Bifurcations on Fully Inhomogeneous Networks

Yangyang Wang (Mathematical Biosciences Institute, The Ohio State University)

(October 16, 2018 10:20 AM - 11:05 AM)

Bifurcations on Fully Inhomogeneous Networks

Abstract

Mathematically, network dynamics can be formalized as coupled cell systems, where

each cell is a system of differential equations. Networks arise naturally in many areas of biology such as gene regulation, ecology and biochemistry. In these fields, networks are often fully inhomogeneous in the sense that all cells are distinct and all couplings are different. Motivated by this, we consider dynamics on fully inhomogeneous networks. Center manifold reduction is a standard technique in bifurcation theory, reducing the essential features of local bifurcations to equations in a small number of variables corresponding to critical eigenvalues. This method can be applied to admissible differential equations for a network, but it bears no obvious relation to the network structure. For fully inhomogeneous networks, however, there are general circumstances in which the center manifold reduced equations inherit a network structure of their own. This observation is used to analyze codimension one and two local bifurcations. For codimension one,  only one critical component is involved and generic local bifurcations are saddle-node and standard Hopf. For codimension two, we focus on the case when one component is downstream from the other in the feedforward structure. Here the generic bifurcations, within the realm of network-admissible equations, differ significantly from generic codimension two bifurcations in a general dynamical system.