WIAS Preprint No. 2447, (2017)

Coexistence of Hamiltonian-like and dissipative dynamics in chains of coupled phase oscillators with skew-symmetric coupling


  • Burylko, Oleksandr
  • Mielke, Alexander
    ORCID: 0000-0002-4583-3888
  • Wolfrum, Matthias
  • Yanchuk, Serhiy

2010 Mathematics Subject Classification

  • 34C14 34C15 34C28, 34C30, 37C80, 37L60


  • Phase oscillators, reversible systems, amplitude equations




We consider rings of coupled phase oscillators with anisotropic coupling. When the coupling is skew-symmetric, i. e. when the anisotropy is balanced in a specific way, the system shows robustly a coexistence of Hamiltonian-like and dissipative regions in the phase space. We relate this phenomenon to the time-reversibility property of the system. The geometry of low-dimensional systems up to five oscillators is described in detail. In particular, we show that the boundary between the dissipative and Hamiltonian-like regions consists of families of heteroclinic connections. For larger chains with skew-symmetric coupling, some sufficient conditions for the coexistence are provided, and in the limit of N → ∞ oscillators, we formally derive an amplitude equation for solutions in the neighborhood of the synchronous solution. It has the form of a nonlinear Schrödinger equation and describes the Hamiltonian-like region existing around the synchronous state similarly to the case of finite rings.

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