Parameter Space Diagram Research Articles

Multistability in a Circulant Dynamical System

In this paper we report on a two parameter four-dimensional dynamical system with cyclic symmetry, namely a circulant dynamical system. This system is a twelve-term polynomial system with four cubic nonlinearities. Reported are some parameter-space diagrams for this system, all of them considering the same range of parameters, but generated from different initial conditions. We show that such diagrams display the occurrence of multistability in this system. Properly generated bifurcation diagrams confirm this finding. Basins of attraction of coexisting attractors in the related phase-space are presented, as well as an example showing phase portraits for periodic and chaotic coexisting attractors.

Contrasting effects of prey refuge on biodiversity of species

Abstract Refugia have been perceived as a major role in structuring species biodiversity, and understanding the impacts of this force in a community assembly with prey–predator species is a difficult task because refuge process can interact with different ecological components and may show counterintuitive effects. To understand this problem, we used a simple two-species model incorporating a functional response inspired by a Holling type-II equation and a prey refuge mechanism that depends on prey and predator population densities (i.e., density-dependent prey refuge). We then perform the co-dimension one and co-dimension two bifurcation analysis to examine steady states and its stability, together with the bifurcation points as different parameters change. As the capacity of prey refuge is varied, there occur critical values i.e., saddle-node and supercritical Hopf bifurcations. The interaction between these two co-dimension one bifurcations engenders distinct outcomes of ecological system such as coexistence of species, bistability phenomena and oscillatory dynamics. Additionally, we construct a parameter space diagram illustrating the dynamics of species interactions as prey refuge intensity and predation pressure vary; as the two saddle-node move nearer to one another, these bifurcations annihilate tangentially in a co-dimension two cusp bifurcation. We also realised several contrasting observations of refuge process on species biodiversity: for instance, while it is believed that some refuge processes (e.g., constant proportion of prey refuge) would result in exclusion of predator species, our findings show that density-dependent prey refuge is beneficial for both predator and prey species, and consequently, promotes the maintenance of species biodiversity.

Concentrated protein solutions investigated using acoustic levitation and small-angle X-ray scattering.

An acoustically levitated droplet has been used to collect synchrotron SAXS data on human serum albumin protein solutions up to a protein concentration of 400 mg ml-1. A careful selection of experiments allows for fast data collection of a large amount of data, spanning a protein concentration/solvent concentration space with limited sample consumption (down to 3 µL per experiment) and few measurements. The data analysis shows data of high quality that are reproducible and comparable with data from standard flow-through capillary-based experiments. Furthermore, using this methodology, it is possible to achieve concentrations that would not be accessible by conventional cells. The protein concentration and ionic strength parameter space diagram may be covered easily and the amount of protein sample is significantly reduced (by a factor of 100 in this work). Used in routine measurements, the benefits in terms of protein cost and time spent are very significant.

Wave propagation reversal for wavy vortices in wide-gap counter-rotating cylindrical Couette flow.

We present a numerical study of wavy supercritical cylindrical Couette flow between counter-rotating cylinders in which the wavy pattern propagates either prograde with the inner cylinder or retrograde opposite the rotation of the inner cylinder. The wave propagation reversals from prograde to retrograde and vice versa occur at distinct values of the inner cylinder Reynolds number when the associated frequency of the wavy instability vanishes. The reversal occurs for both twofold and threefold symmetric wavy vortices. Moreover, the wave propagation reversal only occurs for sufficiently strong counter-rotation. The flow pattern reversal appears to be intrinsic in the system as either periodic boundary conditions or fixed end wall boundary conditions for different system sizes always result in the wave propagation reversal. We present a detailed bifurcation sequence and parameter space diagram with respect to retrograde behavior of wavy flows. The retrograde propagation of the instability occurs when the inner Reynolds number is about two times the outer Reynolds number. The mechanism for the retrograde propagation is associated with the inviscidly unstable region near the inner cylinder and the direction of the global average azimuthal velocity. Flow dynamics, spatio-temporal behavior, global mean angular velocity, and torque of the flow with the wavy pattern are explored.

