Singularity theory for equivariant gradient bifurcation problems

1993 ◽  
pp. 33-62
Author(s):  
Thomas J. Bridges ◽  
Jacques E. Furter
Keyword(s):  
Singularity Theory ◽  
Bifurcation Problems

The numerical analysis of bifurcation problems with application to fluid mechanics

Acta Numerica ◽  
2000 ◽  
Vol 9 ◽  
pp. 39-131 ◽  
Author(s):  
K. A. Cliffe ◽  
A. Spence ◽  
S. J. Tavener
Keyword(s):  
Numerical Analysis ◽  
Fluid Mechanics ◽  
Stokes Equations ◽  
Mixed Finite Element ◽  
Finite Difference Methods ◽  
Singularity Theory ◽  
Mixed Finite Element Method ◽  
Convergence Theory ◽  
Navier Stokes ◽  
Bifurcation Problems

In this review we discuss bifurcation theory in a Banach space setting using the singularity theory developed by Golubitsky and Schaeffer to classify bifurcation points. The numerical analysis of bifurcation problems is discussed and the convergence theory for several important bifurcations is described for both projection and finite difference methods. These results are used to provide a convergence theory for the mixed finite element method applied to the steady incompressible Navier–Stokes equations. Numerical methods for the calculation of several common bifurcations are described and the performance of these methods is illustrated by application to several problems in fluid mechanics. A detailed description of the Taylor–Couette problem is given, and extensive numerical and experimental results are provided for comparison and discussion.

BIFURCATION ANALYSIS ON A SELF-EXCITED HYSTERETIC SYSTEM

2004 ◽  
Vol 14 (08) ◽  
pp. 2825-2842 ◽  
Author(s):  
ZHIQIANG WU ◽  
PEI YU ◽  
KEQI WANG
Keyword(s):  
Mechanical System ◽  
Singularity Theory ◽  
Bifurcation Problem ◽  
Van Der Pol ◽  
Smooth Functions ◽  
Restoring Force ◽  
Symbolic Notation ◽  
Hysteretic Damper ◽  
Bifurcation Problems ◽  
Unconstrained Problems

This paper investigates periodic bifurcation solutions of a mechanical system which involves a van der Pol type damping and a hysteretic damper representing restoring force. This system has recently been studied based on the singularity theory for bifurcations of smooth functions. However, the results do not actually take into account the property of nonsmoothness involved in the system. In particular, the transition varieties due to constraint boundaries were ignored, resulting in failure in finding some important bifurcation solutions. To reveal all possible bifurcation patterns for such systems, a new method is developed in this paper. With this method, a continuous, piecewise smooth bifurcation problem can be transformed into several subbifurcation problems with either single-sided or double-sided constraints. Further, the constrained bifurcation problems are converted to unconstrained problems and then singularity theory is employed to find transition varieties. Explicit formulas are applied to reconsider the mechanical system. Numerical simulations are carried out to verify analytical predictions. Moreover, symbolic notation for a sequence of bifurcations is introduced to easily show the characteristics of bifurcations, and also the comparison of different bifurcations. The method developed in this paper can be easily extended to study bifurcation problems with other types of nonsmoothness.

scholarly journals Dn-forced symmetry breaking of O(2)-equivariant problems

2002 ◽  
Vol 132 (5) ◽  
pp. 1185-1218
Author(s):  
Jacques-Elie Furter ◽  
Angela Maria Sitta
Keyword(s):  
Symmetry Breaking ◽  
Orthogonal Group ◽  
Group Actions ◽  
Singularity Theory ◽  
Periodic Perturbation ◽  
Subharmonic Solutions ◽  
Autonomous Equation ◽  
Bifurcation Problems ◽  
Image Position ◽  
Symmetry Breaking Bifurcation

We use singularity theory to classify forced symmetry-breaking bifurcation problems where f1 is O(2)-equivariant and f2 is Dn-equivariant with the orthogonal group actions on z ∈ R2. Forced symmetry breaking occurs when the symmetry of the equation changes when parameters are varied. We explicitly apply our results to the branching of subharmonic solutions in a model periodic perturbation of an autonomous equation and sketch further applications.

Bifurcation on the hexagonal lattice and the planar Bénard problem

1983 ◽  
Vol 308 (1505) ◽  
pp. 617-667 ◽  
Keyword(s):  
Stable Equilibrium ◽  
Plane Waves ◽  
Hexagonal Lattice ◽  
Singularity Theory ◽  
Boussinesq Equations ◽  
Point Of View ◽  
Equilibrium Solutions ◽  
Bénard Problem ◽  
Benard Problem ◽  
Bifurcation Problems

One of the simplifications, used by Sattinger (1978), in studying the planar Bénard problem is to assume that the solutions are doubly periodic with respect to the hexagonal lattice in the plane. Once one makes this assumption, the generic situation is that the kernel of the linearized Boussinesq equations (linearized about the pure conduction solution) is six-dimensional, the eigenfunctions being superpositions of plane waves along three directions at mutual angles of 120°. In this situation the Liapunov-Schmidt procedure leads to a reduced bifurcation problem of the form g ( x, λ ) = 0 where g: [R6 x R -» R6 is smooth. Here λ represents the Rayleigh number. Moreover, such a g must commute with the symmetry group of the hexagonal lattice. In the paper we study such covariant bifurcation problems from the point of view of singularity theory and group theory, thus refining the work of Sattinger (1978). In particular we are able to classify the simplest such bifurcation problems as well as all of their perturbations. We find that stable rolls and stable hexagons occur as possible solutions. In addition, we find a rich structure of non-stable equilibrium solutions including wavy rolls and false hexagons appearing in the unfoldings of even the simplest degenerate bifurcation problems.

