Angeli, David; Banaji, Murad; Pantea, Casian Combinatorial approaches to Hopf bifurcations in systems of interacting elements. (English) Zbl 1315.15007 Commun. Math. Sci. 12, No. 6, 1101-1133 (2014). The main contribution is the introduction of a new type of labeled digraphs associated to a product of matrices, called DSR\(^{[2]}\) graphs. They reveal to be an important tool for investigating the spectrum of products of real matrices belonging to certain classes. This issue arises in the study of Hopf bifurcations of certain dynamical systems, including systems modeling chemical reactions networks. Since the vector fields of these systems can be written as a product of two matrices, the problems under discussion can be formulated as follows: given a matrix \(A\), does the spectrum of \(AB\) have nonzero pure imaginary eigenvalues, for each matrix \(B\) having the same sign pattern as the transpose \(A^t\)?; or, given a matrix \(A\), does the spectrum of \(AB\) avoid the left or the right complex half-planes for each matrix \(B\) having the same sign pattern as \(A^t\)? Answering these questions is crucial to know if the systems associated to chemical reaction networks admit Hopf bifurcations. Using DSR\(^{[2]}\) graphs, the authors derive some conditions in order to do so. These graphs are compared with the existing DSR graphs and some ideas for further applications are discussed. Several examples are included as well as a helpful background section. Reviewer: João R. Cardoso (Coimbra) Cited in 8 Documents MSC: 15A18 Eigenvalues, singular values, and eigenvectors 15A75 Exterior algebra, Grassmann algebras 34C23 Bifurcation theory for ordinary differential equations 05C90 Applications of graph theory 37C27 Periodic orbits of vector fields and flows 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 05C20 Directed graphs (digraphs), tournaments 37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems 92E20 Classical flows, reactions, etc. in chemistry 80A32 Chemically reacting flows Keywords:Hopf bifurcation; compound matrices; interaction networks; digraph; spectrum; chemical reaction network Software:Maxima PDFBibTeX XMLCite \textit{D. Angeli} et al., Commun. Math. Sci. 12, No. 6, 1101--1133 (2014; Zbl 1315.15007) Full Text: DOI arXiv