Chimeras in Networks Without Delay

Part of the Springer Theses book series (Springer Theses)


The aim of this Chapter is to study chimera states in a network of non-locally coupled Stuart-Landau oscillators. Motivated by former studies, we discuss how a specific set of initial conditions initially separating the network into distinct domains gives rise to a clustered chimera state. Furthermore, the interplay between these initial conditions and non-local coupling is studied. Considering the dynamics of chimera states, our argument shows how “flipped” profiles of the mean phase velocities can be explained by a change of sign of the coupling phase. These profiles are believed to be a distinct feature of (phase) chimeras, at least in the case of non-locally coupled systems. Extending our reasoning, we show that this argument intuitively explains the transition from phase- to amplitude-mediated chimera state as a result of increasing coupling strength.


  1. 1.
    Abrams DM, Strogatz SH (2004) Chimera states for coupled oscillators. Phys Rev Lett 93:174102Google Scholar
  2. 2.
    Abrams DM (2006) Two coupled oscillator models: the Millennium bridge and the Chimera state. Ph.D. thesis, Cornell UniversityGoogle Scholar
  3. 3.
    Atay FM (2003) Distributed delays facilitate amplitude death of coupled oscillators. Phys Rev Lett 91:094101Google Scholar
  4. 4.
    Banerjee T (2015) Mean-field-diffusion-induced chimera death state. Europhys Lett 110:60003Google Scholar
  5. 5.
    Banerjee T, Dutta PS, Zakharova A, Schöll E (2016) Chimera patterns induced by distance-dependent power-law coupling in ecological networks. Phys Rev E 94:032206Google Scholar
  6. 6.
    Banerjee T, Ghosh D, Biswas D, Schöll E, Zakharova A (2018) Networks of coupled oscillators: from phase to amplitude chimeras. Chaos 28:113124Google Scholar
  7. 7.
    Bastidas VM, Omelchenko I, Zakharova A, Schöll E, Brandes T (2015) Quantum signatures of chimera states. Phys Rev E 92:062924Google Scholar
  8. 8.
    Bick C, Martens EA (2015) Controlling chimeras. New J Phys 17:033030Google Scholar
  9. 9.
    Bogomolov S, Strelkova G, Schöll E, Anishchenko VS (2016) Amplitude and phase chimeras in an ensemble of chaotic oscillators. Tech Phys Lett 42:765–768Google Scholar
  10. 10.
    Bogomolov S, Slepnev A, Strelkova G, Schöll E, Anishchenko VS (2017) Mechanisms of appearance of amplitude and phase chimera states in a ring of nonlocally coupled chaotic systems. Commun Nonlinear Sci Numer Simul 43:25Google Scholar
  11. 11.
    Böhm F, Zakharova A, Schöll E, Lüdge K (2015) Amplitude-phase coupling drives chimera states in globally coupled laser networks. Phys Rev E 91:040901(R)Google Scholar
  12. 12.
    Buscarino A, Frasca M, Gambuzza LV, Hövel P (2015) Chimera states in time-varying complex networks. Phys Rev E 91:022817Google Scholar
  13. 13.
    Choe CU, Dahms T, Hövel P, Schöll E (2010) Controlling synchrony by delay coupling in networks: from in-phase to splay and cluster states. Phys Rev E 81:025205(R)Google Scholar
  14. 14.
    Daido H, Nakanishi K (2004) Aging transition and universal scaling in oscillator networks. Phys Rev Lett 93:104101Google Scholar
  15. 15.
    D’Huys O, Vicente R, Danckaert J, Fischer I (2010) Amplitude and phase effects on the synchronization of delay-coupled oscillators. Chaos 20:043127Google Scholar
  16. 16.
    Fiedler B, Flunkert V, Georgi M, Hövel P, Schöll E (2007) Refuting the odd number limitation of time-delayed feedback control. Phys Rev Lett 98:114101Google Scholar
  17. 17.
    Gonzalez-Avella JC, Cosenza MG, Miguel MS (2014) Localized coherence in two interacting populations of social agents. Phys A 399:24–30Google Scholar
  18. 18.
    Gambuzza LV, Buscarino A, Chessari S, Fortuna L, Meucci R, Frasca M (2014) Experimental investigation of chimera states with quiescent and synchronous domains in coupled electronic oscillators. Phys Rev E 90:032905Google Scholar
  19. 19.
    Gjurchinovski A, Zakharova A, Schöll E (2014) Amplitude death in oscillator networks with variable-delay coupling. Phys Rev E 89:032915Google Scholar
  20. 20.
