scholarly journals Computing all Pairs (λ,μ) Such That λ is a Double Eigenvalue of A+μB

2011 ◽  
Vol 32 (3) ◽  
pp. 902-927 ◽  
Author(s):  
Elias Jarlebring ◽  
Simen Kvaal ◽  
Wim Michiels
Keyword(s):  
Double Eigenvalue

Bifurcation to vortex solutions in superconducting films

1997 ◽  
Vol 8 (2) ◽  
pp. 125-148 ◽  
Author(s):  
T. BOECK ◽  
S. J. CHAPMAN
Keyword(s):  
Magnetic Field ◽  
Superconducting Films ◽  
Leading Order ◽  
Nonlinear Stability Analysis ◽  
Slab Geometry ◽  
Weakly Nonlinear ◽  
Double Eigenvalue ◽  
Wide Range ◽  
Vortex Solutions ◽  
Parameter Values

The bifurcation from a normally conducting state to a superconducting state in a decreasing magnetic field is studied for a slab geometry. The leading eigenvalue is a double eigenvalue, leading to a rich structure of possible behaviours. A weakly-nonlinear stability analysis is performed, and the possible responses of the material are classified. Finally, the leading-order equations are solved numerically for a wide range of parameter values to determine which of these behaviours will occur in practice.

Bifurcations near a multiple eigenvalue of the rectangular plate problem

1986 ◽  
Vol 103 (1-2) ◽  
pp. 35-62
Author(s):  
Zoltán Sadovský
Keyword(s):  
Aspect Ratio ◽  
Rectangular Plate ◽  
Small Perturbation ◽  
Perturbation Parameter ◽  
Multiple Eigenvalue ◽  
Bifurcation Problem ◽  
Operator Equations ◽  
Double Eigenvalue ◽  
Elastic Rectangular Plate ◽  
Degenerate Cases

SynopsisWe consider the bifurcation problem of the Föppl–Kármán equations for a thin elastic rectangular plate near a multiple eigenvalue allowing for a small perturbation parameter related to the aspect ratio of the plate. The first step in the study is to introduce equivalent operator equations in the energy spaces of the problem which explicitly contain the perturbation parameter. By dealing partially with a general formulation, we obtain the main results for the double eigenvalue and Z2 ⊓ Z2 symmetry of bifurcation equations. We are chiefly interested in the degenerate cases of bifurcation equations.

On the stability of flow in an elliptic pipe which is nearly circular

1994 ◽  
Vol 281 ◽  
pp. 357-369 ◽  
Author(s):  
A. Davey ◽  
H. Salwen
Keyword(s):  
Cross Section ◽  
Specific Problem ◽  
Sectional Area ◽  
Leading Order ◽  
Straight Pipe ◽  
Cross Sectional ◽  
Double Eigenvalue ◽  
Circular Problem ◽  
Qualitative Structure ◽  
The Stability

In an earlier paper (Davey 1978) the first author investigated the linear stability of flow in a straight pipe whose cross-section was an ellipse, of small ellipticity e, by regarding the flow as a perturbation of Poiseuille flow in a circular pipe. That paper contained some serious errors which we correct herein. We show analytically that for the most important mode n = 1, for which the circular problem has a double eigenvalue c0 as the ‘swirl’ can be in either direction, the ellipticity splits the double eigenvalue into two separate eigenvalues c0 ± e2c12, to leading order, when the cross-sectional area of the pipe is kept fixed. The imaginary part of c12 is non-zero and so the ellipticity always makes the flow less stable. This specific problem is generic to a much wider class of fluid dynamical problems which are made less stable when the symmetry group of the dynamical system is reduced from S1 to Z2.In the Appendix, P. G. Drazin describes simply the qualitative structure of this problem, and other problems with the same symmetries, without technical detail.

Bifurcation from a double eigenvalue in the unstirred chemostat

2013 ◽  
Vol 92 (7) ◽  
pp. 1449-1461 ◽  
Author(s):  
Gaihui Guo ◽  
Jianhua Wu ◽  
Yan'e Wang
Keyword(s):  
Double Eigenvalue

The bifurcation and secondary bifurcation of capillary-gravity waves

1985 ◽  
Vol 399 (1817) ◽  
pp. 391-417 ◽  
Keyword(s):  
Gravity Waves ◽  
Phase Speed ◽  
Implicit Function ◽  
Existence And Multiplicity ◽  
A Value ◽  
Double Eigenvalue ◽  
Bifurcation Phenomena ◽  
Critical Phase ◽  
Secondary Bifurcation ◽  
Special Case

The bifurcation and secondary bifurcation of capillary-gravity waves is analysed when the surface tension is close to or equal to a value where the eigenspace of the critical phase speed has multiplicity two. The existence and multiplicity of solutions is seen, via the implicit function theorem, to be a special case of the secondary bifurcation phenomena, which occur when a double eigenvalue splits, under perturbation, into two simple eigenvalues in the presence of a symmetry in the problem.

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