Mode Interactions With Symmetry

Castro, Sofia B. S. D.

doi:10.1080/02681119508806192

Cited by 4 publications

(3 citation statements)

References 10 publications

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“…In the Hopf-steady-state interaction problem, these correspond to bifurcations to tori. These secondary bifurcations to periodic solutions are to be expected as was shown by the author in [3]. In the present paper, we are concerned with the evolution of the limit cycle arising from this secondary bifurcation.…”

Section: Introductionsupporting

confidence: 65%

The disappearance of the limit cycle in a mode interaction problem with symmetry

Castro¹

1997

Nonlinearity

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For a steady-state mode interaction problem with Z 2 symmetry Golubitsky, Stewart and Schaeffer (1988 Singularities and Groups in Bifurcation Theory vol II (New York: Springer) ch XIX) prove that there are parameter values for which a Hopf bifurcation occurs along a mixed-mode branch. The author has proved that it is always so in mode interaction problems with symmetry (Castro S B S D 1995 Mode interactions with symmetry Dyn. Stability Syst. 10 13-31). In this work, we are concerned with what happens to the limit cycle arising from this Hopf bifurcation. We find that, for certain parameter values, it vanishes through a homoclinic connection. To prove this, we use Melnikov theory since the equations under study are a perturbation of Hamiltonian equations. This completes the bifurcation diagrams in Golubitsky et al (above).

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Section: Introductionsupporting

confidence: 65%

The disappearance of the limit cycle in a mode interaction problem with symmetry

Castro¹

1997

Nonlinearity

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“…Gomes also developed algorithms for finding prenormal forms for domains with additional symmetry (boxes with square cross-section, cubes) and interactions of more than two modes. For multidimensional rectangular domains, Gomes and Stewart (1994b) subsequently extended the ideas to Hopf bifurcation, a process begun by Castro (1990Castro ( , 1995. Field et al (1991) applied the method to bifurcation on hemispheres, described a more general class of domains for which the method applies and proved a basic theorem on the regularity of extensions by reflection, which underpins the method.…”

Section: Reaction-di$usion On An Intervalmentioning

confidence: 98%

Hidden symmetries and pattern formation in Lapwood convection

Impey¹,

Roberts

Stewart

1996

Dynamics and Stability of Systems

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We study Lapwood convection (convection of afluid in aporous medium) on a two-dimensional rectangular domain. The linearized eigenmodes are symmetric p x q cellular patterns, which we call (p, q) modes. Numerical calculations of the branching structure near mode interaction points have derived bifurcation diagrams for the (3, 1)/(1, 1) and (3, 1)/(2, 2) mode interactions which are non-generic, even when the rectangular symmety of the domain is taken into account. This has raised questions about the accuracy of the numerical method used, a finite-element Galerkin approximation implemented using Harwell's E N T W F E code. We show that this apparent lack of genericity is partly a consequence of 'hidden' translational symmetries, which arise when the problem is extended to one with periodic bounda y conditions. This extension procedure has become standard for partial diferential equations (PDEs) with Neumann or Dirichlet bounday conditions, and it reveals restrictions on the Liapunou-Schmidt reduced bifurcation equations and the resulting singulan'ty-theoretic normal forms. Its application to Lapwood convection is unusual in that the PDE involves a mixture of both Neumann and Dirichlet bounday conditions. Specifically, on the vertical sidewalls the stream function satisfies Dirichlet bounda y conditions (is zero), but the temperature satisfies Neumann (no-flux) bounda y conditions. Nevertheless, we show that for abstract group-theoretical reasons the same symmety constraints that occur for purely Neumann bounda y conditions are imposed on the LiapunowSchmidt reduced bzfurcation equations, and therefore the same list of normal forms is valid. The hidden symmetries force certain terms in the reduced bzfurcation equations to be zero and change the generic branching geomety. With the aid of M A C S Y M A , we determine a small number of low-order coeficients of the reduced bzfurcation equations which are needed to find the correct normal form. We show that in some cases the normal form is more degenerate than might be anticipated, but that when these degeneracies are taken into account the resulting branching geomety reproduces that found in the earlier numerical approach. In particular, we obtain an analytic vindication of the numerical method. 0268-1 110/96/030155-38 Q 1996 Journals Oxford Ltd Downloaded by [Virginia Tech Libraries] at 03:40 15 March 2015 156 M. IMPEY ET AL.4

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“…Results on equivariant bifurcation theory can be found in Golubitsky and Stewart's book [10]. For mode interactions we refer the reader to Castro [4,5,6] and [11,Ch XIX and XX].…”

Section: Bifurcation Problems With Euclidean Symmetrymentioning

confidence: 99%