Volume-preserving normal forms of Hopf-zero singularity

A practical method is described for computing the unique generator of the algebra of first integrals associated with a large class of Hopf-zero singularity. The set of all volume-preserving classical normal forms of this singularity is introduced via a Lie algebra description. This is a maximal vector space of classical normal forms with first integral; this is whence our approach works. Systems with a non-zero condition on their quadratic parts are considered. The algebra of all first integrals for any such system has a unique (modulo scalar multiplication) generator. The infinite level volume-preserving parametric normal forms of any non-degenerate perturbation within the Lie algebra of any such system is computed, where it can have rich dynamics. The associated unique generator of the algebra of first integrals are derived. The symmetry group of the infinite level normal forms are also discussed. Some necessary formulas are derived and applied to appropriately modified R\"{o}ssler and generalized Kuramoto--Sivashinsky equations to demonstrate the applicability of our theoretical results. An approach (introduced by Iooss and Lombardi) is applied to find an optimal truncation for the first level normal forms of these examples with exponentially small remainders. The numerically suggested radius of convergence (for the first integral) associated with a hypernormalization step is discussed for the truncated first level normal forms of the examples. This is achieved by an efficient implementation of the results using Maple

Rayleigh and depinning instabilities of forced liquid ridges on heterogeneous substrates

Depinning of two-dimensional liquid ridges and three-dimensional drops on an inclined substrate is studied within the lubrication approximation. The structures are pinned to wetting heterogeneities arising from variations of the strength of the short-range polar contribution to the disjoining pressure. The case of a periodic array of hydrophobic stripes transverse to the slope is studied in detail using a combination of direct numerical simulation and branch-following techniques. Under appropriate conditions the ridges may either depin and slide downslope as the slope is increased, or first breakup into drops via a transverse instability, prior to depinning. The different transition scenarios are examined together with the stability properties of the different possible states of the system.Comment: Physics synopsis link: http://physics.aps.org/synopsis-for/10.1103/PhysRevE.83.01630

The cusp–Hopf bifurcation

The coalescence of a Hopf bifurcation with a codimension-two cusp bifurcation of equilibrium points yields a codimension-three bifurcation with rich dynamic behavior. This paper presents a comprehensive study of this cusp-Hopf bifurcation on the three-dimensional center manifold. It is based on truncated normal form equations, which have a phase-shift symmetry yielding a further reduction to a planar system. Bifurcation varieties and phase portraits are presented. The phenomena include all four cases that occur in the codimension-two fold-Hopf bifurcation, in addition to bistability involving equilibria, limit cycles or invariant tori, and a fold-heteroclinic bifurcation that leads to bursting oscillations. Uniqueness of the torus family is established locally. Numerical simulations confirm the prediction from the bifurcation analysis of bursting oscillations that are similar in appearance to those that occur in the electrical behavior of neurons and other physical systems

Female Mucopolysaccharidosis IIIA Mice Exhibit Hyperactivity and a Reduced Sense of Danger in the Open Field Test

Reliable behavioural tests in animal models of neurodegenerative diseases allow us to study the natural history of disease and evaluate the efficacy of novel therapies. Mucopolysaccharidosis IIIA (MPS IIIA or Sanfilippo A), is a severe, neurodegenerative lysosomal storage disorder caused by a deficiency in the heparan sulphate catabolising enzyme, sulfamidase. Undegraded heparan sulphate accumulates, resulting in lysosomal enlargement and cellular dysfunction. Patients suffer a progressive loss of motor and cognitive function with severe behavioural manifestations and premature death. There is currently no treatment. A spontaneously occurring mouse model of the disease has been described, that has approximately 3% of normal enzyme activity levels. Behavioural phenotyping of the MPS IIIA mouse has been previously reported, but the results are conflicting and variable, even after full backcrossing to the C57BL/6 background. Therefore we have independently backcrossed the MPS IIIA model onto the C57BL/6J background and evaluated the behaviour of male and female MPS IIIA mice at 4, 6 and 8 months of age using the open field test, elevated plus maze, inverted screen and horizontal bar crossing at the same circadian time point. Using a 60 minute open field, we have demonstrated that female MPS IIIA mice are hyperactive, have a longer path length, display rapid exploratory behaviour and spend less time immobile than WT mice. Female MPS IIIA mice also display a reduced sense of danger and spend more time in the centre of the open field. There were no significant differences found between male WT and MPS IIIA mice and no differences in neuromuscular strength were seen with either sex. The altered natural history of behaviour that we observe in the MPS IIIA mouse will allow more accurate evaluation of novel therapeutics for MPS IIIA and potentially other neurodegenerative disorders

