Eliana Manuel Pinho

Spatial hidden symmetries in pattern formation

Partial differential equations that are invariant under Euclidean transformations are traditionally used as models in pattern formation. These models are often posed on finite domains (in particular, multidimensional rectangles). Defining the correct boundary conditions is often a very subtle problem. On the other hand, there is pressure to choose boundary conditions which are attractive to mathematical treatment. Geometrical shapes and mathematically friendly boundary conditions usually imply spatial symmetry. The presence of symmetry effects that are “hidden” in the boundary conditions was noticed and carefully investigated by several researchers during the past 15–20 years. Here we review developments in this subject and introduce a unifying formalism to uncover spatial hidden symmetries (hidden translations and hidden rotations) in multidimensional rectangular domains with Neumann boundary conditions.

Spatially Periodic Patterns of Synchrony in Lattice Networks

We consider n-dimensional Euclidean lattice networks with nearest neighbor coupling architecture. The associated lattice dynamical systems are infinite systems of ordinary differential equations, the cells, indexed by the points in the lattice. A pattern of synchrony is a finite-dimensional flow-invariant subspace for all lattice dynamical systems with the given network architecture. These subspaces correspond to a classification of the cells into k classes, or colors, and are described by a local coloring rule, named balanced coloring. Previous results with planar lattices show that patterns of synchrony can exhibit several behaviors such as periodicity. Considering sufficiently extensive couplings, spatial periodicity appears for all the balanced colorings with k colors. However, there is not a direct way of relating the local coloring rule and the coloring of the whole lattice network. Given an n-dimensional lattice network with nearest neighbor coupling architecture, and a local coloring rule with k c...

Symmetries of projected wallpaper patterns

In this paper we study periodic functions of one and two variables that are invariant under a subgroup of the Euclidean group. Starting with a function defined on the plane we obtain a function of one variable by two methods: we project the values of the function on a strip into its edge, by integrating along the width; and we restrict the function to a line. If the functions had been obtained by solving a partial differential equation equivariant under the Euclidean group, how do their symmetries compare to those of solutions of equations formulated directly in one dimension? Some of the symmetries of projected and of restricted functions can be obtained knowing the symmetries of the original functions only. There are also some extra symmetries arising for special widths of the strip and for some special positions of the line used for restriction. We obtain a general description of the two types of symmetries and discuss how they arise in the wallpaper groups (crystalographic groups of the plane). We show that the projections and restrictions of solutions of p.d.e.s in the plane may have symmetry groups larger than those of solutions of problems formulated in one dimension.

Enumerating periodic patterns of synchrony via finite bidirectional networks

Periodic patterns of synchrony are lattice networks whose cells are coloured according to a local rule, or balanced colouring, and such that the overall system has spatial periodicity. These patterns depict the finite-dimensional flow-invariant subspaces for all the lattice dynamical systems, in the given lattice network, that exhibit those periods. Previous results relate the existence of periodic patterns of synchrony, in n-dimensional Euclidean lattice networks with nearest neighbour coupling architecture, with that of finite coupled cell networks that follow the same colouring rule and have all the couplings bidirectional. This paper addresses the relation between periodic patterns of synchrony and finite bidirectional coloured networks. Given an n-dimensional Euclidean lattice network with nearest neighbour coupling architecture, and a colouring rule with k colours, we enumerate all the periodic patterns of synchrony generated by a given finite network, or graph. This enumeration is constructive and based on the automorphisms group of the graph.

Projected Wallpaper Patterns

Consider a periodic function f of two variables with symmetry Γ and let ℒ ⊂ Γ be the subgroup of translations. The Fourier expansion of a periodic function is a sum over ℒ*, the dual of the set ℒ of all the periods of f. After projecting f, some of its original symmetry remains. We describe the symmetries of the projected function, starting from Γ and from the structure of ℒ*.