., Huang, L., Wang, X.: Topological control of synchronous patterns in systems of networked chaotic oscillators. Phys. Rev. E 87, 032909 (2013)

    Article  Google Scholar 

  11. Kanter, I., Zigzag, M., Englert, A., Geissler, F., Kinzel, W.: Synchronization of unidirectional time delay chaotic networks and the greatest common divisor. Europhys. Lett. 93, 6003 (2011)

    Google Scholar 

  12. Rosin, D.P., Rontani, D., Gauthier, D.J., Schöll, E.: Control of synchronization patterns in neural-like boolean networks. Phys. Rev. Lett. 110, 104102 (2013)

    Article  Google Scholar 

  13. Sorrentino, F., Ott, E.: Network synchronization of groups. Phys. Rev. E 76, 056114 (2007)

    Article  MathSciNet  Google Scholar 

  14. Williams, C.R.S., Murphy, T.E., Roy, R., Sorrentino, F., Dahms, T., Schöll, E.: Experimental observations of group synchrony in a system of chaotic optoelectronic oscillators. Phys. Rev. Lett. 110, 064104 (2013)

    Article  Google Scholar 

  15. Belykh, V.N., Osipov, G.V., Petrov, V.S., Suykens, J.A.K., Vandewalle, J.: Cluster synchronization in oscillatory networks. Chaos 18, 037106 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Pecora, L.M., Sorrentino, F., Hagerstrom, A.M., Murphy, T.E., Roy, R.: Cluster synchronization and isolated desynchronization in complex networks with symmetries. Nat. Commun. 5, 1–8 (2014)

    Article  Google Scholar 

  17. Abrams, D.M., Strogatz, S.H.: Chimera states for coupled oscillators. Phys. Rev. Lett. 93, 174102 (2004)

    Article  Google Scholar 

  18. Hagerstrom, A.M., Murphy, T.E., Roy, R., Hovel, P., Omelchenko, I., Schöll, E.: Experimental observation of chimeras in coupled-map lattices. Nat. Phys. 8, 658–661 (2012)

    Article  Google Scholar 

  19. Golubitsky, M., Stewart, I., Schaeffer, D.G.: Singularities and Groups in Bifurcation Theory, vol. II. Springer, New York (1985)

    Book  MATH  Google Scholar 

  20. Golubitsky, M., Stewart, I.: The Symmetry Perspective: From Equilibrium to Chaos in Phase Space and Physical Space. Berkhäuser, Basel (2002)

    Book  MATH  Google Scholar 

  21. D’Huys, O., Vicente, R., Erneux, T., Danckaert, J., Fischer, I.: Synchronization properties of network motifs: influence of coupling delay and symmetry. Chaos 18, 037116 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Nicosia, V., Valencia, M., Chavez, M., Díaz-Guilera, A., Latora, V.: Remote synchronization reveals network symmetries and functional modules. Phys. Rev. Lett. 110, 174102 (2013)

    Article  Google Scholar 

  23. Russo, G., Slotine, J.-J.E.: Symmetries, stability, and control in nonlinear systems and networks. Phys. Rev. E 84, 041929 (2011)

    Article  Google Scholar 

  24. Judd, K.: Networked dynamical systems with linear coupling: synchronisation patterns, coherence and other behaviours. Chaos 23, 043112 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Motter, A.E., Myers, S.A., Anghel, M., Nishikawa, T.: Spontaneous synchrony in power-grid networks. Nat. Phys. 9, 191–197 (2013)

    Article  Google Scholar 

  26. Fink, K.S., Johnson, G., Carroll, T., Mar, D., Pecora, L.: Three-oscillator systems as universal probes of coupled oscillator stability. Phys. Rev. E 61, 5080–5090 (2000)

    Article  Google Scholar 

  27. Irving, D., Sorrentino, F.: Synchronization of dynamical hypernetworks: dimensionality reduction through simultaneous block-diagonalization of matrices. Phys. Rev. E 86, 056102 (2012)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  29. Newman, M.: Networks, An Introduction, chap. 18 Oxford University Press, Oxford (2011)

    Google Scholar 

  30. Tinkham, M.: Group Theory and Quantum Mechanics. McGraw-Hill, New York (1964)

    MATH  Google Scholar 

  31. Stein, W.: SAGE: Software for Algebra and Geometry Experimentation (2013). http://www.sagemath.org/sage/, http://sage.scipy.org/

