Multistability of traveling waves in an ensemble of harmonic oscillators with long-range couplings

The work is devoted to study of multistability of traveling waves in a ring of harmonic oscillators with a linear non-local couplings. It analyses the influence of the strength and radius of the couplings on stability of spatially periodic regimes with different values of their wavelengths. The system under study is an array of identical van der Pol generators in the approximation of quasi-harmonic oscillations. On the one hand, the chosen model is a very simple one, that allows analytical studies; on the other hand, it applicable to a wide range of oscillatory systems with almost harmonic behavior.The research of the multistability is carried out in the way of the constucting analytical solutions by means of the method of slowly-changing amplitudes and then, by the standard methods of the stability analysis of the linearization matrix eigenvalues. In some cases the analitycal solution are supported by numerical calculations. The study has shown that the number of simultaneously coexisting regimes is bounded by the value of the phase shift between oscillations of the subsystems on the length of the links. In the contrary of the locally coupled oscillators, here the maximum value of the phase shift may exceed the value of $0.5\pi$ and can reach a value of $0.7\pi$. The every coexisting wave is born from the equilibrium in the origin as a saddle limit cycle (excluding the in-phase oscillating mode), which then becomes stable further on the parameter. Regions of stability of spatially periodic regimes represent a set of cones, where regions of shorter wave locate inside of the regions with much longer ones.