Cluster Synchronization of Diffusively Coupled Nonlinear Systems: A Contraction-Based Approach

  • Zahra Aminzare
  • Biswadip Dey
  • Elizabeth N. Davison
  • Naomi Ehrich Leonard
Article
  • 29 Downloads

Abstract

Finding the conditions that foster synchronization in networked nonlinear systems is critical to understanding a wide range of biological and mechanical systems. However, the conditions proved in the literature for synchronization in nonlinear systems with linear coupling, such as has been used to model neuronal networks, are in general not strict enough to accurately determine the system behavior. We leverage contraction theory to derive new sufficient conditions for cluster synchronization in terms of the network structure, for a network where the intrinsic nonlinear dynamics of each node may differ. Our result requires that network connections satisfy a cluster-input-equivalence condition, and we explore the influence of this requirement on network dynamics. For application to networks of nodes with FitzHugh–Nagumo dynamics, we show that our new sufficient condition is tighter than those found in previous analyses that used smooth or nonsmooth Lyapunov functions. Improving the analytical conditions for when cluster synchronization will occur based on network configuration is a significant step toward facilitating understanding and control of complex networked systems.

Keywords

Cluster synchronization Contraction theory for stability Diffusively coupled nonlinear networks Neuronal oscillators 

Notes

Acknowledgements

This work was jointly supported by the National Science Foundation under NSF-CRCNS grant DMS-1430077 and the Office of Naval Research under ONR Grant N00014-14-1-0635. This material is also based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant DGE-1656466. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. The authors thank the anonymous reviewers for their thoughtful and detailed comments.

