Coexistence of Hamiltonian-like and dissipative dynamics in chains of coupled phase oscillators with skew-symmetric coupling
- Burylko, Oleksandr
- Mielke, Alexander
- 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.
- SIAM J. Appl. Dyn. Syst., 17 (2018), pp. 2076--2105, DOI 10.1137/17M1155685 .