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Multiplex Networks

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Delay Controlled Partial Synchronization in Complex Networks

Part of the book series: Springer Theses ((Springer Theses))

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Abstract

Network science presents a unique platform to study various complex real-world systems by analyzing the interactions between their constituent entities and collectively investigating their behaviors. A recent addition to the network science is the multiplex framework. These networks consisting of several interacting layers allow for remote synchronization of distant layers via an intermediate relay layer. In such case there exist synchronization between one layer and a second layer, where these two layers are not directly connected. A simple realization of such a system is a three-layer multiplex network where a relay layer in the middle, which is generally not synchronized, acts as a transmitter between two outer layers. It is the purpose of the present Chapter to extend the notion of relay synchronization from completely synchronized states to partial synchronization patterns in the individual layers and study various scenarios of synchronization of chimera states in a triplex network.

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References

  1. Aguiar MAD, Dias APS, Golubitsky M, Leite MCA (2009) Bifurcations from regular quotient networks: a first insight. Phys D 238:137–155

    Google Scholar 

  2. Ahlborn A, Parlitz U (2007) Controlling spatiotemporal chaos using multiple delays. Phys Rev E 75:65202

    Google Scholar 

  3. Albert R, Barabási AL (2002) Statistical mechanics of complex networks. Rev Mod Phys 74:47–97

    Google Scholar 

  4. Andrzejak RG, Rummel C, Mormann F, Schindler K (2016) All together now: analogies between chimera state collapses and epileptic seizures. Sci Rep 6:23000

    Google Scholar 

  5. Andrzejak RG, Ruzzene G, Malvestio I (2017) Generalized synchronization between chimera states. Chaos 17:053114

    Google Scholar 

  6. Andrzejak RG, Ruzzene G, Malvestio I, Schindler K, Schöll E, Zakharova A (2018) Mean field phase synchronization between chimera states. Chaos 28:091101

    Google Scholar 

  7. Ashwin P, Timme M (2005) Unstable attractors: existence and robustness in networks of oscillators with delayed pulse coupling. Nonlinearity 18:2035

    Google Scholar 

  8. Atay FM (ed) (2010) Complex time-delay systems, understanding complex systems. Springer, Berlin, Heidelberg

    Google Scholar 

  9. Bachmair CA, Schöll E (2014) Nonlocal control of pulse propagation in excitable media. Eur Phys J B 87:276

    Google Scholar 

  10. Banerjee T, Ghosh D, Biswas D, Schöll E, Zakharova A (2018) Networks of coupled oscillators: from phase to amplitude chimeras. Chaos 28:113124

    Google Scholar 

  11. Banerjee T, Bandyopadhyay B, Zakharova A, Schöll E (2019) Filtering suppresses amplitude chimeras. Front Appl Math Stat 5:8

    Google Scholar 

  12. Boccaletti S, Bianconi G, Criado R, del Genio CI, Gómez-Gardeñes J, Romance M, Sendiña Nadal I, Wang Z, Zanin M (2014) The structure and dynamics of multilayer networks. Phys Rep 544:1–122

    Google Scholar 

  13. Bogomolov S, Strelkova G, Schöll E, Anishchenko VS (2016) Amplitude and phase chimeras in an ensemble of chaotic oscillators. Tech Phys Lett 42:765–768

    Google Scholar 

  14. 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 43:25

    Google Scholar 

  15. Bordyugov G, Pikovsky A, Rosenblum M (2010) Self-emerging and turbulent chimeras in oscillator chains. Phys Rev E 82:035205

    Google Scholar 

  16. Bountis T, Kanas V, Hizanidis J, Bezerianos A (2014) Chimera states in a two-population network of coupled pendulum-like elements. Eur Phys J Spec Top 223:721–728

    Google Scholar 

  17. Brandstetter SA, Dahlem MA, Schöll E (2010) Interplay of time-delayed feedback control and temporally correlated noise in excitable systems. Philos Trans Royal Soc A 368:391

    Google Scholar 

  18. Bukh A, Rybalova E, Semenova N, Strelkova G, Anishchenko V (2017) New type of chimera and mutual synchronization of spatiotemporal structures in two coupled ensembles of nonlocally interacting chaotic maps. Chaos 27:111102

