Destabilization patterns in chains of coupled oscillators

We describe the mechanism of destabilization in a chain of identical coupled oscillators. Along with the transition from stationary to oscillatory behavior of the single oscillator, the network undergoes a complicated bifurcation scenario including the coexistence of multiple periodic orbits with different frequencies, spatial patterns, and modulation instabilities. This scenario, which is similar to the well-known Eckhaus scenario in spatially extended systems, occurs here also in the case of purely convective unidirectional coupling, and hence it cannot be explained as a simple discretization of its spatially continuous counterpart. Although the number of coexisting periodic orbits grows with the number of oscillators, we are able to treat this problem independently of the actual size of the network by investigating the limiting equations for the related spectral problems.