Period-Adding Structures in the Parameter-Space of a Driven Josephson Junction

Two-dimensional parameter-space diagrams related to a driven Josephson junction are reported. Three cases are considered, namely those involving the external direct current as one of the parameters. Typical periodic structures embedded in a chaotic region are observed in all diagrams, organized in different ways: (i) As structures with a similar shape to the Arnold tongues of the circle map, in period-adding sequences, and (ii) as structures with other shapes, in arrangements including two mixed sets of period-adding sequences.

Phase diagrams of forced magnetic reconnection in Taylor’s model

Recent progress in the understanding of how externally driven magnetic reconnection evolves is organized in terms of parameter space diagrams. These diagrams are constructed using four pivotal dimensionless parameters: the Lundquist number $S$, the magnetic Prandtl number $P_{m}$, the amplitude of the boundary perturbation $\hat{{\it\Psi}}_{0}$, and the perturbation wave number $\hat{k}$. This new representation highlights the parameter regions of a given system in which the magnetic reconnection process is expected to be distinguished by a specific evolution. Contrary to previously proposed phase diagrams, the diagrams introduced here take into account the dynamical evolution of the reconnection process and are able to predict slow or fast reconnection regimes for the same values of $S$ and $P_{m}$, depending on the parameters that characterize the external drive, which have not been considered until now. These features are crucial to understanding the onset and evolution of magnetic reconnection in diverse physical systems.

Visualizing Estimation Error with Bias-Variance Parameter Space Diagrams

Who would want to be biased? Bias seems obviously, inherently bad. An example of a biased estimator is one that excludes explanatory variables, such as a model of school dropout rates that excludes science test scores. Another example is a model that restricts the allowed parameter values, such as a financial investment model that has a maximum amount that can be invested in any single mutual fund. But there are empirical and analytical demonstrations that biased estimators like these can outperform unbiased estimators. Yet data modelers remain reluctant to use biased estimators because these findings seem so unintuitive.This paper provides a visual demonstration of how biased estimators work so even people with minimal statistics can feel comfortable using them. The bias-variance parameter space in this paper are a simplified version of the parameter space diagrams in standard machine learning and statistics textbooks such as Hastie et. al (2001). The baseline case shows an unbiased estimator using a large parameter space, fit with a large sample or small sample. The bias and variance achieved in this case is compared with a) a small, constrained parameter space such as a unit-weight linear model and b) ridge regression, an example of shrinkage. We then consider these cases again in highly collinear data, a situation in which biased models have an even bigger advantage over unbiased models. The visualization provides a deeper understanding of what it means for an estimator to reduce sampling variance at the cost of additional bias. This should give the reader more confidence in using biased estimators in situations where they are expected to outperform unbiased estimators.

Dynamics in the Parameter Space of a Neuron Model

Some two-dimensional parameter-space diagrams are numerically obtained by considering the largest Lyapunov exponent for a four-dimensional thirteen-parameter Hindmarsh—Rose neuron model. Several different parameter planes are considered, and it is shown that depending on the combination of parameters, a typical scenario can be preserved: for some choice of two parameters, the parameter plane presents a comb-shaped chaotic region embedded in a large periodic region. It is also shown that there exist regions close to these comb-shaped chaotic regions, separated by the comb teeth, organizing themselves in period-adding bifurcation cascades.

Dynamics of a neuron model in different two-dimensional parameter-spaces

We report some two-dimensional parameter-space diagrams numerically obtained for the multi-parameter Hindmarsh–Rose neuron model. Several different parameter planes are considered, and we show that regardless of the combination of parameters, a typical scenario is preserved: for all choice of two parameters, the parameter-space presents a comb-shaped chaotic region immersed in a large periodic region. We also show that exist regions close these chaotic region, separated by the comb teeth, organized themselves in period-adding bifurcation cascades.