scholarly journals Origin Preserving Path Formulation for Multiparameter ℤ2-Equivariant Corank 2 Bifurcation Problems

2020 ◽  
Vol 30 (09) ◽  
pp. 2050140
Author(s):  
Jacques-Elie Furter
Keyword(s):  
Singularity Theory ◽  
Cylindrical Panel ◽  
Qualitative Behavior ◽  
Partial Classification ◽  
Equivariant Bifurcation ◽  
Bifurcation Problems ◽  
Multiparameter Bifurcation ◽  
Generic Bifurcation ◽  
Minimal Modification

A singularity theory, in the form of path formulation, is developed to analyze and organize the qualitative behavior of multiparameter [Formula: see text]-equivariant bifurcation problems of corank 2 and their deformations when the trivial solution is preserved as parameters vary. Path formulation allows for an efficient discussion of different parameter structures with a minimal modification of the algebra between cases. We give a partial classification of one-parameter problems. With a couple of parameter hierarchies, we show that the generic bifurcation problems are 2-determined and of topological codimension-0. We also show that the preservation of the trivial solutions is an important hypotheses for multiparameter bifurcation problems. We apply our results to the bifurcation of a cylindrical panel under axial compression.

𝔻n-forced symmetry breaking of 𝕆(2)-equivariant problems

2002 ◽  
Vol 132 (5) ◽  
pp. 1185-1218
Author(s):  
Jacques-Elie Furter ◽  
Angela Maria Sitta
Keyword(s):  
Symmetry Breaking ◽  
Orthogonal Group ◽  
Group Actions ◽  
Singularity Theory ◽  
Periodic Perturbation ◽  
Subharmonic Solutions ◽  
Autonomous Equation ◽  
Bifurcation Problems ◽  
Image Position ◽  
Symmetry Breaking Bifurcation

We use singularity theory to classify forced symmetry-breaking bifurcation problems where f1 is 𝕆(2)-equivariant and f2 is 𝔻n-equivariant with the orthogonal group actions on z ∈ ℝ2. Forced symmetry breaking occurs when the symmetry of the equation changes when parameters are varied. We explicitly apply our results to the branching of subharmonic solutions in a model periodic perturbation of an autonomous equation and sketch further applications.

scholarly journals Singularity theory and equivariant bifurcation problems with parameter symmetry

1996 ◽  
Vol 120 (3) ◽  
pp. 547-578 ◽  
Author(s):  
Jacques-Élie Furter ◽  
Angela Maria Sitta ◽  
Ian Stewart
Keyword(s):  
Symmetry Breaking ◽  
Singularity Theory ◽  
Equivariant Bifurcation ◽  
Bifurcation Problems ◽  
Parameter Symmetry

The study of equivariant bifurcation problems via singularity theory (Golubitsky and Schaeffer[8], Golubitsky, Stewart and Schaeffer[9]) has been mainly concerned with models exhibiting spontaneous symmetry-breaking. The solutions of such bifurcation problems lose symmetry as the parameters vary, but the equations that they satisfy retain the same symmetry throughout.

scholarly journals Bifurcation Analysis of Composite Laminated Piezoelectric Rectangular Plate Structure in the Case of 1:2 Internal Resonance

2019 ◽  
Vol 2019 ◽  
pp. 1-23 ◽  
Author(s):  
Y. H. Guo ◽  
W. Zhang
Keyword(s):  
Rectangular Plate ◽  
Internal Resonance ◽  
Singularity Theory ◽  
Bifurcation Problem ◽  
Universal Unfolding ◽  
Sign Function ◽  
Plate Structure ◽  
Parametric Plane ◽  
Transition Set ◽  
Bifurcation Problems

In this paper, the authors study the bifurcation problems of the composite laminated piezoelectric rectangular plate structure with three bifurcation parameters by singularity theory in the case of 1:2 internal resonance. The sign function is employed to the universal unfolding of bifurcation equations in this system. The proposed approach can ensure the nondegenerate conditions of the universal unfolding of bifurcation equations in this system to be satisfied. The study presents that the proposed system with three bifurcation parameters is a high codimensional bifurcation problem with codimension 4, and 6 forms of universal unfolding are given. Numerical results show that the whole parametric plane can be divided into several persistent regions by the transition set, and the bifurcation diagrams in different persistent regions are obtained.

scholarly journals Stochastic P-bifurcation in a bistable Van der Pol oscillator with fractional time-delay feedback under Gaussian white noise excitation

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Yajie Li ◽  
Zhiqiang Wu ◽  
Guoqi Zhang ◽  
Feng Wang ◽  
Yuancen Wang
Keyword(s):  
Time Delay ◽  
White Noise ◽  
Gaussian White Noise ◽  
Singularity Theory ◽  
Van Der Pol Oscillator ◽  
Order System ◽  
Damping Force ◽  
Van Der Pol ◽  
Restoring Force ◽  
White Noise Excitation

Abstract The stochastic P-bifurcation behavior of a bistable Van der Pol system with fractional time-delay feedback under Gaussian white noise excitation is studied. Firstly, based on the minimal mean square error principle, the fractional derivative term is found to be equivalent to the linear combination of damping force and restoring force, and the original system is further simplified to an equivalent integer order system. Secondly, the stationary Probability Density Function (PDF) of system amplitude is obtained by stochastic averaging, and the critical parametric conditions for stochastic P-bifurcation of system amplitude are determined according to the singularity theory. Finally, the types of stationary PDF curves of system amplitude are qualitatively analyzed by choosing the corresponding parameters in each area divided by the transition set curves. The consistency between the analytical solutions and Monte Carlo simulation results verifies the theoretical analysis in this paper.

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