Volume 25, Issue 3
Hopf Bifurcation and Time Periodic Orbits with pde2path – Algorithms and Applications

Hannes Uecker

Commun. Comput. Phys., 25 (2019), pp. 812-852.

Published online: 2018-11

Preview Full PDF 150 1658
Export citation
  • 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.

  • Keywords

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

  • AMS Subject Headings

35J47 35B22 37M20

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address
  • BibTex
  • RIS
  • TXT
@Article{CiCP-25-812, author = {Hannes Uecker}, title = {Hopf Bifurcation and Time Periodic Orbits with pde2path – Algorithms and Applications}, journal = {Communications in Computational Physics}, year = {2018}, volume = {25}, number = {3}, pages = {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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2017-0181}, url = {http://global-sci.org/intro/article_detail/cicp/12830.html} }
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://dor.org/10.4208/cicp.OA-2017-0181 UR - https://global-sci.org/intro/cicp/12830.html KW - Hopf bifurcation KW - periodic orbit continuation KW - Floquet multipliers KW - partial differential equations KW - finite element method KW - reaction-diffusion KW - distributed optimal control. AB -

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.

Hannes Uecker. (1970). Hopf Bifurcation and Time Periodic Orbits with pde2path – Algorithms and Applications. Communications in Computational Physics. 25 (3). 812-852. doi:10.4208/cicp.OA-2017-0181
Copy to clipboard
The citation has been copied to your clipboard