    García-Morales V, Krischer K (2012) The complex Ginzburg-Landau equation: an introduction. Contemp Phys 53:79–95Google Scholar
  21. 21.
    Hagerstrom AM, Murphy TE, Roy R, Hövel P, Omelchenko I, Schöll E (2012) Experimental observation of chimeras in coupled-map lattices. Nat Phys 8:658–661Google Scholar
  22. 22.
    Haugland SW, Schmidt L, Krischer K (2015) Self-organized alternating chimera states in oscillatory media. Sci Rep 5:9883Google Scholar
  23. 23.
    Hizanidis J, Kanas V, Bezerianos A, Bountis T (2014) Chimera states in networks of nonlocally coupled Hindmarsh-Rose neuron models. Int J Bifurcat Chaos 24:1450030Google Scholar
  24. 24.
    Hizanidis J, Panagakou E, Omelchenko I, Schöll E, Hövel P, Provata A (2015) Chimera states in population dynamics: networks with fragmented and hierarchical connectivities. Phys Rev E 92:012915Google Scholar
  25. 25.
    Kalle P (2014) Chimera states in Stuart-Landau networks. Master’s thesis, Technische Universität BerlinGoogle Scholar
  26. 26.
    Kalle P, Sawicki J, Zakharova A, Schöll E (2017) Chimera states and the interplay between initial conditions and non-local coupling. Chaos 27:033110Google Scholar
  27. 27.
    Kapitaniak T, Kuzma P, Wojewoda J, Czolczynski K, Maistrenko Y (2014) Imperfect chimera states for coupled pendula. Sci Rep 4:6379Google Scholar
  28. 28.
    Kemeth FP, Haugland SW, Schmidt L, Kevrekidis YG, Krischer K (2016) A classification scheme for chimera states. Chaos 26:094815Google Scholar
  29. 29.
    Ko TW, Ermentrout GB (2008) Partially locked states in coupled oscillators due to inhomogeneous coupling. Phys Rev E 78:016203Google Scholar
  30. 30.
    Kuramoto Y, Battogtokh D (2002) Coexistence of coherence and incoherence in nonlocally coupled phase oscillators. Nonlinear Phenom Complex Syst 5:380–385Google Scholar
  31. 31.
    Kyrychko YN, Blyuss KB, Schöll E (2011) Amplitude death in systems of coupled oscillators with distributed-delay coupling. Eur Phys J B 84:307–315Google Scholar
  32. 32.
    Kyrychko YN, Blyuss KB, Schöll E (2014) Synchronization of networks of oscillators with distributed-delay coupling. Chaos 24:043117Google Scholar
  33. 33.
    Laing CR, Chow CC (2001) Stationary bumps in networks of spiking neurons. Neural Computation 13:1473–1494Google Scholar
  34. 34.
    Laing CR (2009) The dynamics of chimera states in heterogeneous Kuramoto networks. Phys D 238:1569–1588Google Scholar
  35. 35.
    Laing CR (2010) Chimeras in networks of planar oscillators. Phys Rev E 81:066221Google Scholar
  36. 36.
    Laing CR (2015) Chimeras in networks with purely local coupling. Phys Rev E 92:050904(R)Google Scholar
  37. 37.
    Larger L, Penkovsky B, Maistrenko Y (2013) Virtual chimera states for delayed-feedback systems. Phys Rev Lett 111:054103Google Scholar
  38. 38.
    Larger L, Penkovsky B, Maistrenko Y (2015) Laser chimeras as a paradigm for multistable patterns in complex systems. Nat Commun 6:7752Google Scholar
  39. 39.
    Lehnert J, Hövel P, Selivanov AA, Fradkov AL, Schöll E (2014) Controlling cluster synchronization by adapting the topology. Phys Rev E 90:042914Google Scholar
  40. 40.
    Levnajic Z, Pikovsky A (2010) Phase resetting of collective rhythm in ensembles of oscillators. Phys Rev E 82:056202Google Scholar
  41. 41.
    Loos S, Claussen JC, Schöll E, Zakharova A (2016) Chimera patterns under the impact of noise. Phys Rev E 93:012209Google Scholar
  42. 42.
    Maistrenko Y, Sudakov O, Osiv O, Maistrenko VL (2015) Chimera states in three dimensions. New J Phys 17:073037Google Scholar
  43. 43.
    Majhi S, Bera BK, Ghosh D, Perc M (2018) Chimera states in neuronal networks: a review. Phys Life Rev 26Google Scholar
  44. 44.
