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@ARTICLE{Manik:840231,
      author       = {Manik, Debsankha and Timme, Marc and Witthaut, Dirk},
      title        = {{C}ycle flows and multistability in oscillatory networks},
      journal      = {Chaos},
      volume       = {27},
      number       = {8},
      issn         = {1089-7682},
      address      = {Woodbury, NY},
      publisher    = {American Institute of Physics},
      reportid     = {FZJ-2017-07785},
      pages        = {083123 -},
      year         = {2017},
      abstract     = {We study multistability in phase locked states in networks
                      of phase oscillators under both Kuramoto dynamics and swing
                      equation dynamics - a popular model for studying
                      coarse-scale dynamics of an electrical AC power grid. We
                      first establish the existence of geometrically frustrated
                      states in such systems - where although a steady state flow
                      pattern exists, no fixed point exists in the dynamical
                      variables of phases due to geometrical constraints. We then
                      describe the stable fixed points of the system with phase
                      differences along each edge not exceeding pi/2 in terms of
                      cycle flows - constant flows along each simple cycle - as
                      opposed to phase angles or flows. The cycle flow formalism
                      allows us to compute tight upper and lower bounds to the
                      number of fixed points in ring networks. We show that long
                      elementary cycles, strong edge weights, and spatially
                      homogeneous distribution of natural frequencies (for the
                      Kuramoto model) or power injections (for the oscillator
                      model for power grids) cause such networks to have more
                      fixed points. We generalize some of these bounds to
                      arbitrary planar topologies and derive scaling relations in
                      the limit of large capacity and large cycle lengths, which
                      we show to be quite accurate by numerical computation.
                      Finally, we present an algorithm to compute all phase locked
                      states - both stable and unstable - for planar networks.},
      cin          = {IEK-STE},
      ddc          = {530},
      cid          = {I:(DE-Juel1)IEK-STE-20101013},
      pnm          = {153 - Assessment of Energy Systems – Addressing Issues of
                      Energy Efficiency and Energy Security (POF3-153) /
                      VH-NG-1025 - Helmholtz Young Investigators Group
                      "Efficiency, Emergence and Economics of future supply
                      networks" $(VH-NG-1025_20112014)$ / CoNDyNet - Kollektive
                      Nichtlineare Dynamik Komplexer Stromnetze $(PIK_082017)$},
      pid          = {G:(DE-HGF)POF3-153 / $G:(HGF)VH-NG-1025_20112014$ /
                      $G:(Grant)PIK_082017$},
      typ          = {PUB:(DE-HGF)16},
      pubmed       = {pmid:28863499},
      UT           = {WOS:000409112600027},
      doi          = {10.1063/1.4994177},
      url          = {https://juser.fz-juelich.de/record/840231},
}