TY - JOUR
T1 - Hopf Bifurcation and Time Periodic Orbits with pde2path – Algorithms and Applications
AU - Hannes Uecker
JO - Communications in Computational Physics
VL - 3
SP - 812
EP - 852
PY - 2018
DA - 2018/11
SN - 25
DO - http://doi.org/10.4208/cicp.OA-2017-0181
UR - https://global-sci.org/intro/article_detail/cicp/12830.html
KW - Hopf bifurcation, periodic orbit continuation, Floquet multipliers, partial differential
equations, finite element method, reaction-diffusion, distributed optimal control.
AB - <p style="text-align: justify;">We describe the algorithms used in the Matlab continuation and bifurcation
package pde2path for Hopf bifurcation and continuation of branches of periodic orbits
in systems of PDEs in 1, 2, and 3 spatial dimensions, including the computation
of Floquet multipliers. We first test the methods on three reaction diffusion examples,
namely a complex Ginzburg-Landau equation as a toy problem, a reaction diffusion
system on a disk with rotational waves including stable spirals bifurcating out of the
trivial solution, and a Brusselator system with interaction of Turing and Turing-Hopf
bifurcations. Then we consider a system from distributed optimal control, which is
ill-posed as an initial value problem and thus needs a particularly stable method for
computing Floquet multipliers, for which we use a periodic Schur decomposition. The
implementation details how to use pde2path on these problems are given in an accompanying
tutorial, which also includes a number of further examples and algorithms,
for instance on Hopf bifurcation with symmetries, on Hopf point continuation, and on
branch switching from periodic orbits.</p>