TY - JOUR

T1 - Automated bifurcation analysis for nonlinear elliptic partial difference equations on graphs

AU - Neuberger, John M.

AU - Sieben, Nándor

AU - Swift, James W.

N1 - Funding Information:
∗Partially supported by NSF Grant DMS-0074326.

PY - 2009/8

Y1 - 2009/8

N2 - We seek solutions u ∈ 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. 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 NewtonGalerkin 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.

AB - We seek solutions u ∈ 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. 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 NewtonGalerkin 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.

KW - Bifurcation

KW - GNGA

KW - Graphs

KW - Nonlinear difference equations

KW - Symmetry

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U2 - 10.1142/S0218127409024293

DO - 10.1142/S0218127409024293

M3 - Article

AN - SCOPUS:70449336618

VL - 19

SP - 2531

EP - 2556

JO - International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

JF - International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

SN - 0218-1274

IS - 8

ER -