Self-organization in predominantly feedforward oscillator chains

Results on the dynamics of certain predominantly feedforward networks of phase oscillators are presented. We start with a purely feedforward chain and show numerically that in certain parameter ranges, independent of the length of the chain, there is a global attractor that is effectively a two-dimensional torus. Specifically, we find that after the system has reached its steady state, the phases of all the oscillators in the chain at any one moment in time are determined entirely by the phases of the first two oscillators. In the cases tested this phenomenon is found to persist with the addition of feedback couplings whose strengths may be up to a significant fraction of the feedforward strengths.