GROUP THEORETICAL MODE INTERACTIONS WITH DIFFERENT SYMMETRIES

1994 ◽  
Vol 04 (01) ◽  
pp. 177-191 ◽  
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.

FOLLOWING PATHS OF SYMMETRY-BREAKING BIFURCATION POINTS

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.

Amplitude Modulation and Symmetry Breaking Bifurcation in Micro Devices

2012 ◽  
Author(s):  
Z. C. Feng ◽  
Mahmoud Almasri
Keyword(s):  
Symmetry Breaking ◽  
Numerical Simulations ◽  
Amplitude Modulation ◽  
Dynamic Responses ◽  
Symmetric Structure ◽  
Different Types ◽  
Asymmetric Responses ◽  
Symmetric Response ◽  
Symmetry Breaking Bifurcation

Designs of many micro devices take advantage of the symmetry for better performance, immunity to noise, and for simpler analysis. When a symmetric structure is subjected to symmetric forcing, the symmetric response can become unstable leading to asymmetric responses. The occurrence of symmetry breaking bifurcation leads to complicated dynamic responses which often result in less desirable performances. In this paper, we obtain analytical criteria for the onset of symmetry breaking bifurcations. We also conduct numerical simulations to demonstrate different types of asymmetric dynamic responses resulting from the symmetry breaking bifurcation. In particular, we show the occurrence of amplitude modulated motions in such systems.

Symmetry-breaking bifurcation for the one-dimensional Hénon equation

2019 ◽  
Vol 21 (01) ◽  
pp. 1750097
Author(s):  
Inbo Sim ◽  
Satoshi Tanaka
Keyword(s):  
Symmetry Breaking ◽  
Bifurcation Point ◽  
Global Bifurcation ◽  
One Dimensional ◽  
Bifurcation Points ◽  
Connected Set ◽  
Hénon Equation ◽  
The One ◽  
Symmetry Breaking Bifurcation

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.

scholarly journals Automating Symmetry-Breaking Calculations

2004 ◽  
Vol 7 ◽  
pp. 101-119 ◽  
Author(s):  
P. C. Matthews
Keyword(s):  
Group Theory ◽  
Symmetry Breaking ◽  
Symmetry Group ◽  
Irreducible Representations ◽  
Symmetry Groups ◽  
Periodic Patterns ◽  
Alternating Groups ◽  
Spatial Symmetry

AbstractThe process of classifying possible symmetry-breaking bifurcations requires a computation involving the subgroups and irreducible representations of the original symmetry group. It is shown how this calculation can be automated using a group theory package such as GAP. This enables a number of new results to be obtained for larger symmetry groups, where manual computation is impractical. Examples of symmetric and alternating groups are given, and the method is also applied to the spatial symmetry-breaking of periodic patterns observed in experiments.

Numerical Methods for Steady State/Hopf Mode Interactions

1997 ◽  
Vol 07 (03) ◽  
pp. 585-605 ◽  
Author(s):  
F. Amdjadi ◽  
P. J. Aston
Keyword(s):  
Steady State ◽  
Numerical Methods ◽  
Symmetry Breaking ◽  
Mode Interaction ◽  
Extended System ◽  
Test Function ◽  
Alternative Approach ◽  
Mode Interactions ◽  
Sivashinsky Equation ◽  
Two Parameter

Numerical methods for dealing with steady state/Hopf mode interactions using extended systems are considered. In particular, it is shown that such a mode interaction corresponds to a symmetry breaking bifurcation of a Hopf extended system as well as a Hopf bifurcation of a symmetry breaking extended system. Non-degeneracy conditions associated with these bifurcations are derived and interpreted in the context of the mode interaction. The alternative approach of using a single test function instead of a full extended system is considered in detail in one of the cases. Numerical results for a two-parameter version of the Kuramoto–Sivashinsky equation are presented to illustrate the theory.

scholarly journals Numerical computation of symmetry-breaking bifurcation points

1993 ◽  
Vol 35 (1) ◽  
pp. 103-113 ◽  
Author(s):  
B. S. Attili
Keyword(s):  
Symmetry Breaking ◽  
Direct Method ◽  
Pitchfork Bifurcation ◽  
Multiple Shooting ◽  
Dimensional Problem ◽  
Bifurcation Points ◽  
Finite Dimensional ◽  
Parameter Dependent ◽  
A Direct Method ◽  
Symmetry Breaking Bifurcation

AbstractWe consider symmetry-breaking bifurcation points which arise in parameter-dependent nonlinear equations of the form f(x, λ) = 0. These types of bifurcation points are connected to pitchfork bifurcation points. A direct method is used to compute such points. Multiple shooting is used to discretise the two-point boundary-value problems to obtain a finite-dimensional problem.

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