Optical Design of a Normally-Black Twisted-Nematic Liquid Crystal Mode with Achromatic Dark State

We propose an optical configuration to compensate dispersion characteristics in the dark state of a normally-black twisted-nematic (NB-TN) liquid crystal display (LCD). By optimization of parameters with a parameter space diagram (PSD), the high contrast-ratio (CR) over 500:1 could be obtained using two +A plates. Furthermore, excellent dispersion characteristics were also obtained in the bright state. We confirmed the performance of the proposed method and structure through both the numerical calculations and the experiments.

P‐188: Optical Configuration of Transflective Liquid Crystal Display for Dynamic and Memory Operating Mode

AbstractWe propose an optical configuration of transflective liquid crystal display (LCD), which can be operated not only as a bistable device but also as a monostable one. By using the parameter space diagram (PSD) method, the memory mode and the dynamic mode are optimized for a reflective type and a transmissive type, respectively. We demonstrate excellent optical performance of our transflective dual mode, such as an excellent dark state and high contrast ratio.

Transflective Dual Operating Liquid Crystal Display for Memory and Dynamic Modes

We propose a transflective configuration of a tristate liquid crystal display (LCD), which has a unique texture of the splay, π twist, and bend states so that it can be operated not only as a bistable device but also as a monostable one. By using the parameter space diagram method as well as a single-cell gap structure, the memory mode is optimized for a reflective type and the dynamic mode for a transmissive type. We demonstrate excellent optical performance of our transflective dual mode, such as an excellent dark state, a high contrast ratio, and a high transmittance efficiency.

꼬인 네마틱 액정 셀의 Normally Black 모드 광학설계

본 논문에서는 <TEX>$90^{\circ}$</TEX> 꼬인 네마틱 액정 셀에 전계를 인가하지 않았을 때 어두운 상태를 나타내는 normally black 모드 구현 시, 높은 명암비를 제공하는 광학구조를 제안하였다. 제안된 구조에서 +A 필름과 <TEX>${\lambda}/2$</TEX> 필름을 각각 1 장씩 사용하여 <TEX>$90^{\circ}$</TEX> 꼬인 액정층을 통과한 빛의 가시광 파장대에서 편광 상태가 거의 동일하게 만들어 줌으로써, 어두운 상태의 분광특성을 향상시킬 수 있다. PSD(Parameter Space Diagram) 방법을 사용하여 최적조건을 구해 계산한 결과 500: 1 이상의 높은 명암비를 확보할 수 있었으며, 어두운 상태뿐만 아니라 밝은 상태의 분광특성도 향상됨을 시뮬레이션 및 실험을 통해 확인할 수 있었다. We propose an optical configuration to compensate dispersion characteristics in the dark state of normally black twisted nematic (NB-TN) liquid crystal display (LCD). We employed a half-wave plate (HWP) and a +A plate to achieve the superior dark state. By using the parameter space diagram (PSD) method, we could obtain the optimum values of parameters and the high contrast-ratio over 500 : 1 could be obtained. Furthermore, excellent dispersion characteristics were also obtained in the bright state. We could confirm the performance of the proposed structure using both the numerical calculations and the experiments.

Transflective liquid-crystal display using low-twisted vertically aligned mode

We present the design of a transflective and low-power LCD using a low-twist vertically aligned liquid crystal (LC) cell, achieved by blending a chiral additive with a patterned reflector in a single-domain configuration. Unlike the conventional single-domain transflective LCD, in which it is possible to optimize only one of the transmissive and reflective regions, the device suggests that it is possible to optimize both the transmission and reflection design to obtain favorable results in both reflective and transmissive light conditions by optimizing the chiral pitch and twist angle. From the parameter space diagram (PSD) method, which does not include the information on chiral pitch and the nonuniform LC director tilt angle, optimization focused on the transmissive region is performed. By analyzing the relation between the transmittance and the chiral pitch under the applied voltage, the optimized twist angle and chiral pitch are proposed. It is described that the optimized twist angle is also available for the reflective region by the dynamic PSD method by considering the average tilt angle under applied voltage.