References

  1. Abrams, D.M., Pecora, L.M., Motter, A.E.: Introduction to focus issue: patterns of network synchronization. Chaos 26(9), 094601 (2016)CrossRefGoogle Scholar
  2. Aminzare, Z.: On Synchronous Behavior in Complex Nonlinear Dynamical Systems. Dissertation, Rutgers University, The State University of New Jersey (2015)Google Scholar
  3. Aminzare, Z., Sontag, E.D.: Contraction methods for nonlinear systems: A brief introduction and some open problems. In: Proceedings of the 53rd IEEE Conference on Decision and Control (CDC), pp. 3835–3847 (2014a)Google Scholar
  4. Aminzare, Z., Sontag, E.D.: Synchronization of diffusively-connected nonlinear systems: results based on contractions with respect to general norms. IEEE Trans. Netw. Sci. Eng. 1(2), 91–106 (2014b)MathSciNetCrossRefGoogle Scholar
  5. Aminzare, Z., Shafi, Y., Arcak, M., Sontag, E.D.: Guaranteeing spatial uniformity in reaction-diffusion systems using weighted \(L_2\) -norm contractions. In: Kulkarni, V., Stan, G.-B., Raman, K. (eds.) A Systems Theoretic Approach to Systems and Synthetic Biology I: Models and System Characterizations, pp. 73–101. Springer, Berlin (2014)Google Scholar
  6. Arcak, M.: Certifying spatially uniform behavior in reaction-diffusion PDE and compartmental ODE systems. Automatica 47(6), 1219–1229 (2011)MathSciNetCrossRefMATHGoogle Scholar
  7. Belykh, V.N., Osipov, G.V., Petrov, V.S., Suykens, J.A.K., Vandewalle, J.: Cluster synchronization in oscillatory networks. Chaos 18(3), 037106 (2008)MathSciNetCrossRefMATHGoogle Scholar
  8. Belykh, V.N., Petrov, S., Osipov, G.V.: Dynamics of the finite-dimensional Kuramoto model: global and cluster synchronization. Regular Chaotic Dyn. 20(1), 37–48 (2015)MathSciNetCrossRefMATHGoogle Scholar
  9. Brown, E., Holmes, P., Moehlis, J.: Globally coupled oscillator networks. In: Perspectives and Problems in Nonlinear Science: A Celebratory Volume in Honor of Larry Sirovich, pp 183–215 (2003)Google Scholar
  10. Chen, C.C., Litvak, V., Gilbertson, T., Kühn, A., Lu, C.S., Lee, S.T., Tsai, C.H., Tisch, S., Limousin, P., Hariz, M., et al.: Excessive synchronization of basal ganglia neurons at 20 Hz slows movement in Parkinson’s disease. Exp. Neurol. 205(1), 214–221 (2007)CrossRefGoogle Scholar
  11. Chow, C.C., Kopell, N.: Dynamics of spiking neurons with electrical coupling. Neural Computat. 12(7), 1643–1678 (2000)CrossRefGoogle Scholar
  12. Chung, S.-J., Slotine, J.-J.E., Miller, D.W.: Nonlinear model reduction and decentralized control of tethered formation flight. J. Guid. Control Dyn. 30(2), 390–400 (2007)CrossRefGoogle Scholar
  13. Dahlquist, G.: Stability and Error Bounds in the Numerical Integration of Ordinary Differential Equations. Inaugural dissertation, University of Stockholm, Almqvist & Wiksells Boktryckeri AB, Uppsala (1958)Google Scholar
  14. Davison, E.N., Dey, B., Leonard, N.E.: Synchronization bound for networks of nonlinear oscillators. In: Proceedings of the 54th Annual Allerton Conference on Communication, Control, and Computing (Allerton), pp 1110–1115 (2016)Google Scholar
  15. Demidovič, B.P.: On the dissipativity of a certain non-linear system of differential equations. I. Vestnik Moskov University Series in Mathematical. Methods 1961(6), 19–27 (1961)Google Scholar
  16. Demidovič, B.P.: Lektsii po matematicheskoi teorii ustoichivosti. Izdat. Nauka, Moscow (1967)Google Scholar
  17. Desoer, C.A., Vidyasagar, M.: Feedback Systems: Input-Output Properties. Electrical Science. Academic Press [Harcourt Brace Jovanovich, Publishers], New York (1975)MATHGoogle Scholar
  18. Dumas, G., Nadel, J., Soussignan, R., Martinerie, J., Garnero, L.: Inter-brain synchronization during social interaction. PLoS ONE 5(8), 1–10 (2010)Google Scholar
  19. Favaretto, C., Bassett, D.S., Cenedese, A., Pasqualetti, F.: Bode meets kuramoto: synchronized clusters in oscillatory networks. In: Proceedings of American Control Conference (ACC), pp. 2799–2804 (2017a)Google Scholar
  20. Favaretto, C., Cenedese, A., Pasqualetti, F.: Cluster Synchronization in Networks of Kuramoto Oscillators. In: Proceedings of the IFAC 2017 World Congress, pp. 2485–2490 (2017b)Google Scholar
  21. Ferreira, A.S.R., Arcak, M.: A graph partitioning approach to predicting patterns in lateral inhibition systems. SIAM J. Appl. Dyn. Syst. 12(4), 2012–2031 (2013)MathSciNetCrossRefMATHGoogle Scholar
  22. Fiore, D., Russo, G., di Bernardo, M.: Exploiting nodes symmetries to control synchronization and consensus patterns in multiagent systems. IEEE Control Syst. Lett. 1(2), 364–369 (2017)CrossRefGoogle Scholar
  23. 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)MathSciNetCrossRefMATHGoogle Scholar
  24. Hartman, P.: On stability in the large for systems of ordinary differential equations. Can. J. Math. 13, 480–492 (1961)MathSciNetCrossRefMATHGoogle Scholar
  25. Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991)CrossRefMATHGoogle Scholar
  26. Jouffroy, J.: Some ancestors of contraction analysis. In: Proceedings of the 44th IEEE Conference on Decision and Control 2005 and European Control Conference 2005, pp. 5450–5455 (Dec 2005)Google Scholar
  27. Lehnertz, K., Bialonski, S., Horstmann, M.-T., Krug, D., Rothkegel, A., Staniek, M., Wagner, T.: Synchronization phenomena in human epileptic brain networks. J. Neurosci. Methods 183(1), 42–48 (2009)CrossRefGoogle Scholar
  28. Lewis, D.C.: Metric properties of differential equations. Am. J. Math. 71, 294–312 (1949)MathSciNetCrossRefMATHGoogle Scholar
