Nonlinear Growth Rate Research Articles

Trudinger–Moser‐type inequality with logarithmic convolution potentials

We establish Moser–Trudinger-type inequalities in the presence of a logarithmic convolution potential when the domain is a ball or the entire space R 2 $\mathbb {R}^2$ . Moreover, we characterize critical nonlinear growth rates for these inequalities to hold and for the existence of corresponding extremal functions. In addition, we show that extremal functions satisfy corresponding Euler–Lagrange equations, and we derive general symmetry and uniqueness results for solutions of these equations.

Output-Feedback stabilization for stochastic nonlinear systems with Markovian switching and time-varying powers.

This paper investigates the output-feedback stabilization for stochastic nonlinear systems with both Markovian switching and time-varying powers. Specifically, by developing a novel dynamic gain method and using the Itô formula of Markovian switching systems, a reduced-order observer with a dynamic gain and an output-feedback controller are designed. By using advanced stochastic analysis methods, we show that the closed-loop system has an almost surely unique solution and the states are regulated to the origin almost surely. A distinct feature of this paper is that even though there is no Markovian switching, our design is also new since it can deal with nonlinear growth rate, while the existing results can only deal with constant growth rate. Finally, the effectiveness of the design method is verified by a simulation example.

Nonlinear Agglomeration of Bimodal Colloids under Microgravity

Abstract A study of like-charged, bimodal colloidal suspensions was conducted in microgravity aboard the International Space Station as part of NASA's Advanced Colloids Experiments-Heated-2 (ACE-H-2) experiments. Samples comprised of silsesquioxane microparticles (600 nm) and zirconia nanoparticles (5–15 nm) in 1.5 pH nitric acid were mixed and allowed to agglomerate over time while being imaged with NASA's Light Microscopy Module (LMM). The samples contained 1% of microparticles with varying concentrations of nanoparticles in 0.1%, 0.055%, and 0.01% by volume. Digital images were captured periodically by the LMM over 12 days. Image analysis, including cluster size and distribution, was performed in Python using the “Colloidspy” package. The study found that cluster size had increased over time in at least seven of nine samples, but two samples exhibited nonlinear growth rates, while others showed very slow growth with cluster sizes two orders of magnitude greater than the free microparticles. We hypothesize that all samples experienced nonlinear growth, but early transient effects after mixing were missed due to timing limitations in image acquisition. Transport limitations of clusters in these systems may have dominated agglomeration behavior in microgravity, despite the samples being thermodynamically unstable, but more study is required.

Soliton generation and drift wave turbulence spreading via geodesic acoustic mode excitation

The two-field equations that govern fully nonlinear dynamics of the drift wave (DW) and geodesic acoustic mode (GAM) interaction in toroidal geometry are derived within the nonlinear gyrokinetic framework. Two stages with distinctive features are identified and analyzed using both analytical and numerical approaches. In the ‘linear’ growth stage, the derived set of nonlinear equations can be reduced to the intensively studied parametric decay instability, accounting for the spontaneous resonant excitation of GAM by the DW. The main results of previous works on spontaneous GAM excitation, e.g. the greatly enhanced GAM group velocity and the nonlinear growth rate of GAM, are reproduced from the numerical solution of the two-field equations. In the fully nonlinear stage, soliton structures are observed to form due to the balancing of the self-trapping effect by the spontaneously excited GAM and kinetic dispersiveness of the DW. The soliton structures enhance turbulence spreading from the DW linearly unstable region to the stable region, exhibiting convective propagation instead of a typical linear dispersive process, and are thus expected to induce core-edge interaction and nonlocal transport.

Parametric Study of Resistive Plasmoid Instability

By using 2.5-dimensional resistive MHD simulations, dynamics of the plasmoid instability in a Harris current sheet has been studied with taking into account two main controlling parameters: the plasma-β in the range (0 < β < 1) and the amplitude ratio of magnetic guide field to the reconnection plane field in three different cases with zero, uniform, and non-uniform guide field. Varying the plasma-β changes the plasma compressibility which affects significantly on the linear and nonlinear growth rates of the plasmoid instability. For each of three cases, some associated scaling relations between the instability growth rate, the plasma-β and the magnitude of guide field are obtained.

