Combinatorial approaches to Hopf bifurcations in systems of interacting elements

Abstract

Abstract. We describe combinatorial approaches to the question of whether families of real matrices admit pairs of nonreal eigenvalues passing through the imaginary axis. When the matrices arise as Jacobian matrices in the study of dynamical systems, these conditions provide necessary conditions for Hopf bifurcations to occur in parameterised families of such systems. The techniques depend on the spectral properties of additive compound matrices: in particular, we associate with a product of matrices a signed, labelled digraph termed a DSR [2] graph, which encodes information about the second additive compound of this product. A condition on the cycle structure of this digraph is shown to rule out the possibility of nonreal eigenvalues with positive real part. The techniques developed are applied to systems of interacting elements termed "interaction networks", of which networks of chemical reactions are a special case.

Adding species to chemical reaction networks: preserving rank preserves nondegenerate behaviours

“…There is a large classical and modern literature on conditions which preclude certain dynamical behaviours in CRNs. Examples include [14,15,16,17,18,19,20,21]. Each new result in either direction helps to narrow the gap between conditions guaranteeing, and conditions ruling out, nontrivial behaviours in CRNs.…”

Mathematical Methods for Modeling Chemical Reaction Networks

“…Another powerful software is the Chemical Reaction Network Toolbox [11]. Other questions in reaction network theory have also witnessed notable progress: oscillations [12,13]; global stability and global convergence, i.e., informally speaking, the question of when all initial conditions lead to the same stable behavior [14][15][16][17][18]; persistence, i.e., whether some reactant concentrations can decrease to zero despite positive initial concentrations [14,15,19,20].…”

CoNtRol: an open source framework for the analysis of chemical reaction networks

“…An example is illustrated in Figure 1. Easily checked structural conditions on the DSR graph can imply, or contribute towards, powerful conclusions regarding multistationarity, asymptotic stability, or oscillatory behavior in a CRN (Craciun et al, 2006;Banaji and Craciun, 2009;Angeli et al, 2010Angeli et al, , 2013Shinar and Feinberg, 2013). Several of these are implemented or lend themselves naturally to future implementation.…”

“…Such parameter-free approaches to the analysis of CRNs fall into the scope of chemical reaction network theory (Horn and Jackson, 1972;Feinberg, 1972Feinberg, , 1979. Fuelled in part by its implications to systems biology (Bailey, 2001;Shinar and Feinberg, 2010), chemical reaction network theory has seen a surge of interest in recent years, attacking questions about multistationarity (Craciun et al, 2006;Conradi et al, 2007;Banaji et al, 2007;Banaji and Craciun, 2009;Shinar and Feinberg, 2012), global stability (Craciun et al, 2009;Angeli et al, 2010;Anderson, 2011;Donnell and Banaji, 2013), oscillatory behavior (Angeli et al, 2013) and persistence (Angeli et al, 2007;Pantea, 2012).…”