Modal and wave synchronization in coupled self-excited oscillators.

In addition to a common synchronization and/or localization behavior, a system of linearly coupled identical bistable van der Pol (BVdP) oscillators can exhibit a "non-conventional" or "modal" synchronization. In a two degree-of-freedom case, one can observe stable beatings attractor with synchronized amplitudes of the symmetric and antisymmetric modes. The current study demonstrates that this unusual behavior is generic and quite ubiquitous, if the system is explored in an appropriate parametric regime. In the absence of the self- excitation, the system of coupled linear oscillators possesses a complete degenerate set of non-interacting eigenmodes. If the system is symmetric and the coupling is weak enough, appropriate initial conditions will result in a continuous multi-parametric family of stationary beatings or beating waves. If the self-excitation terms are small enough, i.e., even weaker than the weak coupling, one can expect that the degeneracy will be removed and stable attractors very close to some special stationary or wave beating responses will appear. This phenomenon is demonstrated in chains of ring-coupled BVdP oscillators. In particular, we observe simple two-wave synchronization for N=2,3,5,6,7 coupled oscillators; for N=2, it corresponds to the modal synchronization observed previously. The case N=4 is special: due to internal resonances in the slow flow, the two-wave synchronization turns unstable, and more complicated patterns of the multi-wave synchronization are revealed. Analytic models manage to capture the shapes of the observed beat waves. All results are verified by direct numeric simulations.