  29. Lohmiller, W., Slotine, J.-J.E.: On contraction analysis for non-linear systems. Automatica 34(6), 683–696 (1998)MathSciNetCrossRefMATHGoogle Scholar
  30. Lohmiller, W., Slotine, J.: Contraction analysis of nonlinear distributed systems. Int. J. Control 78, 678–688 (2005)CrossRefMATHGoogle Scholar
  31. Lu, W., Liu, B., Chen, T.: Cluster synchronization in networks of coupled nonidentical dynamical systems. Chaos 20(1), 013120 (2010)MathSciNetCrossRefMATHGoogle Scholar
  32. MacLeod, K., Laurent, G.: Distinct mechanisms for synchronization and temporal patterning of odor-encoding neural assemblies. Science 274(5289), 976–979 (1996)CrossRefGoogle Scholar
  33. Mirollo, R.E., Strogatz, S.H.: Synchronization of pulse-coupled biological oscillators. SIAM J. Appl. Math. 50(6), 1645–1662 (1990)MathSciNetCrossRefMATHGoogle Scholar
  34. Motter, A.E., Myers, S.A., Anghel, M., Nishikawa, T.: Spontaneous synchrony in power-grid networks. Nature Phys. 9, 191–197 (2013)CrossRefGoogle Scholar
  35. Nair, S., Leonard, N.E.: Stable synchronization of mechanical system networks. SIAM J. Control Optim. 47(2), 661–683 (2008)MathSciNetCrossRefMATHGoogle Scholar
  36. Orosz, G., Moehlis, J., Ashwin, P.: Designing the dynamics of globally coupled oscillators. Prog. Theor. Phys. 122(3), 611–630 (2009)CrossRefMATHGoogle Scholar
  37. Pavlov, A., Pogromvsky, A., van de Wouv, N., Nijmeijer, H.: Convergent dynamics, a tribute to Boris Pavlovich Demidovich. Syst. Control Lett. 52, 257–261 (2004)MathSciNetCrossRefMATHGoogle Scholar
  38. 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, 4079 (2014)CrossRefGoogle Scholar
  39. Pham, Q.-C., Slotine, J.-J.: Stable concurrent synchronization in dynamic system networks. Neural Netw. 20(1), 62–77 (2007)CrossRefMATHGoogle Scholar
  40. Pikovsky, A., Rosenblum, M., Kurths, J.: Synchronization: A Universal Concept in Nonlinear Sciences, volume 12 of Cambridge Nonlinear Science Series. Cambridge University Press, Cambridge (2003)MATHGoogle Scholar
  41. Russo, G., Di Bernardo, M.: Contraction theory and master stability function: linking two approaches to study synchronization of complex networks. IEEE Trans. Circuits Syst. 56(2), 177–181 (2009)Google Scholar
  42. Russo, G., Slotine, J.-J.E.: Global convergence of quorum-sensing networks. Phys. Rev. E 82(4), 041919 (2010)MathSciNetCrossRefGoogle Scholar
  43. 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 Interdiscip. J. Nonlinear Sci. 26(9), 094821 (2016)MathSciNetCrossRefMATHGoogle Scholar
  44. Sepulchre, R., Paley, D., Leonard, N.E.: Stabilization of planar collective motion with limited communication. IEEE Trans. Autom. Control 53(3), 706–719 (2008)MathSciNetCrossRefMATHGoogle Scholar
  45. Sivrikaya, F., Yener, B.: Time synchronization in sensor networks: a survey. IEEE Netw. 18(4), 45–50 (2004)CrossRefGoogle Scholar
  46. Smith, H.M.: Synchronous flashing of fireflies. Science 82(2120), 151–152 (1935)CrossRefGoogle Scholar
  47. Soderlind, G.: The logarithmic norm. History and modern theory. BIT Numer. Math. 46(3), 631–652 (2006)MathSciNetCrossRefMATHGoogle Scholar
  48. Sorrentino, F., Ott, E.: Network synchronization of groups. Phys. Rev. E 76(5), 056114 (2007)MathSciNetCrossRefGoogle Scholar
  49. Sorrentino, F., Pecora, L.: Approximate cluster synchronization in networks with symmetries and parameter mismatches. Chaos 26(9), 094823 (2016)MathSciNetCrossRefMATHGoogle Scholar
  50. Sorrentino, F., Pecora, L.M., Hagerstrom, A.M., Murphy, T.E., Roy, R.: Complete characterization of the stability of cluster synchronization in complex dynamical networks. Sci. Adv. 2(4), e1501737 (2016)CrossRefGoogle Scholar
  51. 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)MathSciNetCrossRefMATHGoogle Scholar
  52. Tiberi, L., Favaretto, C., Innocenti, M., Bassett, D.S., Pasqualetti, F.: Synchronization patterns in networks of Kuramoto oscillators: A geometric approach for analysis and control. In: Proceedings of the 56th IEEE Conference on Decision and Control (CDC), pp. 481–486 (2017)Google Scholar
  53. Wang, W., Slotine, J.J.E.: On partial contraction analysis for coupled nonlinear oscillators. Biol. Cybern. 92, 38–53 (2005)MathSciNetCrossRefMATHGoogle Scholar
  54. Wang, K., Fu, X., Li, K.: Cluster synchronization in community networks with nonidentical nodes. Chaos 19(2), 023106 (2009)MathSciNetCrossRefMATHGoogle Scholar
  55. Wilson, D., Moehlis, J.: Clustered desynchronization from high-frequency deep brain stimulation. PLoS Comput. Biol. 11(12), e1004673 (2015)CrossRefGoogle Scholar
  56. Winfree, A.T.: Biological rhythms and the behavior of populations of coupled oscillators. J. Theor. Biol. 16(1), 15–42 (1967)CrossRefGoogle Scholar
  57. Xia, W., Cao, M.: Clustering in diffusively coupled networks. Automatica 47(11), 2395–2405 (2011)MathSciNetCrossRefMATHGoogle Scholar
  58. Yoshizawa, T.: Stability Theory by Liapunov’s Second Method. Publications of the Mathematical Society of Japan, No. 9. The Mathematical Society of Japan, Tokyo (1966)Google Scholar
  59. Yoshizawa, T.: Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions. Applied Mathematical Sciences, vol. 14. Springer, New York (1975)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Zahra Aminzare
    • 1
  • Biswadip Dey
    • 2
  • Elizabeth N. Davison
    • 2
  • Naomi Ehrich Leonard
    • 2
  1. 1.The Program in Applied and Computational MathematicsPrinceton UniversityPrincetonUSA
  2. 2.Department of Mechanical and Aerospace EngineeringPrinceton UniversityPrincetonUSA

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