93 research outputs found
Observing the Symmetry of Attractors
We show how the symmetry of attractors of equivariant dynamical systems can
be observed by equivariant projections of the phase space. Equivariant
projections have long been used, but they can give misleading results if used
improperly and have been considered untrustworthy. We find conditions under
which an equivariant projection generically shows the correct symmetry of the
attractor.Comment: 28 page LaTeX document with 9 ps figures included. Supplementary
color figures available at http://odin.math.nau.edu/~jws
Broken symmetries and pattern formation in two-frequency forced Faraday waves
We exploit the presence of approximate (broken) symmetries to obtain general
scaling laws governing the process of pattern formation in weakly damped
Faraday waves. Specifically, we consider a two-frequency forcing function and
trace the effects of time translation, time reversal and Hamiltonian structure
for three illustrative examples: hexagons, two-mode superlattices, and two-mode
rhomboids. By means of explicit parameter symmetries, we show how the size of
various three-wave resonant interactions depends on the frequency ratio m:n and
on the relative temporal phase of the two driving terms. These symmetry-based
predictions are verified for numerically calculated coefficients, and help
explain the results of recent experiments.Comment: 4 pages, 6 figure
Towards an experimental von Karman dynamo: numerical studies for an optimized design
Numerical studies of a kinematic dynamo based on von Karman type flows
between two counterrotating disks in a finite cylinder are reported. The flow
has been optimized using a water model experiment, varying the driving
impellers configuration. A solution leading to dynamo action for the mean flow
has been found. This solution may be achieved in VKS2, the new sodium
experiment to be performed in Cadarache, France. The optimization process is
described and discussed, then the effects of adding a stationary conducting
layer around the flow on the threshold, on the shape of the neutral mode and on
the magnetic energy balance are studied. Finally, the possible processes
involved into kinematic dynamo action in a von Karman flow are reviewed and
discussed. Among the possible processes we highlight the joint effect of the
boundary-layer radial velocity shear and of the Ohmic dissipation localized at
the flow/outer-shell boundary
Resonance bifurcations from robust homoclinic cycles
We present two calculations for a class of robust homoclinic cycles with
symmetry Z_n x Z_2^n, for which the sufficient conditions for asymptotic
stability given by Krupa and Melbourne are not optimal.
Firstly, we compute optimal conditions for asymptotic stability using
transition matrix techniques which make explicit use of the geometry of the
group action.
Secondly, through an explicit computation of the global parts of the Poincare
map near the cycle we show that, generically, the resonance bifurcations from
the cycles are supercritical: a unique branch of asymptotically stable period
orbits emerges from the resonance bifurcation and exists for coefficient values
where the cycle has lost stability. This calculation is the first to explicitly
compute the criticality of a resonance bifurcation, and answers a conjecture of
Field and Swift in a particular limiting case. Moreover, we are able to obtain
an asymptotically-correct analytic expression for the period of the bifurcating
orbit, with no adjustable parameters, which has not proved possible previously.
We show that the asymptotic analysis compares very favourably with numerical
results.Comment: 24 pages, 3 figures, submitted to Nonlinearit
Symmetry Reduction by Lifting for Maps
We study diffeomorphisms that have one-parameter families of continuous
symmetries. For general maps, in contrast to the symplectic case, existence of
a symmetry no longer implies existence of an invariant. Conversely, a map with
an invariant need not have a symmetry. We show that when a symmetry flow has a
global Poincar\'{e} section there are coordinates in which the map takes a
reduced, skew-product form, and hence allows for reduction of dimensionality.
We show that the reduction of a volume-preserving map again is volume
preserving. Finally we sharpen the Noether theorem for symplectic maps. A
number of illustrative examples are discussed and the method is compared with
traditional reduction techniques.Comment: laTeX, 31 pages, 5 figure
Point vortices on the sphere: a case with opposite vorticities
We study systems formed of 2N point vortices on a sphere with N vortices of
strength +1 and N vortices of strength -1. In this case, the Hamiltonian is
conserved by the symmetry which exchanges the positive vortices with the
negative vortices. We prove the existence of some fixed and relative
equilibria, and then study their stability with the ``Energy Momentum Method''.
