Hopf Bifurcation and Time Periodic Orbits with pde2path – Algorithms and Applications (Communications in Computational Physics, 2018)
We describe the algorithms used in the Matlab continuation and bifurcationpackage pde2path for Hopf bifurcation and continuation of branches of periodic orbitsin systems of PDEs in 1, 2, and 3 spatial dimensions, including the computationof Floquet multipliers. We first test the methods on three reaction diffusion examples,namely a complex Ginzburg-Landau equation as a toy problem, a reaction diffusionsystem on a disk with rotational waves including stable spirals bifurcating out of thetrivial solution, and a Brusselator system with interaction of Turing and Turing-Hopfbifurcations. Then we consider a system from distributed optimal control, which isill-posed as an initial value problem and thus needs a particularly stable method forcomputing Floquet multipliers, for which we use a periodic Schur decomposition. Theimplementation details how to use pde2path on these problems are given in an accompanyingtutorial, which also includes a number of further examples and algorithms,for instance on Hopf bifurcation with symmetries, on Hopf point continuation, and onbranch switching from periodic orbits.