Automated Bifurcation Analysis for Nonlinear Elliptic Partial Difference Equations on Graphs

Abstract

Abstract. We seek solutions u ∈ R n 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.

Synchrony and Antisynchrony for Difference-Coupled Vector Fields on Graph Network Systems

“…Example 2.3. In [20], we approximate zeros of f ∈ D G, defined by g(x i ) = sx i + x 3 i , and h(y) = y on a graph network system with V = R. That is, we approximate solutions to…”

“…An automorphism of the graph G is a permutation σ of the cells that preserves the edges of G, that is, ij is an edge exactly when σ(i)σ(j) is an edge. The automorphisms of G form a group Aut(G), which acts on V n by σ · [20]. The fixed point subspace Fix(Σ, V n ) of an isotropy subgroup Σ is the polydiagonal subspace ∆ A , where A is the set of group orbits of the Σ action on C. The partition A obtained this way is always balanced.…”

“…When the vector field f in Equation (1) is odd, more invariant subspaces exist. For example, the trivial subspace x = 0 is invariant for any odd g and h. When f is odd, the system (1) is equivariant under the group Aut [20]. For many graph networks there are invariant subspaces for all f ∈ D G,odd that are not fixed point subspaces.…”

“…This invariant subspace is discussed in [10] and is a discrete analog of the hidden symmetry described for ODEs in [5]. Note that this invariant subspace is not a fixed point subspace of the Aut(P 6 ) × Z 2 action on the space of functions defined on the cells of P 6 [20]. (i) (ii) (iii) Figure 7.…”

“…The dynamics on an invariant synchrony subspace or anti-synchrony subspace is described by an easily constructed reduced system on the lower dimensional invariant subspace. This makes reduced systems a useful computational tool for efficiently finding solutions to partial difference equations (PdE, [20]) with given local symmetry. The reduced system should correspond to some sort of quotient cell network system.…”