Areas Of Parameter Space Research Articles

High order instabilities of the Poincaré solution for precessionally driven flow

Sloudsky in 1895 and Poincaré in 1910 were the first to derive solutions for the flow driven in the Earth's fluid core by the luni-solar precession. In 1993, Kerswell investigated the stability of this so-called “Poincaré flow” by applying a method devised in 1992 by Ponomarev and Gledzer to study the instability of flows with elliptical streamlines. They represented the components of the perturbed flow by sums of polynomials. Kerswell restricted attention to the linear and quadratic cases. Here cubic, quartic, quintic, and sextic generalizations are developed. Instabilities are located in new areas of parameter space, including some that verge on the small oblateness of the Earth's core

Synchronization of macrospins arranged into a linear chain

One of the important topics in nanomagnetics concerns the use of spin valves for generation of the microwave GHz-frequency oscillations. The power output of an individual spin valve can be increased by synchronization of several devices, which was successfully confirmed experimentally. To achieve such synchronization, the values of applied magnetic field and injected spin current should be properly adjusted. We made numerical simulations for a linear chain of macrospins aimed to determine the parameters required for an efficient propagation of magnetization oscillations through a sandwich structure of ten thin ferromagnetic layers separated by non-magnetic spacers. The individual layers were assumed coupled only by spin-polarized current flowing through the system. We report the presence of considerable areas in parameter space, for which magnetic motion in a macrospin chain becomes synchronized after a short period of time, which may have significant importance for the possible device applications.

Algal defenses, population stability, and the risk of herbivore extinctions: a chemostat model and experiment

AbstractThe effects of inducible defenses and constitutive defenses on population dynamics were investigated in a freshwater plankton system with rotifers as predators and different algal strains as prey. We made predictions for these systems using a chemostat predator–prey model and focused on population stability and predator persistence as a function of flow‐through rate. The model exhibits three major types of behavior at a high nutrient concentration: (1) at high dilution rates, only algae exist; (2) at intermediate dilution rates, algae and rotifers show stable coexistence; (3) at low dilution rates, large population fluctuations occur, with low minimum densities entailing a risk of stochastic rotifer extinctions. The size and location of the corresponding areas in parameter space critically depend on the type of algal defense strategy. In an 83‐day high‐nutrient chemostat experiment we changed the dilution rate every 3 weeks, from 0.7 to 0.5 to 0.3 to 0.1 per day. Within this range of dilution rates, rotifers and algae coexisted, and population fluctuations of algae clearly increased as dilution rates decreased. The CV of herbivore densities was highest at the end of the experiment, when the dilution rate was low. On day 80, herbivorous rotifers had become undetectable in all three chemostats with permanently defended algae (where rotifer densities had already been low) and in two out of three chemostats where rotifers had been feeding on algae with inducible defenses (that represented more edible food). We interpret our results in relation to the paradox of enrichment.

Subharmonic Pattern Formation in a Nematic Cell – A 1D Flexoelectric Model Approach

The application of an AC electric field to a nematic liquid crystal cell produces two distinct instability pattern regimes, conductive or dielectric. Recently, asymmetric electric field waveforms have been shown to cause a period-doubling subharmonic bifurcation. We extend these calculations to include flexoelectricity in the one-dimensional case. We calculate the threshold values for the electric field strength determining the optimum oblique angle at which the roll patterns form. We find, in some areas of parameter space, the formation of oblique rolls is favoured over normal rolls. In addition we find cross-over points from normal to oblique rolls called Lifshitz points. These have been found in the subharmonic and dielectric regions of the linear stability diagram.

Modelling gap junctions in a neural field model

We study a nonlinear, one-dimensional neural field model based upon a partial integro-differential equation, that is used to model spatial patterns in working memory. Through the application of Fourier transforms to the PIDE, steady states of spatially localised areas of high activity can be represented by solutions of a fourth order ODE. Recent research has shown a high density of gap junctions in areas of the brain that experience epileptic events. We extend the model by including a diffusion-like term to model gap junctions and derive a sixth order ODE which we use to investigate changes in the dynamics of spatially localised solutions. We find that symmetric homoclinic orbits to a zero steady state exist for a wide area of parameter space. Numerical work shows families of solutions are destroyed as the strength of the term modelling gap junctions increases.

Diffusion and dispersion characterization of a numerical tsunami model

The Method Of Splitting Tsunami numerical model is being developed at the NOAA Center for Tsunami Research for use in the tsunami forecasting system in the United States. The ability to produce fast and accurate forecast of tsunami is critical, and the tradeoff between high speed model runs and numerical errors that are introduced due to finite grid parameters must be evaluated. The details of the underlying numerics of this model are examined in an effort to understand the numerical diffusion and dispersion inherent in finite difference models. Diffusion and dispersion are quantified in a simplified linear case, and extended in a qualitative manner to general cases. This knowledge is valuable in the development of efficient grids that will provide accurate solutions in a tsunami forecasting framework, as well as in choosing free parameters in a manner that allows a fit with theoretical dispersion. A variable space grid scheme that uses numerical dispersion to mimic theoretical dispersion is outlined. Also, areas in parameter space that will result in unrealistic wave fields and thus should be avoided are highlighted.