An energy balance model for paleoclimate transitions

A new energy balance model (EBM) is presented and is used to study paleoclimate transitions. While most previous EBMs only dealt with the globally averaged climate, this new EBM has three variants: Arctic, Antarctic and tropical climates. The EBM incorporates the greenhouse warming effects of both carbon dioxide and water vapour, and also includes ice–albedo feedback and evapotranspiration. The main conclusion to be inferred from this EBM is that the climate system may possess multiple equilibrium states, both warm and frozen, which coexist mathematically. While the actual climate can exist in only one of these states at any given time, the EBM suggests that climate can undergo transitions between the states via mathematical saddle-node bifurcations. This paper proposes that such bifurcations have actually occurred in Paleoclimate transitions. The EBM is applied to the study of the Pliocene paradox, the glaciation of Antarctica and the so-called warm, equable climate problem of both the mid-Cretaceous Period and the Eocene Epoch. In all cases, the EBM is in qualitative agreement with the geological record.</p

Symmetry-breaking transitions in networks of nonlinear circuit elements

We investigate a nonlinear circuit consisting of N tunnel diodes in series, which shows close similarities to a semiconductor superlattice or to a neural network. Each tunnel diode is modeled by a three-variable FitzHugh-Nagumo-like system. The tunnel diodes are coupled globally through a load resistor. We find complex bifurcation scenarios with symmetry-breaking transitions that generate multiple fixed points off the synchronization manifold. We show that multiply degenerate zero-eigenvalue bifurcations occur, which lead to multistable current branches, and that these bifurcations are also degenerate with a Hopf bifurcation. These predicted scenarios of multiple branches and degenerate bifurcations are also found experimentally.Comment: 32 pages, 11 figures, 7 movies available as ancillary file

Minimum error discrimination of Pauli channels

We solve the problem of discriminating with minimum error probability two given Pauli channels. We show that, differently from the case of discrimination between unitary transformations, the use of entanglement with an ancillary system can strictly improve the discrimination, and any maximally entangled state allows to achieve the optimal discrimination. We also provide a simple necessary and sufficient condition in terms of the structure of the channels for which the ultimate minimum error probability can be achieved without entanglement assistance. When such a condition is satisfied, the optimal input state is simply an eigenstate of one of the Pauli matrices.Comment: 8 pages, no figure

Enhancing multiphoton rates with quantum memories

Single photons are a vital resource for optical quantum information processing. Efficient and deterministic single photon sources do not yet exist, however. To date, experimental demonstrations of quantum processing primitives have been implemented using non-deterministic sources combined with heralding and/or postselection. Unfortunately, even for eight photons, the data rates are already so low as to make most experiments impracticable. It is well known that quantum memories, capable of storing photons until they are needed, are a potential solution to this `scaling catastrophe'. Here, we analyze in detail the benefits of quantum memories for producing multiphoton states, showing how the production rates can be enhanced by many orders of magnitude. We identify the quantity $\eta B$ as the most important figure of merit in this connection, where $\eta$ and $B$ are the efficiency and time-bandwidth product of the memories, respectively.Comment: Just over 4 pages, 2 figure

Temporal Modulation of Traveling Waves in the Flow Between Rotating Cylinders With Broken Azimuthal Symmetry

The effect of temporal modulation on traveling waves in the flows in two distinct systems of rotating cylinders, both with broken azimuthal symmetry, has been investigated. It is shown that by modulating the control parameter at twice the critical frequency one can excite phase-locked standing waves and standing-wave-like states which are not allowed when the system is rotationally symmetric. We also show how previous theoretical results can be extended to handle patterns such as these, that are periodic in two spatial direction.Comment: 17 pages in LaTeX, 22 figures available as postscript files from http://www.esam.nwu.edu/riecke/lit/lit.htm