    Google Scholar 

  19. Bukh AV, Slepnev A, Anishchenko VS, Vadivasova TE (2018) Stability and noise-induced transitions in an ensemble of nonlocally coupled chaotic maps. Reg Chaotic Dyn 23:325–338

    Google Scholar 

  20. Buldú JM, Porter MA (2018) Frequency-based brain networks: from a multiplex framework to a full multilayer description. Netw Neurosci 2:418–441

    Google Scholar 

  21. Cholvin T, Hok V, Giorgi L, Chaillan FA, Poucet B (2018) Ventral midline thalamus is necessary for hippocampal place field stability and cell firing modulation. J Neurosci 38:158

    Google Scholar 

  22. Chouzouris T, Omelchenko I, Zakharova A, Hlinka J, Jiruska P, Schöll E (2018) Chimera states in brain networks: empirical neural versus modular fractal connectivity. Chaos 28:045112

    Google Scholar 

  23. De Domenico M (2017) Multilayer modeling and analysis of human brain networks. Gigascience 6:1–8

    Google Scholar 

  24. Feng YE, Li HH (2015) The dependence of chimera states on initial conditions. Chin Phys Lett 32:060502

    Google Scholar 

  25. Fischer I, Vicente R, Buldú JM, Peil M, Mirasso CR, Torrent MC, García-Ojalvo J (2006) Zero-lag long-range synchronization via dynamical relaying. Phys Rev Lett 97:123902

    Google Scholar 

  26. Flunkert V, D’Huys O, Danckaert J, Fischer I, Schöll E (2009) Bubbling in delay-coupled lasers. Phys Rev E 79:065201(R)

    Google Scholar 

  27. Flunkert V (2011) Delay-coupled complex systems, Springer Theses. Springer, Heidelberg

    Google Scholar 

  28. Geffert PM, Zakharova A, Vüllings A, Just W, Schöll E (2014) Modulating coherence resonance in non-excitable systems by time-delayed feedback. Eur Phys J B 87:291

    Google Scholar 

  29. Ghosh S, Kumar A, Zakharova A, Jalan S (2016) Birth and death of chimera: interplay of delay and multiplexing. Europhys Lett 115:60005

    Google Scholar 

  30. Ghosh S, Jalan S (2018) Engineering chimera patterns in networks using heterogeneous delays. Chaos 28:071103

    Google Scholar 

  31. Gjurchinovski A, Schöll E, Zakharova A (2017) Control of amplitude chimeras by time delay in dynamical networks. Phys Rev E 95:042218

    Google Scholar 

  32. Gollo LL, Mirasso CR, Atienza M, Crespo-Garcia M, Cantero JL (2011) Theta band zero-lag long-range cortical synchronization via hippocampal dynamical relaying. PLoS ONE 6:e17756

    Google Scholar 

  33. Guillery RW, Sherman SM (2002) Thalamic relay functions and their role in corticocortical communication: generalizations from the visual system. Neuron 33:163–175

    Google Scholar 

  34. Hövel P, Schöll E (2005) Control of unstable steady states by time-delayed feedback methods. Phys Rev E 72:046203

    Google Scholar 

  35. Hagerstrom AM, Murphy TE, Roy R, Hövel P, Omelchenko I, Schöll E (2012) Experimental observation of chimeras in coupled-map lattices. Nat Phys 8:658–661

    Google Scholar 

  36. Halverson E, Poremba A, Freeman JH (2015) Medial auditory thalamus is necessary for acquisition and retention of eyeblink conditioning to cochlear nucleus stimulation. Learn Mem 22:258

    Google Scholar 

  37. Halassa MM, Kastner S (2017) Thalamic functions in distributed cognitive control. Nat Neurosci 20:1669–1679

    Google Scholar 

  38. Havlin S, Kenett DY, Ben-Jacob E, Bunde A, Cohen R, Hermann H, Kantelhardt JW, Kertész J, Kirkpatrick S, Kurths J, Portugali J, Solomon S (2012) Challenges in network science: applications to infrastructures, climate, social systems and economics. Eur Phys J Spec Top 214:273–293

