Hopf bifurcation and time periodic orbits in reaction-diffusion systems with pde2path - algorithms and applications

TL;DR

This paper details algorithms implemented in pde2path for analyzing Hopf bifurcations and periodic orbits in PDE systems across multiple dimensions, demonstrated through various reaction-diffusion examples and an optimal control application.

Contribution

It introduces new algorithms for Hopf bifurcation and periodic orbit continuation in PDEs, including stable Floquet multiplier computation, with practical implementation guidance.

Findings

Validated methods on reaction-diffusion examples

Successfully computed bifurcations and periodic orbits in complex systems

Demonstrated stable Floquet multiplier calculation in ill-posed problems

Abstract

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 (anti) 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, together with all other downloads (function libraries, demos and further documentation) can be found at http://www.staff.uni-oldenburg.de/hannes.uecker/pde2path.