62 research outputs found

### Simple heteroclinic cycles in R^4

In generic dynamical systems heteroclinic cycles are invariant sets of
codimension at least one, but they can be structurally stable in systems which
are equivariant under the action of a symmetry group, due to the existence of
flow-invariant subspaces. For dynamical systems in R^n the minimal dimension
for which such robust heteroclinic cycles can exist is n=3. In this case the
list of admissible symmetry groups is short and well-known. The situation is
different and more interesting when n=4. In this paper we list all finite
groups Gamma such that an open set of smooth Gamma-equivariant dynamical
systems in R^4 possess a very simple heteroclinic cycle (a structurally stable
heteroclinic cycle satisfying certain additional constraints). This work
extends the results which were obtained by Sottocornola in the case when all
equilibria in the heteroclinic cycle belong to the same Gamma-orbit (in this
case one speaks of homoclinic cycles).Comment: 43 pages; submitted to "Nonlinearity

### Pattern formation for the Swift-Hohenberg equation on the hyperbolic plane

We present an overview of pattern formation analysis for an analogue of the
Swift-Hohenberg equation posed on the real hyperbolic space of dimension two,
which we identify with the Poincar\'e disc D. Different types of patterns are
considered: spatially periodic stationary solutions, radial solutions and
traveling waves, however there are significant differences in the results with
the Euclidean case. We apply equivariant bifurcation theory to the study of
spatially periodic solutions on a given lattice of D also called H-planforms in
reference with the "planforms" introduced for pattern formation in Euclidean
space. We consider in details the case of the regular octagonal lattice and
give a complete descriptions of all H-planforms bifurcating in this case. For
radial solutions (in geodesic polar coordinates), we present a result of
existence for stationary localized radial solutions, which we have adapted from
techniques on the Euclidean plane. Finally, we show that unlike the Euclidean
case, the Swift-Hohenberg equation in the hyperbolic plane undergoes a Hopf
bifurcation to traveling waves which are invariant along horocycles of D and
periodic in the "transverse" direction. We highlight our theoretical results
with a selection of numerical simulations.Comment: Dedicated to Klaus Kirchg\"assne

### Bifurcation of hyperbolic planforms

Motivated by a model for the perception of textures by the visual cortex in
primates, we analyse the bifurcation of periodic patterns for nonlinear
equations describing the state of a system defined on the space of structure
tensors, when these equations are further invariant with respect to the
isometries of this space. We show that the problem reduces to a bifurcation
problem in the hyperbolic plane D (Poincar\'e disc). We make use of the concept
of periodic lattice in D to further reduce the problem to one on a compact
Riemann surface D/T, where T is a cocompact, torsion-free Fuchsian group. The
knowledge of the symmetry group of this surface allows to carry out the
machinery of equivariant bifurcation theory. Solutions which generically
bifurcate are called "H-planforms", by analogy with the "planforms" introduced
for pattern formation in Euclidean space. This concept is applied to the case
of an octagonal periodic pattern, where we are able to classify all possible
H-planforms satisfying the hypotheses of the Equivariant Branching Lemma. These
patterns are however not straightforward to compute, even numerically, and in
the last section we describe a method for computation illustrated with a
selection of images of octagonal H-planforms.Comment: 26 pages, 11 figure

### The Motion of the Spherical Pendulum Subjected to a D_n Symmetric Perturbation

The motion of a spherical pendulum is characterized by the fact that all trajectories are relative periodic orbits with respect to its circle group of symmetry (invariance by rotations around the vertical axis). When the rotational symmetry is broken by some mechanical effect, more complicated, possibly chaotic behavior is expected. When, in particular, the symmetry reduces to the dihedral group D_n of symmetries of a regular n-gon, n > 2, the motion itself undergoes dramatic changes even when the amplitude of oscillations is small, which we intend to explain in this paper. Numerical simulations confirm the validity of the theory and show evidence of new interesting effects when the amplitude of the oscillations is larger (symmetric chaos)

### Some theoretical results for a class of neural mass equations

We study the neural field equations introduced by Chossat and Faugeras in
their article to model the representation and the processing of image edges and
textures in the hypercolumns of the cortical area V1. The key entity, the
structure tensor, intrinsically lives in a non-Euclidean, in effect hyperbolic,
space. Its spatio-temporal behaviour is governed by nonlinear
integro-differential equations defined on the Poincar\'e disc model of the
two-dimensional hyperbolic space. Using methods from the theory of functional
analysis we show the existence and uniqueness of a solution of these equations.
In the case of stationary, i.e. time independent, solutions we perform a
stability analysis which yields important results on their behavior. We also
present an original study, based on non-Euclidean, hyperbolic, analysis, of a
spatially localised bump solution in a limiting case. We illustrate our
theoretical results with numerical simulations.Comment: 35 pages, 7 figure

### Hamiltonian Hopf bifurcation with symmetry

In this paper we study the appearance of branches of relative periodic orbits
in Hamiltonian Hopf bifurcation processes in the presence of compact symmetry
groups that do not generically exist in the dissipative framework. The
theoretical study is illustrated with several examples.Comment: 35 pages, 3 figure

### A spatialized model of textures perception using structure tensor formalism

International audienceThe primary visual cortex (V1) can be partitioned into fundamental domains or hypercolumns consisting of one set of orientation columns arranged around a singularity or ''pinwheel'' in the orientation preference map. A recent study on the specific problem of visual textures perception suggested that textures may be represented at the population level in the cortex as a second-order tensor, the structure tensor, within a hypercolumn. In this paper, we present a mathematical analysis of such interacting hypercolumns that takes into account the functional geometry of local and lateral connections. The geometry of the hypercolumn is identified with that of the Poincaré disk \D. Using the symmetry properties of the connections, we investigate the spontaneous formation of cortical activity patterns. These states are characterized by tuned responses in the feature space, which are doubly-periodically distributed across the cortex

### Heteroclinic cycles in Hopfield networks

Learning or memory formation are associated with the strengthening of the
synaptic connections between neurons according to a pattern reflected by the
input. According to this theory a retained memory sequence is associated to a
dynamic pattern of the associated neural circuit. In this work we consider a
class of network neuron models, known as Hopfield networks, with a learning
rule which consists of transforming an information string to a coupling
pattern. Within this class of models we study dynamic patterns, known as robust
heteroclinic cycles, and establish a tight connection between their existence
and the structure of the coupling

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