Multistability in an ensemble of phase oscillatorswith long-distance couplings

The work is devoted to investigation of multistability of running waves in a ring of periodic oscillators with diffusive non-local couplings. It analyzes the influence of long-range couplings and their change with distance on the stability of spatially-periodic regimes with different wave numbers.
The research are carried out by numerical (computer) experiments. The system under study is an ensemble of identical phase oscillators. It is, on the one hand the most simple model providing opportunities for analytical studies; on the other hand its properties can be generalized to an arbitrary ensemble of almost harmonic self sustained oscillators.
The studies have shown that multistability is observed in the ensemble of phase oscillators as a set of coexisting modes of waves running along the ring, the every of which characterizing by its own phase-shift between oscillations in the neighboring sites. In the case of stationary or slowly varying couplings a running wave mode remains stable while the total phase shift on the interval of interaction keeps to be less than $\pi/2$. When the couplings are decreased sufficiently fast, the stabilization of the maximum value of the phase shift is observed, so the further increase of range of interaction no longer affects the number of coexisting modes. Thus, the multistability keeps to exist in an ensemble with an arbitrarily distance of interconnection. The study the basins of attraction of the running waves have demonstrated that basins of long-wavelength modes are increased and simultaneously basins of short-wavelength modes are decreased while the range of interconnection grows.