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Synchronization and Chimera States in Complex Networks

Lehrveranstaltung 3233 L 525 im Wintersemester 2015/16

Dozent: Prof. Yuri Maistrenko

Vorlesung: (Beginn 14.10.)
Mi 10:15-11:45 (EW 731)

Für Studierende der Physik (Master und Diplom-Hauptstudium), des Masterstudiengangs Computational Neuroscience und interessierte Mitglieder des SFB 910.

When attending this course 3 credit points within the ECTS system can be obtained. The course will be given in English.

Zusammen mit der Lehrveranstaltung LV 3232 L 152 „Theoretische Physik VI (Vertiefung): Nichtgleichgewichtsstatistik“ kann diese Vorlesung zu einem physikalischen Wahlpflichtmodul (grundlagenorientierte Studienrichtung) mit 12 ECTS kombiniert werden.


The lectures cover major ideas of complex network theory, with emphasis on nonlinear dynamics and applications in real-world systems. A focus is put on characterization of emergent collective behavior when the network nodes are coupled over various configurations, including the Kuramoto model of coupled phase oscillators and coupled chaotic maps. Conditions for different types of synchronization and clustering will be derived with combination of analytical methods and numerical simulations. The main attention will be paid to chimera states as one of the most fascinating recent discoveries in nonlinear science, illuminating self-organized spatial coexistence of synchronization and desynchronization. Experimental evidence of chimera states will be presented for such fields as neuroscience, chemistry, optics, fluid dynamics, and mechanical systems.

More specifically, the following topics will be discussed in the lectures:

  • Synchronization in complex networks, theory and application
  • Nonlinear dynamics, bifurcations and chaos
  • Kuramoto model: collective dynamics and phase chaos
  • Chaotic synchronization, riddling and blowout bifurcations
  • Chimera states as a new paradigm of network science
  • Different types of chimera states in Kuramoto, FitzHugh-Nagumo and coupled map models
  • Kuramoto model with inertia, solitary states and spatial chaos
  • Self-propelled chimeras in the Vicsek model
  • Chimera states in delayed-feedback systems
  • Chimera states in 2 and 3 spatial dimensions: spiral and scroll wave chimeras
  • Experimental evidence of chimera states

The lecture slides are available here:


Prof. Yuri Maistrenko
Mi 13:00 - 14:00


Books and review papers:

  • S. Strogatz, Nonlinear Dynamics and Chaos: With Applications To Physics Biology, Chemistry And Engineering, Westview Press (2000)
  • S. Strogatz, Exploring complex networks, Nature 410, 268–276 (2001)
  • E. Mosekilde, Y. Maistrenko, and D. Postnov, Chaotic Synchronization: Application to Living Systems, World Scientific (2002)
  • Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence, Springer (1984)
  • E. Ott, Chaos in dynamical systems, 2nd Edition. Cambridge Univ. Press (2002)
  • S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, and D. Hwang, Complex networks: Structure and dynamics, Physics Reports 424, 175–308 (2006)
  • M. Panaggio and D. Abrams, Chimera states: coexistence of coherence and incoherence in networks of coupled oscillators, Nonlinerity 28, 67–87 (2015)

Journal papers:

  • Y. Maistrenko and T. Kapitaniak, Different types of chaos synchronization in two coupled piecewise linear maps, Phys. Rev. E. 54, 3285–3292 (1996)
  • M. Hasler and Y. Maistrenko, An introduction to the synchronization of chaotic systems: coupled skew tent maps, IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 44, 856–866 (1997)
  • Y. Maistrenko, O. V. Popovych, and P. A. Tass, Desynchronization and Chaos in the Kuramoto Model, Lect. Notes Phys. 671, 285–306 (2005)
  • O. V. Popovych, Y. Maistrenko, and P. A. Tass, Phase chaos in coupled oscillators, Phys. Rev. E. 71 065201 (2005)
  • Y. Kuramoto and D. Battogtokh, Coexistence of coherence and incoherence in nonlocally coupled phase oscillators, Nonlinear Phenom. Complex Syst. 5, 380 (2002)
  • S. Shima and Y. Kuramoto, Rotating spiral waves with phase-randomized core in nonlocally coupled oscillators, Phys. Rev. E 69, 036213 (2004)
  • D. Abrams and S. Strogatz, Chimera states for coupled oscillators, Phys. Rev. Lett. 93, 174102 (2004)
  • O. Omelchenko, Y. Maistrenko, and P. A. Tass, Chimera states: the natural link between coherence and incoherence, Phys. Rev. Lett. 100, 044105 (2008)
  • O. Omelchenko, M. Wolfrum, and Y. Maistrenko, Chimera states as chaotic spatiotemporal patterns, Phys. Rev. E 81, 065201 (2010)
  • M. Wolfrum, O. Omelchenko, S. Yanchuk, and Y. Maistrenko, Spectral properties of chimera states, Chaos 21, 013112 (2011)
  • I. Omelchenko, Y. Maistrenko, P. Hövel, E. Schöll, Loss of coherence in dynamical networks: spatial chaos and chimera states, Phys. Rev. Lett. 106, 234102 (2011)
  • A. Zakharova , M. Kapeller, and E.Schöll, Chimera death: symmetry breaking in dynamical networks, Phys. Rev. Lett. 112, 154101 (2014)
  • D. Dudkowski, Y. Maistrenko, and T. Kapitaniak, Different types of chimera states: An interplay between spatial and dynamical chaos, Phys. Rev. E 90, 032920 (2014)
  • O. Omelchenko, M. Wolfrum, S. Yanchuk, Y. Maistrenko, and O. Sudakov, Stationary patterns of coherence and incoherence in two-dimensional arrays of non-locally coupled phase oscillators, Phys. Rev. E 85, 036210 (2012)
  • M. Panaggio and D. Abrams, Chimera states on a flat torus, Phys. Rev. Lett. 110 094102 (2013)
  • L. Larger, B. Penkovsky, and Y. Maistrenko, Virtual chimera state for delayed-feedback systems, Phys. Rev.Lett. 111, 054103 (2013)
  • L. Larger, B. Penkovsky, and Y. Maistrenko, Laser Chimeras as a paradigm for multi-stable patterns in complex systems, Nat. Comm. 6, 7752 (2015)
  • Y. Maistrenko, B. Penkovsky, and M. Rosenblum, Solitary state at the edge of synchrony in ensembles with attractive and repulsive interactions, Phys. Rev. E 89, 060901(R) (2014)
  • Y. Maistrenko, A. Vasylenko, O. Sudakov, R. Levchenko, and V. Maistrenko, Cascades of multi-headed chimera states for coupled phase oscillators, Int. J. Bif. and Chaos 24, 1440014 (2014)
  • T. Kapitaniak, P. Kuzma, J. Wojewoda, K. Czolczynski, and Y. Maistrenko, Imperfect chimera states for coupled pendula, Sci. Rep. 4, 6379 (2014)
  • P. Jaros, Y. Maistrenko, and T. Kapitaniak, Chimera states on the route from coherence to rotating waves, Phys. Rev. E 91, 022907 (2015)
  • Y. Maistrenko, O. Sudakov, O. Osiv, and V. Maistrenko, Chimera states in three dimensions, New J. Phys. 17, 073037 (2015)

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