Automated bifurcation Analysis for Nonlinear Elliptic Partial Difference Equations on Graphs

We seek solutions u ∈ R to the semilinear elliptic partial difference equation −Lu+ fs(u) = 0, where L is the matrix corresponding to the Laplacian operator on a graph G and fs is a one-parameter family of nonlinear functions. This article combines the ideas introduced by the authors in two papers: a) Nonlinear Elliptic Partial Difference Equations on Graphs (J. Experimental Mathematics, 2006), which introduces analytical and numerical techniques for solving such equations, and b) Symmetry and Automated Branch Following for a Semilinear Elliptic PDE on a Fractal Region (SIAM J. of Dynamical Systems, 2006), wherein we present some of our recent advances concerning symmetry, bifurcation, and automation for PDE. We apply the symmetry analysis found in the SIAM paper to arbitrary graphs in order to obtain better initial guesses for Newton’s method, create informative graphics, and better understand the role of symmetry in the underlying variational structure. We use two modified implementations of the gradient Newton-Galerkin algorithm (GNGA, Neuberger and Swift) to follow bifurcation branches in a robust way. By handling difficulties that arise when encountering accidental degeneracies and higher-dimensional critical eigenspaces, we can find many solutions of many symmetry types to the discrete nonlinear system. We present a selection of experimental results which demonstrate our algorithm’s capability to automatically generate bifurcation diagrams and solution graphics starting with only an edgelist of a graph. We highlight interesting symmetry and variational phenomena.