Recognition of symmetries in reversible maps

We deal with germs of diffeomorphisms that are reversible under an involution. We establish that this condition implies that, in general, both the family of reversing symmetries and the group of symmetries are not finite, in contrast with continuous-time dynamics, where typically there are finitely many reversing symmetries. From this we obtain two chains of fixed-points subspaces of involutory reversing symmetries that we use to obtain geometric information on the discrete dynamics generated by a given diffeomorphism. The results are illustrated by the generic case in arbitrary dimension, when the diffeomorphism is the composition of transversal linear involutions.

Quotients and lifts of symmetric directed graphs

Given a directed graph, an equivalence relation on the graph vertex set is said to be balanced if, for every two vertices in the same equivalence class, the number of directed edges from vertices of each equivalence class directed to each of the two vertices is the same. In this paper we describe the quotient and lift graphs of symmetric directed graphs associated with balanced equivalence relations on the associated vertex sets. In particular, we characterize the quotients and lifts which are also symmetric. We end with an application of these results to gradient and Hamiltonian coupled cell systems, in the context of the coupled cell network formalism of Golubitsky, Stewart and Torok(Patterns of synchrony in coupled cell networks with multiple arrows. {SIAM Journal of Applied Dynamical Systems, 4 (1) (2005) 78-100).

On equivariant binary differential equations

This paper introduces the study of occurrence of symmetries in binary differential equations (BDEs). These are implicit differential equations given by the zeros of a quadratic 1-form, $a(x,y)dy^2 + b(x,y)dxdy + c(x,y)dx^2 = 0,$ for $a, b, c$ smooth real functions defined on an open set of $mathbb{R}^2$. Generically, solutions of a BDE are given as leaves of a pair of foliations, and the appropriate way to define the action of a symmetry must depend not only whether it preserves or inverts the plane orientation, but also whether it preserves or interchanges the foliations. The first main result reveals this dependence, which is given algebraically by a formula relating three group homomorphisms defined on the symmetry group of the BDE. The second main result adapts algebraic methods from invariant theory for representations of compact Lie groups on the space of quadratic forms on $mathbb{R}^n$, $n geq 2$. With that we obtain an algorithm to compute general expressions of quadratic forms. Now, symmetric quadratic 1-forms are in one-to-one corrspondence with equivariant quadratic forms on the plane, so these are treated here as a particular case. We then apply the result to obtain the general forms of equivariant quadratic 1-forms under each compact subgroup of the orthogonal group $mathbf{O}(2)$.

Gradient systems on coupled cell networks

For networks of coupled dynamical systems we characterize admissible functions, that is, functions whose gradient is an admissible vector field. The schematic representation of a gradient network dynamical system is of an undirected cell graph, and we use tools from graph theory to deduce the general form of such functions, relating it to the topological structure of the graph defining the network. The coupling of pairs of dynamical systems cells is represented by edges of the graph, and from spectral graph theory we detect the existence and nature of equilibria of the gradient system from the critical points of the coupling function. In particular, we study fully synchronous and 2-state patterns of equilibria on regular graphs.These are two special types of equilibrium configurations for gradient networks. We also investigate equilibrium configurations of S1-invariant admissible functions on a ring of cells.

Relative equivariants under compact Lie groups

In this work we obtain the general form of polynomial mappings that commute with a linear action of a relative symmetry group. The aim is to give results for relative equivariant polynomials that correspond to the results for relative invariants obtained in a previous paper [P.H. Baptistelli, M. Manoel (2013) Invariants and relative invariants under compact Lie groups, J. Pure Appl. Algebra 217, 2213{2220]. We present an algorithm to compute generators for relative equivariant submodules from the invariant theory applied to the subgroup formed only by the symmetries. The same method provides, as a particular case, generators for equivariants under the whole group from the knowledge of equivariant generators by a smaller subgroup, which is normal of finite index.

Complete transversals of reversible equivariant singularities of vector fields

We use group representation theory to give algebraic formulae to compute complete transversals of singularities of vector fields, either in the nonsymmetric or in the reversible equivariant contexts. This computation produces normal forms directly, which are used sistematically in the local analysis of symmetric dynamics.

Invariants and relative invariants under compact Lie groups

This paper presents algebraic methods for the study of polynomial relative invariants, when the group G formed by the symmetries and relative symmetries is a compact Lie group. We deal with the case when the subgroup H of symmetries is normal in G with index m, m greater or equal to 2. For this, we develop the invariant theory of compact Lie groups acting on complex vector spaces. This is the starting point for the study of relative invariants and the computation of their generators. We first obtain the space of the invariants under the subgroup $H$ of $Gamma$ as a direct sum of $m$ submodules over the ring of invariants under the whole group. Then, based on this decomposition, we construct a Hilbert basis of the ring of G-invariants from a Hilbert basis of the ring of H-invariants. In both results the knowledge of the relative Reynolds operators defined on H-invariants is shown to be an essential tool to obtain the invariants under the whole group. The theory is illustrated with some examples.

The Wigner caustic on shell and singularities of odd functions

We study the Wigner caustic on shell of a Lagrangian submanifold L of affine symplectic space. We present the physical motivation for studying singularities of the Wigner caustic on shell and present its mathematical definition in terms of a generating family. Because such a generating family is an odd deformation of an odd function, we study simple singularities in the category of odd functions and their odd versal deformations, applying these results to classify the singularities of the Wigner caustic on shell, interpreting these singularities in terms of the local geometry of L.