Variational and Topological Methods: Theory, Applications,
Numerical Simulations, and Open Problems.
Electron. J. Diff. Eqns., Conference 21 (2014), pp. 6176.
Symmetry analysis and numerical solutions for semilinear elliptic systems
C. Tyler Diggans, John M. Neuberger, James W. Swift
Abstract:
We study a twoparameter family of socalled Hamiltonian
systems defined on a region
in
with the bifurcation parameters
and
of the form:
taking
to be a function of two variables satisfying certain
conditions. We use numerical methods adapted from Automated Bifurcation
Analysis for Nonlinear Elliptic Partial Difference Equations on Graphs
(Inter. J. Bif. Chaos, 2009) to approximate solution pairs.
After providing a symmetry analysis of the solution space of pairs of
functions defined on the unit square, we numerically approximate bifurcation
surfaces over the two dimensional parameter space. A cusp catastrophe
is found on the diagonal in the parameter space where
and is
explained in terms of symmetry breaking bifurcation. Finally, we suggest
a more theoretical direction for our future work on this topic.
Published February 10, 2014.
Math Subject Classifications: 35J15, 65N30.
Key Words: Nonlinear elliptic PDE; elliptic systems; Newton's method;
GNGA; bifurcation.
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C. Tyler Diggans
Department of Mathematics
Northern Arizona University, 86005
Flagstaff, AZ 86011, USA
email: Tyler.Diggans@nau.edu


John M. Neuberger
Department of Mathematics
Northern Arizona University, 86005
Flagstaff, AZ 86011, USA
email: John.Neuberger@nau.edu


Jim W. Swift
Department of Mathematics
Northern Arizona University, 86005
Flagstaff, Arizona, USA
email: Jim.Swift@nau.edu

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