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Volume 16, Issue 5
Computation of Hopf Branches Bifurcating from a Hopf/Pitchfork Point for Problems with $Z_2$-Symmetry

Baisheng Wu & Tassilo Küpper

J. Comp. Math., 16 (1998), pp. 403-416.

Published online: 1998-10

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  • Abstract

This paper is concerned with the computation of Hopf branches emanating from a Hopf/Pitchfork point in a two-parameter nonlinear problem satisfying a $Z_2$-symmetry condition. Our aim is to present a new approach to the theoretical and computational analysis of the bifurcating Hopf branches at this singular point by using the system designed to calculate Hopf points and exploring its symmetry. It is shown that a Hopf/Pitchfork point is a pitchfork bifurcation point in the system. Hence standard continuation and branch-switching can be used to compute these Hopf branches. In addition, an effect method based on the extended system of the singular points is developed for the computation of branch of secondary (non-symmetric) Hopf points. The implementation of Newton's method for solving the extended system is also discussed. A numerical example is given.

  • Keywords

Hopf/pitchfork point, $Z_2$-symmetry, Hopf point, bifurcation, Extended system.

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COPYRIGHT: © Global Science Press

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@Article{JCM-16-403, author = {Wu , Baisheng and Küpper , Tassilo}, title = {Computation of Hopf Branches Bifurcating from a Hopf/Pitchfork Point for Problems with $Z_2$-Symmetry}, journal = {Journal of Computational Mathematics}, year = {1998}, volume = {16}, number = {5}, pages = {403--416}, abstract = {

This paper is concerned with the computation of Hopf branches emanating from a Hopf/Pitchfork point in a two-parameter nonlinear problem satisfying a $Z_2$-symmetry condition. Our aim is to present a new approach to the theoretical and computational analysis of the bifurcating Hopf branches at this singular point by using the system designed to calculate Hopf points and exploring its symmetry. It is shown that a Hopf/Pitchfork point is a pitchfork bifurcation point in the system. Hence standard continuation and branch-switching can be used to compute these Hopf branches. In addition, an effect method based on the extended system of the singular points is developed for the computation of branch of secondary (non-symmetric) Hopf points. The implementation of Newton's method for solving the extended system is also discussed. A numerical example is given.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/9171.html} }
TY - JOUR T1 - Computation of Hopf Branches Bifurcating from a Hopf/Pitchfork Point for Problems with $Z_2$-Symmetry AU - Wu , Baisheng AU - Küpper , Tassilo JO - Journal of Computational Mathematics VL - 5 SP - 403 EP - 416 PY - 1998 DA - 1998/10 SN - 16 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/9171.html KW - Hopf/pitchfork point, $Z_2$-symmetry, Hopf point, bifurcation, Extended system. AB -

This paper is concerned with the computation of Hopf branches emanating from a Hopf/Pitchfork point in a two-parameter nonlinear problem satisfying a $Z_2$-symmetry condition. Our aim is to present a new approach to the theoretical and computational analysis of the bifurcating Hopf branches at this singular point by using the system designed to calculate Hopf points and exploring its symmetry. It is shown that a Hopf/Pitchfork point is a pitchfork bifurcation point in the system. Hence standard continuation and branch-switching can be used to compute these Hopf branches. In addition, an effect method based on the extended system of the singular points is developed for the computation of branch of secondary (non-symmetric) Hopf points. The implementation of Newton's method for solving the extended system is also discussed. A numerical example is given.

Wu , Baisheng and Küpper , Tassilo. (1998). Computation of Hopf Branches Bifurcating from a Hopf/Pitchfork Point for Problems with $Z_2$-Symmetry. Journal of Computational Mathematics. 16 (5). 403-416. doi:
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