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  • 1
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    Hopf bifurcation in the presence of symmetry (1985)
    In:  Other Sources
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    Publication Date: 2019-06-28
    Description: Group theory is applied to obtain generalized differential equations from the Hopf bifurcation theory on branching to periodic solutions. The conditions under which the symmetry group will admit imaginary eigenvalues are delimited. The action of the symmetry group on the circle group are explored and the Liapunov-Schmidt reduction is used to prove the Hopf theorem in the symmetric case. The emphasis is on simplifying calculations of the stability of bifurcating branches. The resulting general theory is demonstrated in terms of O(2) acting on a plane, O(n) in n-space, and O(3) and an irreducible model for spherical harmonics.
    Keywords: NUMERICAL ANALYSIS
    Type: Archive for Rational Mechanics and Analysis (ISSN 0003-9527); 87; 2 19
    Format: text
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    NASA TECHNICAL REPORTS
  • 2
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    Symmetry and stability in Taylor-Couette flow (1986)
    In:  Other Sources
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    Publication Date: 2019-07-12
    Description: The flow of a fluid between concentric rotating cylinders (the Taylor problem) is studied by exploiting the symmetries of the system. The Navier-Stokes equations, linearized about Couette flow, possess two zero and four purely imaginary eigenvalues at a suitable value of the speed of rotation of the outer cylinder. There is thus a reduced bifurcation equation on a six-dimensonal space which can be shown to commute with an action of the symmetry group 0(2) x S0(2). The group structure is used to analyze this bifurcation equation in the simplest (nondegenerate) case, and to compute the stabilities of solutions. In particular, when the outer cylinder is counterrotated, transitions which seem to agree with recent experiments of Andereck, Liu, and Swinney (1984) are obtained. It is also possible to obtain the 'main sequence' in this model. This sequence is normally observed in experiments when the outer cylinder is held fixed.
    Keywords: FLUID MECHANICS AND HEAT TRANSFER
    Type: SIAM Journal on Mathematical Analysis (ISSN 0036-1410); 17; 249-288
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    NASA TECHNICAL REPORTS
  • 3
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    A torus bifurcation theorem with symmetry (1989)
    In:  CASI
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    Publication Date: 2019-07-12
    Description: Hopf bifurcation in the presence of symmetry, in situations where the normal form equations decouple into phase/amplitude equations is described. A theorem showing that in general such degeneracies are expected to lead to secondary torus bifurcations is proved. By applying this theorem to the case of degenerate Hopf bifurcation with triangular symmetry it is proved that in codimension two there exist regions of parameter space where two branches of asymptotically stable two-tori coexist but where no stable periodic solutions are present. Although a theory was not derived for degenerate Hopf bifurcations in the presence of symmetry, examples are presented that would have to be accounted for by any such general theory.
    Keywords: NUMERICAL ANALYSIS
    Type: NASA-CR-185907 , NAS 1.26:185907 , MEMO-760 , (ISSN 0169-2690)
    Format: application/pdf
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    NASA TECHNICAL REPORTS
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