Benoit Dionne

Stability results for steady, spatially periodic planforms

Equivariant bifurcation theory has been used extensively to study pattern formation via symmetry-breaking steady-state bifurcation in various physical systems modelled by E(2)-equivariant partial differential equations. Much attention has been focused on solutions that are doubly periodic with respect to a square or hexagonal lattice, for which the bifurcation problem can be restricted to a finite-dimensional centre manifold. Previous studies have used four- and six-dimensional representations for the square and hexagonal lattice symmetry groups respectively, which in turn allows the relative stability of squares and rolls or hexagons and rolls to be determined. Here we consider the countably infinite set of eight- and 12-dimensional irreducible representations for the square and hexagonal cases, respectively. This extends earlier relative stability results to include a greater variety of bifurcating planforms, and also allows the stability of rolls, squares and hexagons to be established to a countably infinite set of perturbations. In each case we derive the Taylor expansion of the equivariant bifurcation problem and compute the linear, orbital stability of those solution branches guaranteed to exist by the equivariant branching lemma. In both cases we find that many of the stability results are established at cubic order in the Taylor expansion, although to completely determine the stability of certain states, higher-order terms are required. For the hexagonal lattice, all of the solution branches guaranteed by the equivariant branching lemma are, generically, unstable due to the presence of a quadratic term in the Taylor expansion. For this reason we consider two special cases: the degenerate bifurcation problem that is obtained by setting the coefficient of the quadratic term to zero, and the bifurcation problem when an extra reflection symmetry is present.

Planforms in two and three dimensions

SummaryWhen solving systems of PDE with two space dimensions it is often assumed that the solution is spatially doubly periodic. This assumption is usually made in systems such as the Boussinesq equation or reaction-diffusion equations where the equations have Euclidean invariance. In this article we use group theoretic techniques to determine a large class of spatially doubly periodic solutions that are forced to existence near a steady-state bifurcation from a translation-invariant equilibrium.This type of bifurcation problem has been considered by many authors when studying a number of different systems of PDE. Typically, these studies focus at the beginning on equilibria that are spatially periodic with respect to a fixed planar lattice type-such as square or hexagonal. Our focus is different in that we attempt to find all spatially periodic equilibria that bifurcate on all lattices. This point of view leads to some technical simplifications such as being able to restrict to translation free irreducible representations.Of course, many of the types of solutions that we find are well-known-such as hexagon and roll solutions on a hexagonal lattice. This coordinated group theoretic approach does lead, however, to solutions which seem not to have been discussed previously (antisquare solutions on a square lattice) as well as to a more complete classification of the symmetry types of possible solutions. Moreover, our methods extend to triply periodic solutions of PDE with three spatial variables. Some of these results, namely those concerned with primitive cubic lattices, are presented here. The complete results on triply periodic solutions may be found in [6, 7].

Coupled cells with internal symmetry: II. Direct products

We continue the study of arrays of coupled identical cells that possess both global and internal symmetries, begun in part I. Here we concentrate on the ‘direct product’ case, for which the symmetry group of the system decomposes as the direct product L×G of the internal group L and the global groupG. Again, the main aim is to find general existence conditions for symmetry-breaking steady-state and Hopf bifurcations by reducing the problem to known results for systems with symmetry L or G separately. Unlike the wreath product case, the theory makes extensive use of the representation theory of compact Lie groups. Again the central algebraic task is to classify axial and C-axial subgroups of the direct product and to relate them to axial and C-axial subgroups of the two groups L andG. We demonstrate how the results lead to efficient classification by studying both steady state and Hopf bifurcation in rings of coupled cells, where L = O(2) andG = Dn. In particular we show that for Hopf bifurcation the case n = 4 modulo 4 is exceptional, by exhibiting two extra types of solution that occur only for those values of n. AMS classification scheme numbers: 20xx, 57T05

Coupled Cells: Wreath Products and Direct Products

In this note we discuss the structure of systems of coupled cells (which we view as systems of ordinary differential equations) where symmetries of the system are obtained through the group L of global permutations of the cells and the group ℒof local internal symmetries of the dynamics in each cell. We show that even when the cells are assumed to be identical with identical coupling, the way that L and ℒ combine to form the total symmetry group of the system Γ depends on properties of the coupling. We illustrate this point by showing how the combination of ℒ with L can lead to a symmetry group Γ that is either a direct product or a wreath product. The symmetry group has strong implications for the dynamics of the system of cells, and the distinction between the two cases is substantial. This has important implications for the modeling of systems by coupled cells.

TIME-PERIODIC SPATIALLY PERIODIC PLANFORMS IN EUCLIDEAN EQUIVARIANT PARTIAL DIFFERENTIAL EQUATIONS

In Rayleigh-Bénard convection, the spatially uniform motionless state of a fluid loses stability as the Rayleigh number is increased beyond a critical value. In the simplest case of convection in a pure Boussinesq fluid, the instability is a symmetry-breaking steady-state bifurcation that leads to the formation of spatially periodic patterns. However, in many double-diffusive convection systems the heat-conduction solution actually loses stability via Hopf bifurcation. These hydrodynamic systems provide motivation for the present study of spatiotemporally periodic pattern formation in Euclidean equivariant systems. We call such patterns planforms. We classify, according to spatio-temporal symmetries and spatial periodicity, many of the time-periodic solutions that may be obtained through equivariant Hopf bifurcation from a group-invariant equilibrium. Instead of focusing on plan- forms periodic with respect to a specified planar lattice, as has been done in previous investigations, we consider all planforms that are spatially periodic with respect to some planar lattice. Our classification results rely only on the existence of Hopf bifurcation and planar Euclidean symmetry and not on the particular differential equation.

An avian-only Filippov model incorporating culling of both susceptible and infected birds in combating avian influenza

Depopulation of birds has always been an effective method not only to control the transmission of avian influenza in bird populations but also to eliminate influenza viruses. We introduce a Filippov avian-only model with culling of susceptible and/or infected birds. For each susceptible threshold level

Zero-Hopf bifurcation in the Van der Pol oscillator with delayed position and velocity feedback

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HETEROCLINIC CYCLES AND WREATH PRODUCT SYMMETRIES

Sb, we derive the phase portrait for the dynamical system as we vary the infected threshold level