    Google Scholar 

  39. Jalan S, Sarkar C, Madhusudanan A, Dwivedi SK (2014) Uncovering randomness and success in society. PloS one 9:1–8

    Google Scholar 

  40. Kemeth FP, Haugland SW, Schmidt L, Kevrekidis YG, Krischer K (2016) A classification scheme for chimera states. Chaos 26:094815

    Google Scholar 

  41. Kivelä M, Arenas A, Barthélemy M, Gleeson JP, Moreno Y, Porter MA (2014) Multilayer networks. J Complex Netw 2:203–271

    Google Scholar 

  42. Kuramoto Y, Battogtokh D (2002) Coexistence of coherence and incoherence in nonlocally coupled phase oscillators. Nonlin Phen Complex Sys 5:380–385

    Google Scholar 

  43. Leyva I, Sendiña-Nadal I, Sevilla-Escoboza R, Vera-Avila VP, Chholak P, Boccaletti S (2018) Relay synchronization in multiplex networks. Sci Rep 8:8629

    Google Scholar 

  44. Majhi S, Perc M, Ghosh D (2017) Chimera states in a multilayer network of coupled and uncoupled neurons. Chaos 27:073109

    Google Scholar 

  45. Martens EA, Laing CR, Strogatz SH (2010) Solvable model of spiral wave chimeras. Phys Rev Lett 104:044101

    Google Scholar 

  46. Masoliver M, Malik N, Schöll E, Zakharova A (2017) Coherence resonance in a network of FitzHugh-Nagumo systems: interplay of noise, time-delay and topology. Chaos 27:101102

    Google Scholar 

  47. Motter AE (2010) Nonlinear dynamics: spontaneous synchrony breaking. Nat Phys 6:164–165

    Google Scholar 

  48. Omel’chenko OE, Wolfrum M, Maistrenko Y (2010) Chimera states as chaotic spatiotemporal patterns. Phys Rev E 81:065201(R)

    Google Scholar 

  49. Omelchenko I, Maistrenko Y, Hövel P, Schöll E (2011) Loss of coherence in dynamical networks: spatial chaos and chimera states. Phys Rev Lett 106:234102

    Google Scholar 

  50. Omelchenko I, Riemenschneider B, Hövel P, Maistrenko Y, Schöll E (2012) Transition from spatial coherence to incoherence in coupled chaotic systems. Phys Rev E 85:026212

    Google Scholar 

  51. 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 85:036210

    Google Scholar 

  52. Panaggio MJ, Abrams DM (2015) Chimera states: coexistence of coherence and incoherence in networks of coupled oscillators. Nonlinearity 28:R67

    Google Scholar 

  53. Rothkegel A, Lehnertz K (2014) Irregular macroscopic dynamics due to chimera states in small-world networks of pulse-coupled oscillators. New J Phys 16:055006

    Google Scholar 

  54. Ruzzene G, Omelchenko I, Schöll E, Zakharova A, Andrzejak RG (2019) Controlling chimera states via minimal coupling modification. Chaos (preprint)

    Google Scholar 

  55. Sarkar C, Yadav A, Jalan S (2016) Multilayer network decoding versatility and trust. Europhys Lett 113:18007

    Google Scholar 

  56. Sawicki J, Omelchenko I, Zakharova A, Schöll E (2018) Synchronization scenarios of chimeras in multiplex networks. Eur Phys J Spec Top 227:1161

    Google Scholar 

  57. Sawicki J, Omelchenko I, Zakharova A, Schöll E (2018) Delay controls chimera relay synchronization in multiplex networks. Phys Rev E 98:062224

    Google Scholar 

  58. Sawicki J, Ghosh S, Jalan S, Zakharova A (2019) Chimeras in multiplex networks: interplay of inter- and intra-layer delays. Front Appl Math Stat 5:19

    Google Scholar 

  59. Schöll E, Hiller G, Hövel P, Dahlem MA (2009) Time-delayed feedback in neurosystems. Phil Trans R Soc A 367:1079–1096

    Google Scholar 

  60. Schöll E, Hövel P, Flunkert V, Dahlem MA (2010) Time-delayed feedback control: from simple models to lasers and neural systems. In: Atay FM (ed) Complex time-delay systems: theory and applications. Springer, Berlin, pp 85–150

