We consider a one-dimensional directional array of diffusively coupled oscillators. They are perturbed by the injection of small additive noise, typically orders of magnitude smaller than the oscillation amplitude, and the system is studied in a region of the parameters that would yield deterministic synchronization. Non-normal directed couplings seed a coherent amplification of the perturbation: this latter manifests as a modulation, transversal to the limit cycle, which gains in potency node after node. If the lattice extends long enough, the initial synchrony gets eventually lost, and the system moves toward a nontrivial attractor, which can be analytically characterized as an asymptotic splay state. The noise assisted instability, ultimately vehiculated and amplified by the non-normal nature of the imposed couplings, eventually destabilizes also this second attractor. This phenomenon yields spatiotemporal patterns, which cannot be anticipated by a conventional linear stability analysis.
Zankoc, C., Fanelli, D., Ginelli, F., & Livi, R. (2019). Desynchronization and pattern formation in a noisy feed-forward oscillator network. Physical Review E, 99(1). https://doi.org/10.1103/PhysRevE.99.012303