  32. The GAP Group, GAP: Groups, Algorithms, and Programming, Version 4.4 (2005). http://www.gap-system.org

  33. Sagan, B.E.: The Symmetric Group. Wadsworth Brooks, Pacific Grove, CA (1991)

    MATH  Google Scholar 

  34. Walen, A., Brennany, S., Sauer, T., Schiff, S.: Observability and controllability of nonlinear networks: the role of symmetry. Phys. Rev. X 5, 011005-1–17 (2015)

    Google Scholar 

  35. Sorrentino, F., Pecora, L.M., Hagerstrom, A.M., Murphy, T.E., Roy, R.: Complete characterization of stability of cluster synchronization in complex dynamical networks. Sci. Adv. 5, 011005-1–17 (2015)

    Google Scholar 

  36. Hart, J.D., Bansal, K., Murphy, T.E., Roy, R.: Experimental observation of chimera and cluster states in a minimal globally coupled network. Chaos 26, 094801 (2016)

    Article  MathSciNet  Google Scholar 

  37. Belykh, I., Hasler, M.: Mesoscale and clusters of synchrony in networks of bursting neurons. Chaos 21, 016106 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  38. Kamei, H., Cock, P.: Computation of balanced equivalence relations and their lattice for a coupled cell network. SIAM J. Appl. Dyn. Syst. 12, 352–382 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  39. Golubitsky, M., Stewart, I., Török, A.: Coupled cell networks with multiple arrows. SIAM J. Appl. Dyn. Syst. 4, 78–100 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  40. Schaub, M.T., O’Clery, N. Billeh, Y.N., Delvenne, J.-C. Lambiotte, R., Barahona, M.: Graph partitions and cluster synchronization in networks of oscillators. Chaos 26, 094821 (2016)

    Article  MathSciNet  Google Scholar 

  41. Sorrentino, F., Pecora, L.: Approximate cluster synchronization in networks with symmetries and parameter mismatches. CHAOS 26, 094823 (2016). http://doi.org/10.1063/1.4961967

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. U. S. Naval Research Laboratory, Washington, DC, 20375, USA

    Louis M. Pecora

  2. Department of Mechanical Engineering, University of New Mexico, Albuquerque, NM, 87131, USA

    Francesco Sorrentino

  3. Department of Physics, University of Maryland, College Park, MD, 20742, USA

    Aaron M. Hagerstrom & Rajarshi Roy

  4. Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, MD, 20742, USA

    Aaron M. Hagerstrom & Rajarshi Roy

  5. Department of Electrical and Computer Engineering, University of Maryland, College Park, MD, 20742, USA

    Thomas E. Murphy

  6. Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, MD, 20742, USA

    Thomas E. Murphy

  7. Institute for Physical Science and Technology, University of Maryland, College Park, MD, 20742, USA

    Rajarshi Roy

Authors
  1. Louis M. Pecora
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Francesco Sorrentino
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Aaron M. Hagerstrom
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Thomas E. Murphy
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Rajarshi Roy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Louis M. Pecora .

Editor information

Editors and Affiliations

  1. Pennsylvania State University, University Park, Pennsylvania, USA

    Igor S. Aranson

  2. Institute for Physics and Astronomy, University of Potsdam, Potsdam, Germany

    Arkady Pikovsky

  3. BioCircuits Institute, University of California San Diego, La Jolla, California, USA

    Nikolai F. Rulkov

  4. BioCircuits Institute, University of California San Diego, La Jolla, California, USA

    Lev S. Tsimring

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Pecora, L.M., Sorrentino, F., Hagerstrom, A.M., Murphy, T.E., Roy, R. (2017). Discovering, Constructing, and Analyzing Synchronous Clusters of Oscillators in a Complex Network Using Symmetries. In: Aranson, I., Pikovsky, A., Rulkov, N., Tsimring, L. (eds) Advances in Dynamics, Patterns, Cognition. Nonlinear Systems and Complexity, vol 20. Springer, Cham. https://doi.org/10.1007/978-3-319-53673-6_10

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-53673-6_10

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-53672-9

  • Online ISBN: 978-3-319-53673-6

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation