Introduction to bifurcation-theory

The theory of bifurcation from equilibria based on center-manifold reductio, and Poincare-Birkhoff normal forms is reviewed at an introductory level. Both differential equations and maps are discussed, and recent results explaining the symmetry of the normal form are derived. The emphasis is on the simplest generic bifurcations in one-parameter systems. Two applications are developed in detail: a Hopf bifurcation occurring in a model of three-wave mode coupling and steady-state bifurcations occurring in the real Landau-Ginzburg equation. The former provides an example of the importance of degenerate bifurcations in problems with more than one parameter and the latter illustrates new effects introduced into a bifurcation problem by a continuous symmetry.Physic

Electrochemistry and Spectroscopy of Sulfate and Thiosulfate Complexes of Iron Porphyrins

The electrochemical and spectroscopic properties of the complex formed by the addition of thiosulfate to ferric porphyrins were examined. The NMR spectrum of the thiosulfate–ferric porphyrin complex was consistent with a high-spin ferric complex, while the EPR spectrum at liquid nitrogen temperatures indicated that the complex under these conditions was low-spin. Such behavior has been previously observed for other ferric porphyrin complexes. The visible spectra were characterized by a shift in the Soret band to higher energies, with smaller changes in the longer wavelength region. The complex was reasonably stable in DMF, but slowly reduced over several hours to FeII(TPP) and S4O6 2−. The voltammetric behavior of the thiosulfate complex in DMF consists of two waves, the first of which was irreversible. The ferric/ferrous reduction in the presence of thiosulfate was shifted negatively about 400 mV, compared to the Fe(TPP)(Cl) reduction. The visible, NMR and EPR spectra were most consistent with a Fe–S bonded ferric porphyrin–thiosulfate complex, Fe(P)(SSO3)−. The kinetics of the reduction of ferric porphyrin by thiosulfate in DMSO indicated an autocatalytic mechanism, where the first step is the formation of the catalyst. The identity of the catalyst could not be determined because it must be present at low concentrations, but it is formed from the reaction of the ferric complex with thiosulfate. Coordination of thiosulfate to the porphyrin was not necessary for the reduction to occur, and the reduction of Fe(TPP)(Cl) by thiosulfate was accelerated by the addition of sulfate. Under these conditions, sulfate had replaced thiosulfate as the axial ligand for the ferric porphyrin. In the presence of sulfate, the reduction occurred in a single kinetic pseudo-first order step. The voltammetry, spectroelectrochemistry and kinetics for the reaction of thiosulfate with ferric porphyrins were examined. A rapid reaction between ferric porphyrins and thiosulfate was observed in DMF. The reaction was complex, involving the formation of a catalytic intermediate. Window factor analysis and multivariate curve resolution were used to deconvolute the kinetic data

Homoclinic snaking in bounded domains

Homoclinic snaking is a term used to describe the back and forth oscillation of a branch of time-independent spatially localized states in a bistable, spatially reversible system as the localized structure grows in length by repeatedly adding rolls on either side. On the real line this process continues forever. In finite domains snaking terminates once the domain is filled but the details of how this occurs depend critically on the choice of boundary conditions. With periodic boundary conditions the snaking branches terminate on a branch of spatially periodic states. However, with non-Neumann boundary conditions they turn continuously into a large amplitude filling state that replaces the periodic state. This behavior, shown here in detail for the Swift-Hohenberg equation, explains the phenomenon of “snaking without bistability”, recently observed in simulations of binary fluid convection by Mercader, Batiste, Alonso and Knobloch (preprint)

Impacts of hemispherical granular targets: Implications for global impacts

As impact excavation diameters subtend a nontrivial fraction of a planetary body, both the excavation process and ejecta emplacement may depart form the classical description of impacts into a planar surface. Hemispherical particulate targets were impacted at the NASA-Ames Vertical Gun Range in order to trace the evolution of the ejecta curtain and to document the effects of slope and surface curvature on crater shape and cratering efficiency. The experiments suggest that basin size impacts or large craters on small bodies may be shallower than their counterparts on a planar surface but may have displaced a larger relative mass. Moreover, the increased ejecta curtain angle with distance may result in a change in ejecta emplacement style with distance. Although the ejecta curtain is vertical, ejecta within the curtain impact the surface at 45 deg and the time between first and last arrival within the curtain increases. This increased interaction time as the ejecta curtain density decreases should result in a more chaotic style of implacement

Apparatus and method of inserting a microelectrode in body tissue or the like using vibration means

An arrangement for and method of inserting a glass microelectrode having a tip in the micron range into body tissue is presented. The arrangement includes a microelectrode. The top of the microelectrode is attached to the diaphragm center of a first speaker. The microelectrode tip is brought into contact with the tissue by controlling a micromanipulator. Thereafter, an audio signal is applied to the speaker to cause the microelectrode to vibrate and thereby pierce the tissue surface without breaking the microelectrode tip. Thereafter, the tip is inserted into the tissue to the desired depth by operating the micromanipulator with the microelectrode in a vibratory or non-vibratory state

The spatial distribution and time evolution of impact-generated magnetic fields

The production of magnetic fields was revealed by laboratory hypervelocity impacts in easily vaporized targets. As quantified by pressure measurements, high frame-rate photography, and electrostatic probes, these impacts tend to produce large quantities of slightly ionized vapor, which is referred to as impact-generated plasma. Nonaligned electron density and temperature gradients within this plasma may lead to production of the observed magnetic fields. Past experiments were limited to measuring a single component of the impact-generated magnetic fields at only a few locations about the developing impact crater and consequently gave little information about the field production mechanism. To understand this mechanism, the techniques were extended to map the three components of the magnetic field both in space and time. By conducting many otherwise identical experiments with arrayed magnetic detectors, a preliminary 3-D picture was produced of impact-generated magnetic fields as they develop through time

Improving the quality of mental health services using patient outcome data: Making the most of HoNOS

Efforts to assess and improve the quality of mental health services are often hampered by a lack of information on patient outcomes. Most mental health services in England have been routinely collecting Health of the Nation Outcome Scales (HoNOS) data for some time. In this article we illustrate how clinical teams have used HoNOS data to identify areas where performance could be improved. HoNOS data have the potential to give clinical teams the information they need to assess the quality of care they deliver, as well as develop and test initiatives aimed at improving the services they provide

Radio-Continuum Observations Of Small, Radially Polarised Supernova Remnant J0519-6902 In The Large Magellanic Cloud

We report on new Australian Telescope Compact Array (ATCA) observations of SNR J0519-6902. The Supernova Remnant (SNR) is small in size (~8 pc) and exhibits a typical SNR spectrum of alpha = -0.53 +- 0.07, with steeper spectral indices found towards the northern limb of the remnant. SNR J0519-6902 contains a low level of radially orientated polarisation at wavelengths of 3 & 6 cm, which is characteristic of younger SNRs. A fairly strong magnetic field was estimated of ~171 microG. The remnant appears to be the result of a typical Type Ia supernovae, sharing many properties as another small and young Type Ia LMC SNR, J0509-6731.Comment: 10 pages 7 figures, submitted to Serbian Astronomical Journa