Phase Propagations in a Coupled Oscillator–Excitor System ofFitzHugh–Nagumo Models

A one-dimensional array of 2N+1 automata with FitzHugh–Nagumodynamics, in which one is set to be oscillatory and the others areexcitable, is investigated with bi-directional interactions. We findthat 1:1 rhythm propagation in the array depends on the appropriatecouple strength and the excitability of the system. On the two sides ofthe 1:1 rhythm area in parameter space, two different kinds ofdynamical behaviour of the pacemaker, i.e. phase-locking phenomena andcanard-like phenomena, are shown. The latter is found in company withchaotic pattern and period doubling bifurcation. When the couplingstrength is larger than a critical value, the whole system ends to asteady state.

Periodic dynamics in a two-stage Allee effect model are driven by tension between stage equilibria

Single species difference population models can show complex dynamics such as periodicity and chaos under certain circumstances, but usually only when rates of intrinsic population growth or other life history parameter are unrealistically high. Single species models with Allee effects (positive density dependence at low density) have also been shown to exhibit complex dynamics when combined with overcompensatory density dependence or a narrow fertility window. Here we present a simple two-stage model with Allee effects which shows large amplitude periodic fluctuations for some initial conditions, without these requirements. Periodicity arises out of a tension between the critical equilibrium of each stage, i.e. when the initial population vector is such that the adult stage is above the critical value, while the juvenile stage is below the critical value. Within this area of parameter space, the range of initial conditions giving rise to periodic dynamics is driven mainly by adult mortality rates. Periodic dynamics become more important as adult mortality increases up to a certain point, after which periodic dynamics are replaced by extinction. This model has more realistic life history parameter values than most ‘chaotic’ models. Conditions for periodic dynamics might arise in some marine species which are exploited (high adult mortality) leading to recruitment limitation (low juvenile density) and might be an additional source of extinction risk.

Heavy-ion target physics and design in the USA

We present data and simulations of our first experiment to test shims as a method for improving symmetry in indirect drive targets. In the first set of experiments, we succeeded in reversing a P 2 from a capsule waist high drive to a capsule pole high drive. Shims are an important target design feature in the heavy-ion “hybrid” target. To help with optimizing a heavy-ion fusion power plant based on the hybrid target, we also calculated gain curves for this target. In addition, we are using analytic theory to test our computational models of nonlinear Rayleigh–Taylor instability bubble growth in converging geometry. This will give confidence in computational models as we push the capsule into areas of parameter space that are easier for the other parts of the power plant (accelerator, target fabrication) but have less margin and more instability growth.

Chaos in the softening duffing system under multi-frequency periodic forces

The chaotic dynamics of the softening-spring Duffing system with multi-frequency external periodic forces is studied. It is found that the mechanism for chaos is the transverse heteroclinic tori. The Poincare map, the stable and the unstable manifolds of the system under two incommensurate periodic forces were set up on a two-dimensional torus. Utilizing a global perturbation technique of Melnikov the criterion for the transverse interaction of the stable and the unstable manifolds was given. The system under more but finite incommensurate periodic forces was also studied. The Melnikov's global perturbation technique was therefore generalized to higher dimensional systems. The region in parameter space where chaotic dynamics may occur was given. It was also demonstrated that increasing the number of forcing frequencies will increase the area in parameter space where chaotic behavior can occur.

DYNAMICAL SYSTEMS OF DIFFERENT CLASSES AS MODELS OF THE KICKED NONLINEAR OSCILLATOR

Using the nonlinear dissipative kicked oscillator as an example, the correspondence between the descriptions provided by model dynamical systems of different classes is discussed. A detailed study of the approximate 1D map is undertaken: the period doubling is examined and the possibility of non-Feigenbaum period doubling is shown. Illustrations in the form of bifurcation diagrams and sets of iteration diagrams are given, the scaling properties are demonstrated, and the tricritical points (the terminal points of the Feigenbaum critical curves) in parameter space are found. The congruity with the properties of the corresponding 2D map, the Ikeda map, is studied. A description in terms of tricritical dynamics is found to be adequate only in particular areas of parameter space.

Comparative analysis of approximate and precise mapping for a «bouncing ball»

Approximate and precise mappings for a bouncing ball are presented. The results of comparison of dynamics showed by these mappings are given: phase portraits, dynamical regimes topography charts, bifurcation parameter values etc. It is shown that the mapping of a bouncing ball in its traditional form occupies an intermediate position between the physically motivated and formal models, and in certain areas of parameter space it should be regarded as a formal model.