Most of the results obtained are nonlinear stability results. To end, some
bifurcations are described.Comment: 35 pages, 9 figure
Tools in the orbit space approach to the study of invariant functions: rational parametrization of strata
Functions which are equivariant or invariant under the transformations of a
compact linear group acting in an euclidean space , can profitably
be studied as functions defined in the orbit space of the group. The orbit
space is the union of a finite set of strata, which are semialgebraic manifolds
formed by the -orbits with the same orbit-type. In this paper we provide a
simple recipe to obtain rational parametrizations of the strata. Our results
can be easily exploited, in many physical contexts where the study of
equivariant or invariant functions is important, for instance in the
determination of patterns of spontaneous symmetry breaking, in the analysis of
phase spaces and structural phase transitions (Landau theory), in equivariant
bifurcation theory, in crystal field theory and in most areas where use is made
of symmetry adapted functions.
A physically significant example of utilization of the recipe is given,
related to spontaneous polarization in chiral biaxial liquid crystals, where
the advantages with respect to previous heuristic approaches are shown.Comment: Figures generated through texdraw package; revised version appearing
in J. Phys. A: Math. Ge
Hyperbolic planforms in relation to visual edges and textures perception
We propose to use bifurcation theory and pattern formation as theoretical
probes for various hypotheses about the neural organization of the brain. This
allows us to make predictions about the kinds of patterns that should be
observed in the activity of real brains through, e.g. optical imaging, and
opens the door to the design of experiments to test these hypotheses. We study
the specific problem of visual edges and textures perception and suggest that
these features may be represented at the population level in the visual cortex
as a specific second-order tensor, the structure tensor, perhaps within a
hypercolumn. We then extend the classical ring model to this case and show that
its natural framework is the non-Euclidean hyperbolic geometry. This brings in
the beautiful structure of its group of isometries and certain of its subgroups
which have a direct interpretation in terms of the organization of the neural
populations that are assumed to encode the structure tensor. By studying the
bifurcations of the solutions of the structure tensor equations, the analog of
the classical Wilson and Cowan equations, under the assumption of invariance
with respect to the action of these subgroups, we predict the appearance of
characteristic patterns. These patterns can be described by what we call
hyperbolic or H-planforms that are reminiscent of Euclidean planar waves and of
the planforms that were used in [1, 2] to account for some visual
hallucinations. If these patterns could be observed through brain imaging
techniques they would reveal the built-in or acquired invariance of the neural
organization to the action of the corresponding subgroups.Comment: 34 pages, 11 figures, 2 table
Management of MDR-TB in HIV co-infected patients in Eastern Europe: Results from the TB:HIV study
Objectives Mortality among HIV patients with tuberculosis (TB) remains high in Eastern Europe (EE), but details of TB and HIV management remain scarce. Methods In this prospective study, we describe the TB treatment regimens of patients with multi-drug resistant (MDR) TB and use of antiretroviral therapy (ART). Results A total of 105 HIV-positive patients had MDR-TB (including 33 with extensive drug resistance) and 130 pan-susceptible TB. Adequate initial TB treatment was provided for 8% of patients with MDR-TB compared with 80% of those with pan-susceptible TB. By twelve months, an estimated 57.3% (95%CI 41.5\u201374.1) of MDR-TB patients had started adequate treatment. While 67% received ART, HIV-RNA suppression was demonstrated in only 23%. Conclusions Our results show that internationally recommended MDR-TB treatment regimens were infrequently used and that ART use and viral suppression was well below the target of 90%, reflecting the challenging patient population and the environment in which health care is provided. Urgent improvement of management of patients with TB/HIV in EE, in particular for those with MDR-TB, is needed and includes widespread access to rapid TB diagnostics, better access to and use of second-line TB drugs, timely ART initiation with viral load monitoring, and integration of TB/HIV care
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