    Martens EA, Thutupalli S, Fourriere A, Hallatschek O (2013) Chimera states in mechanical oscillator networks. Proc Natl Acad Sci USA 110:10563Google Scholar
  45. 45.
    Motter AE, Myers SA, Anghel M, Nishikawa T (2013) Spontaneous synchrony in power-grid networks. Nat Phys 9:191–197Google Scholar
  46. 46.
    Nkomo S, Tinsley MR, Showalter K (2013) Chimera states in populations of nonlocally coupled chemical oscillators. Phys Rev Lett 110:244102Google Scholar
  47. 47.
    Omelchenko I, Maistrenko Y, Hövel P, Schöll E (2011) Loss of coherence in dynamical networks: spatial chaos and chimera states. Phys Rev Lett 106:234102Google Scholar
  48. 48.
    Omelchenko I, Riemenschneider B, Hövel P, Maistrenko Y, Schöll E (2012) Transition from spatial coherence to incoherence in coupled chaotic systems. Phys Rev E 85:026212Google Scholar
  49. 49.
    Omel’chenko OE, Wolfrum M, Yanchuk S, Maistrenko Y, Sudakov O (2012) Stationary patterns of coherence and incoherence in two-dimensional arrays of non-locally-coupled phase oscillators. Phys Rev E 85:036210Google Scholar
  50. 50.
    Omelchenko I, Omel’chenko OE, Hövel P, Schöll E (2013) When nonlocal coupling between oscillators becomes stronger: patched synchrony or multichimera states. Phys Rev Lett 110:224101Google Scholar
  51. 51.
    Omel’chenko OE (2013) Coherence-incoherence patterns in a ring of non-locally coupled phase oscillators. Nonlinearity 26:2469Google Scholar
  52. 52.
    Omelchenko I, Provata A, Hizanidis J, Schöll E, Hövel P (2015) Robustness of chimera states for coupled FitzHugh-Nagumo oscillators. Phys Rev E 91:022917Google Scholar
  53. 53.
    Omelchenko I, Zakharova A, Hövel P, Siebert J, Schöll E (2015) Nonlinearity of local dynamics promotes multi-chimeras. Chaos 25:083104Google Scholar
  54. 54.
    Omelchenko I, Omel’chenko OE, Zakharova A, Wolfrum M, Schöll E (2016) Tweezers for chimeras in small networks. Phys Rev Lett 116:114101Google Scholar
  55. 55.
    Omel’chenko OE (2018) The mathematics behind chimera states. Nonlinearity 31:R121Google Scholar
  56. 56.
    Panaggio MJ, Abrams DM (2015) Chimera states: Coexistence of coherence and incoherence in networks of coupled oscillators. Nonlinearity 28:R67Google Scholar
  57. 57.
    Rattenborg NC, Amlaner CJ, Lima SL (2000) Behavioral, neurophysiological and evolutionary perspectives on unihemispheric sleep. Neurosci Biobehav Rev 24:817–842Google Scholar
  58. 58.
    Rattenborg NC, Voirin B, Cruz SM, Tisdale R, Dell’Omo G, Lipp HP, Wikelski M, Vyssotski AL (2016) Evidence that birds sleep in mid-flight. Nat Commun 7:12468Google Scholar
  59. 59.
    Rosin DP, Rontani D, Gauthier DJ (2014) Synchronization of coupled Boolean phase oscillators. Phys Rev E 89:042907Google Scholar
  60. 60.
    Rothkegel A, Lehnertz K (2014) Irregular macroscopic dynamics due to chimera states in small-world networks of pulse-coupled oscillators. New J Phys 16:055006Google Scholar
  61. 61.
    Sakaguchi H (2006) Instability of synchronized motion in nonlocally coupled neural oscillators. Phys Rev E 73:031907Google Scholar
  62. 62.
    Schmidt L, Schönleber K, Krischer K, García-Morales V (2014) Coexistence of synchrony and incoherence in oscillatory media under nonlinear global coupling. Chaos 24:013102Google Scholar
  63. 63.
    Schmidt L, Krischer K (2015) Chimeras in globally coupled oscillatory systems: from ensembles of oscillators to spatially continuous media. Chaos 25:064401Google Scholar
  64. 64.
    Schmidt L, Krischer K (2015) Clustering as a prerequisite for chimera states in globally coupled systems. Phys Rev Lett 114:034101Google Scholar
  65. 65.
    Schöll E (2016) Synchronization patterns and chimera states in complex networks: interplay of topology and dynamics. Eur Phys J Spec Top 225:891–919Google Scholar
  66. 66.
