One of the most powerful methods of bifurcation theory is centre manifold reduction, in which a judicious coordinate change greatly simplifies the analysis of dynamical systems in the vicinity of a bifurcation point. This technique is widely used in the study of non-linear systems, but is less well suited to networks of coupled dynamical systems. Naively, there seems to be little connection between the network structure of a dynamical system and the structure of a centre manifold reduction. In a recent paper, however, LML External Fellow Claire Postlethwaite and colleagues show that such a connection does often exist, and sometimes leads to unexpected bifurcations and dynamics.
Other researchers have recently developed an elegant approach to understanding synchrony in networks, as well as other technical issues concerning centre manifold reduction, based on so-called graph fibrations. Their results show that in some cases a centre manifold reduction of a network system has a network structure of its own, inherited from the original network. Their viewpoint is algebraic, and most effective for homogeneous networks in which every node receives exactly one input of each of a specific list of types. In the new work, Postlethwaite and colleagues consider a class of networks that is almost the exact opposite – fully inhomogeneous networks, in which all nodes and arrows have distinct types. Examples include biochemical or gene regulatory networks, as well as food webs.
The authors consider two standard types of local bifurcations – steady-state and Hopf – on any dynamical system that is admissible for a fully inhomogeneous network. Overall, their analysis reveals how it is possible to apply an appropriate version of singularity theory, adapted to the network of the centre manifold and the type of local bifurcation. This leads to a determination of a normal form for either class of bifurcation.
The paper is available at here.