ALEX 2018 Workshop: Abstracts

Coexistence of Hamiltonian-like and dissipative dynamics in chains of coupled oscillators

Serhiy Yanchuk${}^{(1)}$, Oleksandr Burylko${}^{\mathrm{(}\mathrm{2}\mathrm{)}}$, Alexander Mielke${}^{\mathrm{(}\mathrm{3}\mathrm{,}\mathrm{4}\mathrm{)}}$, and Matthias Wolfrum${}^{\mathrm{(}\mathrm{3}\mathrm{)}}$

(1) Institute of Mathematics, Technische Universität Berlin (Germany)

(2) Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv (Ukraine)

(3) Weierstraß-Institut Berlin (Germany)

(4) Institute of Mathematics, Humboldt-Universität zu Berlin (Germany)

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 $N\to \mathrm{\infty }$ 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.

Acknowledgments: OB acknowledges financial support from Erasmus Mundus (Grant MID2012
B895) for the work in Humboldt University. AM was partially supported by DFG within the Collaborative Research Center 910 through Project A5. MW and SY were partially supported by DFG within the Collaborative Research Center 910 through Project A3.