Thermodynamic assessment of a nitro-organic plasma with an experimental design method

The modelling of the plasma ignition of gun propellants requires the thermodynamic properties of a plasma generated by a CHON organic compound. The electrothermal launcher produces pressures up to some hundreds of megapascals and temperatures up to some tens of thousands Kelvin. We have developed a code, ALJAN-EP, to calculate the thermodynamical equilibrium for a CHON plasma by extension of the conventional method of minimisation of the enthalpy. We have presented results of calculation of the enthalpy and the isobaric thermal capacity for a nitrocellulose up to 500 MPa and 50000 K. Furthermore, we have used ALJAN-EP to determine polynomial expressions for a model of the second order in pressure and in temperature by experimental design. These expressions allow thermodynamic state functions and the composition of the gaseous phase to be calculated quickly and accurately. We present results obtained for nitrocellulose in a small temperature and pressure range from 100 to 500 MPa and 5000 to 10000 K. These results show that to cover a wider area, considering reactions in gaseous phase, it is necessary to consider a mosaic of adjacent elementary areas in parameter space or a more complex expression.

A study of the dynamics of a continuously stirred tank reactor with feedback control of the temperature

Feedback control of the temperature in the continuously stirred tank reactor (CSTR) will be studied in this paper. The control is achieved by continuously changing the input flow rate in response to the temperature. In particular the effect the control has on the temperature-dependent oscillations will be investigated. The steady-state solutions and emergence of periodic solutions through the Hopf bifurcation will be analyzed both theoretically and numerically. The region into which all trajectories are ultimately drawn will be defined in the phase plane^ as well as the areas in parameter space to which oscillations will be confined. Finally several nonlinear controlling functions are presented which ensure the system does not exhibit oscillations.

Host-Parasitoid Food Webs: Dynamics, Persistence, and Invasion

We study the population dynamics of a model host-parasitoid community containing two hosts that interact only through a shared generalist parasitoid. Each host is also attacked by a specialist parasitoid, making a community of up to five species of insects. Information from host-parasitoid community ecology is used in deciding how these species interact. By considering both community stability and whether absent species can invade, we predict which communities occur in different regions of parameter space. A crucial determinant of community structure is the relative searching efficiencies of specialists and generalists and the nature of the density-dependent reduction in parasitoid searching efficiency. In some areas of parameter space, there is no dynamically stable end point, and complex cycles occur where the food web structure is constantly changing, species being added through invasion and lost through direct and indirect competitive interactions. We also find that whether a species is able to invade can depend on the interaction between its dynamics and that of the resident species. We conclude that some communities of hosts and parasitoids may display complex dynamic behavior determined by both local population interactions and the frequency of species invasions.

Snowline instability in a general circulation model: application to Carboniferous glaciation

For over twenty years it has been known that energy balance models (EBMs) with snow-albedo feedback are characterized by unstable behavior in some areas of parameter space. This behavior leads to rapid changes in snow area due to small changes in forcing, and has been termed the small ice cap instability (SICI). It has never been clarified whether this behaviour reflects a real feature of the climate system or a limitation in EBMs. In this study we demonstrate that evidence for similar unstable behavior can also be found in an atmospheric general circulation model (GCM), using a realistic set of boundary conditions for the Carboniferous (300 Ma), one of the most extensive periods of glaciation in Earth history. When solar luminosity is sequentially lowered to near values appropriate for the Carboniferous, there is a discontinuous increase in summer snow area. The instability occurs in approximately the same area of parameter space as one previously found in an EBM. Analysis of selected fields indicates that the circulation is primarily affected in the area of snow increase; far-field effects are minimal. There is good agreement between model-generated summer snowcover and one reconstruction of Carboniferous ice cover. Although more work is required on this topic, our results provide increased support for the possibility that the snowline instability represents a real feature of the climate system, and that it may help explain some cases of glacial inception and abrupt transitions in Earth history.

Self-intersections in a three-parameter space of cosmic strings.

We present a three-parameter space of cosmic strings with first and third harmonics which includes the one-parameter Kibble-Turok and the two-parameter Turok spaces as subsets. A numerical examination shows that, while the area of parameter space representing self-intersections of the Turok string is smaller than previously thought, the percentage of self-intersecting strings increases with the larger parameter space. There exists, however, a sufficient fraction of non-self-intersecting strings in the three-parameter space so as to remain compatible with galaxy formation theories.

Analysis of orientationally disordered structures. III. Positive definiteness of orientational distribution functions

In previous publications [Press & Hüller (1973). Acta Cryst. A29, 252-256; Press (1973). Acta Cryst. A29, 257-263], the analysis of molecular crystalline structures by symmetry-adapted surface harmonics has been discussed. Implications of both molecular and site symmetry have been emphasized. Here the question is raised whether the magnitude of symmetry-allowed expansion coefficients is subject to constraints or not. Obvious restrictions result from the requirement that the density distribution at a molecular site has to be positive definite. Even stronger conditions are imposed by the requirement that the orientational distribution function g(τ) has to be positive definite, too. The area in parameter space for parameter values which are in agreement with the strong conditions is shown. Physically meaningful expansion coefficients are guaranteed by starting from a symmetry-allowed rotational potential V(τ).