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Hopf Bifurcation and Time Periodic Orbits with pde2path – Algorithms and Applications

Hopf Bifurcation and Time Periodic Orbits with pde2path – Algorithms and Applications
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Year:    2019

Communications in Computational Physics, Vol. 25 (2019), Iss. 3 : pp. 812–852

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 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.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/cicp.OA-2017-0181

Communications in Computational Physics, Vol. 25 (2019), Iss. 3 : pp. 812–852

Published online:    2019-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    41

Keywords:    Hopf bifurcation periodic orbit continuation Floquet multipliers partial differential equations finite element method reaction-diffusion distributed optimal control.

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