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COMPUTATION OF HOPF BRANCHES BIFURCATING FROM A HOPF/PITCHFORK POINT FOR PROBLEMS WITH Z_2-SYMMETRY

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Journal of Computational Mathematics 1998年第5期16卷 403-416页

作者： Bai－sheng Wu (Department of Mathematics,Jilin University, Changchun 130023, China)Tassilo Kupper (Institute of Mathematics, University of Cologne, D-50931 Koln, Germany) 吉林大学数学系吉林长春 Univeristy of Cologne 德国

This paper is concerned with the computation of Hopf branches emanating from a Hopf/Pitchfork point in a two-parametor nonlinear problem satisfying a Z2symnletry condition. Our aim is to present a new al,proach 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 *** 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 (nonsymmetric) Hopf points. The implementation of Newton's method for solving the extended system is also discussed. A numerical example is ***: Hopf/Pitchfork point, Z2-symmetry, Hopf point, bifurcation, extended system

关键词： Hopf/pitchfork point Z_2-symmetry Hopf point bifurcation extended system

Bifurcation method for solving multiple positive solutions to boundary value problem of p-Henon equation on the unit disk

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Applied Mathematics and Mechanics(English Edition) 2010年第4期31卷 511-520页

作者：李昭祥杨忠华 Department of Mathematics Scientific Computing Key Laboratory of Shanghai UniversityShanghai Normal University

In this paper, an algorithm is proposed to solve the 0（2） symmetric positive solutions to the boundary value problem of the p-Henon equation. Taking 1 in the p- Henon equation as a bifurcation parameter, the symmetry-breaking bifurcation point on the branch of the O（2） symmetric positive solutions is found via the extended systems. The other symmetric positive solutions are computed by the branch switching method based on the Liapunov-Schmidt reduction.

关键词： p-Henon equation symmetry-breaking bifurcation extended system branch switching multiple solutions

THE　NUMERICAL　COMPUTATION　OF　SYMMETRY-BREAKING　TAKENS-BOGDANOV　POINTS

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Applied Mathematics(A Journal of Chinese Universities) 1997年第2期12卷 195-204页

作者： YE RUISONG Institute of Mathematics Shantou University Shantou

A splitting iteration method is proposed to compute symmetry breaking Takens Bogdanov points. The method could reduce the computational work and storage, it could also converge linearly with an adjustable speed. Num... 详细信息

关键词： Symmetry-breaking Takens-Bogdanov point splitting iteration method extended system

Classification of Simple Higher-Order Symmetry-Breaking Bifurcations and Their Computation

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Advances in Manufacturing 1997年第3期 175-183页

作者：叶瑞松杨忠华

This paper deals with the classification of the simple higher-order symmetry-breaking bifurcation in multiparameter nonlinear problems with Z2-symmetry. The regular extended systems for computing the simple higher-order symmetry-breaking bifurcation points with different singularities are proposed. An etficient algorithm for solving the extended systems is given. Finally, some numerical examples are shown to demonstrate the efficiency of the algorithm.

关键词： Z_2-symmetry strongly Z_2-equivalent symmetry-breaking bifurcation extended system

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计算二维零空间的非线性分歧问题的新途径

高等学校计算数学学报 2000年第3期22卷 279-287页

作者：杨忠华郭谦 200234 上海师范大学数学科学学院!上海

For the nonlinear bifurcation problems with two-dimensional null space,we propose an extel1ded system f which is generally not regular’ Applylng the pseudoarc-length homotopy continuatlon algorlthm on the singular problems, we give a newapproach to solving the nonlinear bifurcation problems with two-dimensional nullspace. The numerical exarnples show the effectiveness of the method.

关键词： Nonlinear bifurcation problems two-dimensional null space extended system pseudo arc-length homotopy continuation algorithm

Z2对称非线性方程的X0—分裂双重奇异折叠点附近的奇异点

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Northeastern Mathematical Journal 1994年第3期10卷 385-395页

作者：苏毅吴微

We consider singular Points near an X0-breaking double singular fold (DSF) Point in Z_2-symmetric nonlinear equations with two parameters, where the linearitazion has a two dimensional null space spanned by a symmetric null vector and an anti-symmetric null vector. In particular, we show that there exist two kinds of singular points passing through the DSF point, a pitchfork point Path and two simple X0-breaking bifurcaiton point paths. Their nondegeneracy is confirmed, which guarantees secondary bifurcations. A numerical example is also given. The main tools for our analysis as well as computation are some extended systems.

关键词： Double Singular fold Points Z_2-symmetry Singular Point X_0-breaking Bifurcation Point Ritchfork Point extended system Brusselator