    Selivanov AA, Lehnert J, Dahms T, Hövel P, Fradkov AL, Schöll E (2012) Adaptive synchronization in delay-coupled networks of Stuart-Landau oscillators. Phys Rev E 85:016201Google Scholar
  67. 67.
    Semenov V, Feoktistov A, Vadivasova T, Schöll E, Zakharova A (2015) Time-delayed feedback control of coherence resonance near subcritical Hopf bifurcation: theory versus experiment. Chaos 25:033111Google Scholar
  68. 68.
    Semenov V, Zakharova A, Maistrenko Y, Schöll E (2016) Delayed-feedback chimera states: forced multiclusters and stochastic resonance. Europhys Lett 115:10005Google Scholar
  69. 69.
    Semenova N, Zakharova A, Anishchenko VS, Schöll E (2016) Coherence-resonance chimeras in a network of excitable elements. Phys Rev Lett 117:014102Google Scholar
  70. 70.
    Sethia GC, Sen A, Atay FM (2008) Clustered chimera states in delay-coupled oscillator systems. Phys Rev Lett 100:144102Google Scholar
  71. 71.
    Sethia GC, Sen A, Johnston GL (2013) Amplitude-mediated chimera states. Phys Rev E 88:042917Google Scholar
  72. 72.
    Sethia GC, Sen A (2014) Chimera states: the existence criteria revisited. Phys Rev Lett 112:144101Google Scholar
  73. 73.
    Shima S, Kuramoto Y (2004) Rotating spiral waves with phase-randomized core in nonlocally coupled oscillators. Phys Rev E 69:036213Google Scholar
  74. 74.
    Sieber J, Omel’chenko OE, Wolfrum M (2014) Controlling unstable chaos: stabilizing chimera states by feedback. Phys Rev Lett 112:054102Google Scholar
  75. 75.
    Tsigkri-DeSmedt ND, Hizanidis J, Hövel P, Provata A (2016) Multi-chimera states and transitions in the leaky integrate-and-fire model with excitatory coupling and hierarchical connectivity. Eur Phys J Spec Top 225:1149Google Scholar
  76. 76.
    Teramae JN, Tanaka D (2004) Robustness of the noise-induced phase synchronization in a general class of limit cycle oscillators. Phys Rev Lett 93:204103Google Scholar
  77. 77.
    Tinsley MR, Nkomo S, Showalter K (2012) Chimera and phase cluster states in populations of coupled chemical oscillators. Nat Phys 8:662–665Google Scholar
  78. 78.
    Ulonska S, Omelchenko I, Zakharova A, Schöll E (2016) Chimera states in networks of Van der Pol oscillators with hierarchical connectivities. Chaos 26:094825Google Scholar
  79. 79.
    Vadivasova TE, Strelkova G, Bogomolov SA, Anishchenko VS (2016) Correlation analysis of the coherence-incoherence transition in a ring of nonlocally coupled logistic maps. Chaos 26:093108Google Scholar
  80. 80.
    Viktorov EA, Habruseva T, Hegarty SP, Huyet G, Kelleher B (2014) Coherence and incoherence in an optical comb. Phys Rev Lett 112:224101Google Scholar
  81. 81.
    Vüllings A, Schöll E, Lindner B (2014) Spectra of delay-coupled heterogeneous noisy nonlinear oscillators. Eur Phys J B 87:31Google Scholar
  82. 82.
    Wickramasinghe M, Kiss IZ (2013) Spatially organized dynamical states in chemical oscillator networks: synchronization, dynamical differentiation, and chimera patterns. PLoS ONE 8:e80586Google Scholar
  83. 83.
    Wille C, Lehnert J, Schöll E (2014) Synchronization-desynchronization transitions in complex networks: an interplay of distributed time delay and inhibitory nodes. Phys Rev E 90:032908Google Scholar
  84. 84.
    Wolfrum M, Omel’chenko OE (2011) Chimera states are chaotic transients. Phys Rev E 84:015201Google Scholar
  85. 85.
    Xie J, Knobloch E, Kao HC (2014) Multicluster and traveling chimera states in nonlocal phase-coupled oscillators. Phys Rev E 90:022919Google Scholar
  86. 86.
    Yeldesbay A, Pikovsky A, Rosenblum M (2014) Chimeralike states in an ensemble of globally coupled oscillators. Phys Rev Lett 112:144103Google Scholar
  87. 87.
    Zakharova A, Kapeller M, Schöll E (2014) Chimera death: symmetry breaking in dynamical networks. Phys Rev Lett 112:154101Google Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institut für Theoretische PhysikTechnische Universität BerlinBerlinGermany

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