The computation of symmetry-breaking bifurcation points inZ 2×Z 2-symmetric nonlinear problems

We show the existence of a symmetry-breaking bifurcation point for the one-dimensional Hénon equation [Formula: see text] where [Formula: see text] and [Formula: see text]. Moreover, employing a variant of Rabinowitz’s global bifurcation, we obtain the unbounded connected set (the first of the alternatives about Rabinowitz’s global bifurcation), which emanates from the symmetry-breaking bifurcation point. Finally, we give an example of a bounded branch connecting two symmetry-breaking bifurcation points (the second of the alternatives about Rabinowitz’s global bifurcation) for the problem [Formula: see text] where [Formula: see text] is a specified continuous parametrization function.

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GROUP THEORETICAL MODE INTERACTIONS WITH DIFFERENT SYMMETRIES

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127494000137 ◽

1994 ◽

Vol 04 (01) ◽

pp. 177-191 ◽

Cited By ~ 5

Author(s):

KARIN GATERMANN ◽

BODO WERNER

Keyword(s):

Group Theory ◽

Symmetry Breaking ◽

Irreducible Representations ◽

Bifurcation Points ◽

Mode Interactions ◽

Different Types ◽

Secondary Bifurcation ◽

Two Parameter ◽

Symmetry Breaking Bifurcation

In two-parameter systems two symmetry breaking bifurcation points of different types coalesce generically within one point. This causes secondary bifurcation points to exist. The aim of this paper is to understand this phenomenon with group theory and the inner-connectivity of irreducible representations of supergroup and subgroups. Colored pictures of examples are included.

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Detecting symmetry-breaking bifurcation points of symmetric structures

International Journal for Numerical Methods in Engineering ◽

10.1002/nme.1620361710 ◽

1993 ◽

Vol 36 (17) ◽

pp. 3019-3039 ◽

Cited By ~ 1

Author(s):

C. Carstensen ◽

W. Wagner

Keyword(s):

Symmetry Breaking ◽

Bifurcation Points ◽

Symmetric Structures ◽

Symmetry Breaking Bifurcation

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FOLLOWING PATHS OF SYMMETRY-BREAKING BIFURCATION POINTS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127492000707 ◽

1992 ◽

Vol 02 (03) ◽

pp. 559-576

Author(s):

JOHN H. BOLSTAD

Keyword(s):

Symmetry Breaking ◽

Bifurcation Point ◽

Axisymmetric Flow ◽

Neutral Curve ◽

Bifurcation Points ◽

Nonlinear Elliptic ◽

Codimension One ◽

Aspect Ratios ◽

Two Parameter ◽

Symmetry Breaking Bifurcation

We propose a pseudo-arclength continuation algorithm for computing paths of Z 2-symmetry-breaking bifurcation points for two-parameter nonlinear elliptic problems. The algorithm consists of an Euler predictor step and a solution step composed of a sequence of Newton iterations. This work generalizes the algorithm of Werner and Spence for locating a one-parameter symmetry-breaking bifurcation point by using the approach of Keller and Fier for following a (two-parameter) path of limit points (a "fold"). By repeated use of the bordering algorithm, we solve linear systems whose matrix is the "symmetric" Jacobian or "antisymmetric" Jacobian, thus fully exploiting any (block tridiagonal) structures present. We give numerical results for the steady, axisymmetric flow between rotating coaxial cylinders (Taylor–Couette flow). For finite cylinders, we compute the fold curve and path of symmetry-breaking bifurcation points for small aspect ratios, and illustrate a new method to accurately locale the two Z 2-symmetric codimension one singularities. For infinite cylinders, we show the projections on the (aspect ratio, Reynolds number) plane of the folds and bifurcation point paths in the neighborhood of the two-cell/four-cell neutral curve crossing. We numerically verify a conjecture of Meyer–Spasche and Wagner concerning the connection of two neutral-curve crossings by a path of secondary subharmonic bifurcations.

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