Genesis of Ectopic Waves: Role of Coupling, Automaticity, and Heterogeneity

Many arrhythmias are believed to be triggered by ectopic sources arising from the border of the ischemic tissue. However, the development of ectopic activity from individual sources to a larger mass of cardiac tissue remains poorly understood. To address this critical issue, we used monolayers of neonatal rat cardiomyocytes to create conditions that promoted progression of ectopic activity from single cells to the network that consisted of hundreds of cells. To explain complex spatiotemporal patterns observed in these experiments we introduced a new theoretical framework. The framework’s main feature is a parameter space diagram, which uses cell automaticity and coupling as two coordinates. The diagram allows one to depict network behavior, quantitatively address the heterogeneity factor, and evaluate transitions between different regimes. The well-organized wave trains were observed at moderate and high cell coupling values and network heterogeneity was found to be qualitatively unimportant for these regimes. In contrast, at lower values of coupling, spontaneous ectopic activity led to the appearance of fragmented ectopic waves. For these regimes, network heterogeneity played an essential role. The ectopic waves occasionally gave rise to spiral activity in two different regions within the parameter space via two distinct mechanisms. Together, our results suggest that localized ectopic waves represent an essential step in the progression of ectopic activity. These studies add to the understanding of initiation and progression of arrhythmias and can be applied to other phenomena that deal with assemblies of coupled oscillators.

Topology of Windows in the High-Dimensional Parameter Space of Chaotic Maps

Periodicity is ubiquitous in nature. In this work, we analyze the dynamical reasons for which periodic windows, that appear in parameter space diagrams, have different shapes and structures. For that, we make use of a dynamical quantity, called spine — the skeleton of the window, in order to explain a conjecture that describes the presence of periodic windows in the parameter space of high-dimensional chaotic systems.

Universality and scaling in the circle map under external periodic driving

We investigate scaling properties of а neighborhood of the «golden mean» critical point in the circle map in presence of external periodic forcing upon the system. We consider such perturbations of the fixed point of the Feigenbaum-Kadanoff-Shenker renormalization group equation, which are initiated by periodic forcing. We show that, depending upon the frequency of external forcing, two types of scaling behavior can be observed. The first (P-type) 18 associated with periodic repetition of the structures of dynamical regimes in parameter space under subsequent scaling transformations. For the second case (Q-type of scaling) quasiperiodic behavior of structures in the parameter space takes place. The both types of scaling are illustrated by parameter-space diagrams for re-scaled Lyapunov exponents.

Generalized mixed‐mode reflective LCDs with large cell gaps and high contrast ratios

Abstract— A generalized mixed twisted‐nematic/birefringent‐effect mode reflective LCD is presented. This new mode encompasses all previously published reflective nematic LCD modes and is clearly shown on the parameter space diagrams. We also discuss how to obtain operating conditions of this generalized mixed‐mode display with large cell gaps, low color dispersion, and high contrast at low voltages. This operating mode is particularly suitable for reflective crystalline‐silicon‐backplane active‐matrix LCDs.

Reflective twisted nematic liquid crystal displays. II. Elimination of retardation film and rear polarizer

Reflective twisted nematic liquid crystal displays consisting of just one input polarizer, a liquid crystal cell, and a rear reflector are discussed. We first show that all such display modes reported in the literature can be depicted systematically in a series of parameter space diagrams. Then by making use of such diagrams as a guide, we show that other new high quality reflective display modes can be obtained. The three parameters that are varied are the input polarizer angle α, the liquid crystal twist angle φ, and the cell gap–liquid crystal birefringence product dΔn. A full numerical simulation which takes into account of the liquid crystal director deformation under the application of an electric field was used to optimize such displays. Experimental results agree well with the theoretical predictions.

Parameter space representation of liquid crystal display operating modes

A parameter space approach is used to represent the operation of all liquid crystal displays. The Jones matrix nematic is used to analyze the static optical properties of liquid crystal displays with any twist angle, cell thickness, natural birefringence, and polarizer angles. All the usual display modes such as 90° TN, ECB, OMI, SBE, and STN can be visualized on a simple parameter space diagram. Besides its pedagogical values, this formulation has the advantage of simplicity in showing the physics and operating conditions of supertwist as well as low twist liquid crystal displays. It can also be used to analyze novel situations such as reflective displays which do not require any back polarizer.