The concept of a semi-linear feedback system was introduced in chapter 7 of my Pitman notes Lectures on bifurcations, dynamics and symmetry. The concept was developed, partly in collaboration with Peter Ashwin, over the period 2003-2006 and mainly in connection with work on heteroclinic cycles in coupled cell systems. A selection of pdf files of talks, warts and all, given over the period 2004-2005, give some of the general ideas. Almost everything said here is elucidated and expanded on in the preprint Heteroclinic networks in homogeneous and heterogeneous identical cell systems (see under Recent Publications - available from May 1, 2014).
Research supported by NSF Grants DMS-0806321, DMS-1210262 and (current) DMS-1265253.