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

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

Enhanced magnetic field production during oblique hypervelocity impacts

The natural remanent magnetization of the lunar surface as displayed in returned lunar samples and the data returned by the Apollo subsatellite magnetometer has an unexpectedly high magnitude and exhibits spatial variation at all scales. The origin of the lunar remanent fields may be due to crustal remanence of a core dynamo field occurring early in lunar history prior to extensive modification by impact or remanence of transient fields, particularly associated with impacts, occurring on a local scale throughout lunar history. The presence of an early core dynamo field would have strong consequences for the formation and early evolution of the Moon, yet to deconvolve the role that an internally generated core dynamo field may have had, it is necessary to understand how the magnetic state of the lunar surface has developed through time. Impact-induced magnetism may be an important component of the present magnetic state of the lunar surface. New theoretical considerations suggest that transient magnetic fields within plasma produced by hypervelocity meteorite impacts may have greater significance at larger scales than previously thought

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

Conceptual analyses of extensible booms to support a solar sail

Extensible booms which could function as the diagonal spars and central mast of an 800 meter square, non-rotating Solar Sailing Vehicle were conceptually designed and analyzed. The boom design concept that was investigated is an extensible lattice boom which is stowed and deployed by elastically coiling and uncoiling its continuous longerons. The seven different free-span lengths in each spar which would minimize the total weights of the spars and mast were determined. Boom weights were calculated by using a semi-empirical formulation which related the overall weight of a boom to the weight of its longerons

Radio Continuum Study of Supernova Remnants in the Large Magellanic Cloud - SNR J0519-6926

We present the results of new high resolution ATCA observations of SNR J0519-6926. We found that this SNR exhibits a typical "horseshoe" appearance with alpha = -0.55 +- 0.08 and D=28+-1 pc. No polarization (or magnetic fields) are detected to a level of 1%. This is probably due to a relatively poor sampling of the uv plane caused be observing in "snap-shot" mode.Comment: 6 pages 4 figures, to be published in Serbian Astronomical Journa