    MATH  Google Scholar 

  61. 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 25:033111

    Google Scholar 

  62. Semenova N, Zakharova A, Schöll E, Anishchenko VS (2015) Does hyperbolicity impede emergence of chimera states in networks of nonlocally coupled chaotic oscillators? Europhys Lett 112:40002

    Google Scholar 

  63. Semenov V, Zakharova A, Maistrenko Y, Schöll E (2016) Delayed-feedback chimera states: forced multiclusters and stochastic resonance. Europhys Lett 115:10005

    Google Scholar 

  64. Semenova N, Zakharova A, Anishchenko VS, Schöll E (2016) Coherence-resonance chimeras in a network of excitable elements. Phys Rev Lett 117:014102

    Google Scholar 

  65. Semenova N, Strelkova G, Anishchenko VS, Zakharova A (2017) Temporal intermittency and the lifetime of chimera states in ensembles of nonlocally coupled chaotic oscillators. Chaos 27:061102

    Google Scholar 

  66. Sethia GC, Sen A (2014) Chimera states: the existence criteria revisited. Phys Rev Lett 112:144101

    Google Scholar 

  67. Shepelev IA, Bukh AA, Vadivasova TE, Anishchenko VS, Zakharova A (2017) Double-well chimeras in 2D lattice of chaotic bistable elements. Commun Nonlinear Sci Numer Simul 54:50–61

    Google Scholar 

  68. Shepelev IA, Bukh AV, Strelkova GI, Vadivasova TE, Anishchenko VS (2017) Chimera states in ensembles of bistable elements with regular and chaotic dynamics. Nonlinear Dyn 90:2317

    Google Scholar 

  69. Singh A, Jalan S, Kurths J (2013) Role of delay in the mechanism of cluster formation. Phys Rev E 87:030902(R)

    Google Scholar 

  70. Singh A, Ghosh S, Jalan S, Kurths J (2015) Synchronization in delayed multiplex networks. Europhys Lett 111:30010

    Google Scholar 

  71. Soriano MC, García-Ojalvo J, Mirasso CR, Fischer I (2013) Complex photonics: dynamics and applications of delay-coupled semiconductors lasers. Rev Mod Phys 85:421–470

    Google Scholar 

  72. 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 26:093108

    Google Scholar 

  73. Vann SD, Nelson AJD (2015) The mammillary bodies and memory: more than a hippocampal relay. In: Progress in brain research, vol 219. Elsevier, Amsterdam, pp 163–185

    Google Scholar 

  74. Wang X, Vaingankar V, Sanchez CS, Sommer FT, Hirsch JA (2011) Thalamic interneurons and relay cells use complementary synaptic mechanisms for visual processing. Nat Neurosci 14:224

    Google Scholar 

  75. Winkler M (2018) Synchronization of chimera states in multiplex networks of logistic maps. Bachelor’s thesis, Technische Universität Berlin

    Google Scholar 

  76. Winkler M, Sawicki J, Omelchenko I, Zakharova A, Anishchenko V, Schöll E (2019) Relay synchronization in multiplex networks of discrete maps. Europhys Lett 126:50004

    Google Scholar 

  77. Wolfrum M, Omel’chenko OE, Yanchuk S, Maistrenko Y (2011) Spectral properties of chimera states. Chaos 21:013112

    Google Scholar 

  78. Yanchuk S, Wolfrum M, Hövel P, Schöll E (2006) Control of unstable steady states by long delay feedback. Phys Rev E 74:026201

    Google Scholar 

  79. Yanchuk S, Perlikowski P (2009) Delay and periodicity. Phys Rev E 79:046221

    Google Scholar 

  80. Zakharova A, Kapeller M, Schöll E (2014) Chimera death: symmetry breaking in dynamical networks. Phys Rev Lett 112:154101

    Google Scholar 

  81. Zakharova A, Semenova N, Anishchenko VS, Schöll E (2017) Time-delayed feedback control of coherence resonance chimeras. Chaos 27:114320

    Google Scholar 

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  1. Institut für Theoretische Physik, Technische Universität Berlin, Berlin, Germany

    Jakub Sawicki

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Sawicki, J. (2019). Multiplex Networks. In: Delay Controlled Partial Synchronization in Complex Networks. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-030-34076-6_7

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