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

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

@Article{JCM-16-403, author = {}, 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 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 KW - $Z_2$-symmetry KW - Hopf point KW - bifurcation KW - 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.