Stability and Bifurcation Analysis of a Commensal Model with Additive Allee Effect and Nonlinear Growth Rate

In this paper, a commensal model with additive Allee effect on the first species and nonlinear growth rate on the second species is proposed. The classification, type and stability of all possible equilibria of the system are investigated. We also prove the presence of saddle-node bifurcation and transcritical bifurcation by Sotomayor’s theorem. At the end of this study, an example and its simulations are given to illustrate our main results.

Linear and nonlinear benchmarks between the CLT code and the M3D-C1 code for the 2/1 resistive tearing mode and the 1/1 resistive kink mode

The linear and nonlinear benchmarks between the CLT code and the M3D-C1 code for the 2/1 resistive tearing mode and the 1/1 resistive kink mode are presented. CLT is an explicit finite difference code, while M3D-C1 is an implicit finite element code. Although the implementations of CLT and M3D-C1 are totally different, we find that the simulation results of the resistive-kink mode and the m/n=2/1 tearing mode from M3D-C1 and CLT are almost the same, including the linear and nonlinear growth rates, the mode structures, the nonlinear saturation levels, the Poincare plots, and the scaling laws. This confirms that the nonlinear results for the 1/1 resistive-kink mode and 2/1 tearing mode are accurate and reliable.

Global Dynamics of a Holling-II Amensalism System with Nonlinear Growth Rate and Allee Effect on the First Species

Of concern is the global dynamics of a two-species Holling-II amensalism system with nonlinear growth rate. The existence and stability of trivial equilibrium, semi-trivial equilibria, interior equilibria and infinite singularity are studied. Under different parameters, there exist two stable equilibria which means that this model is not always globally asymptotically stable. Together with the existence of all possible equilibria and their stability, saddle connection and close orbits, we derive some conditions for transcritical bifurcation and saddle-node bifurcation. Furthermore, the global dynamics of the model is performed. Next, we incorporate Allee effect on the first species and offer a new analysis of equilibria and bifurcation discussion of the model. Finally, several numerical examples are performed to verify our theoretical results.

Nonlinear trapping in wave–particle interactions in tokamaks

The physics and transport consequences of the nonlinear trapping of the wave–particle interactions on the transit and bounce time scale are investigated by solving drift kinetic equation including collisions. The nonlinear particle dynamics is governed by a new set of coupled evolution equations for the phase of the wave, the toroidal component of the canonical momentum (radial motion), and the particle energy. The resonant particles are trapped nonlinearly in the phase space in a pendulum-like motion to resolve the linear resonance. The trapping mechanism is similar to that of the superbananas in the neoclassical theory. The nonlinear particle dynamics is employed to facilitate the solution of the drift kinetic equation. It is shown analytically that the particle distribution in the nonlinearly trapped region is flattened when a term of the order of is neglected. Here, δA is the typical magnitude of the normalized perturbed wave field. The trapped and barely circulating particles diffuse collisionally in the phase space. Transport fluxes, wave–particle energy transfer rate and nonlinear damping rate are calculated. The transport coefficients have a new analytic scaling U in the context of the wave–particle interaction, and can be significant even for a small amplitude wave in tokamaks. Here, ν is the collision frequency, r is the minor radius, and U is a function of the wave frequency, aspect ratio, magnetic shear, and mode numbers. It is estimated that when δA ∼ 10−4 for the magnetic perturbation, alpha particle energy loss rate is comparable to that of the neoclassical theory. The energy transfer rate is proportional to part of the particle flux, a new relation, and scales as ν ; the nonlinear damping or growth rate ν .

Influence of Kappa Distributions on the Whistler Mode Instability

AbstractKappa distributions possess an enhanced high‐energy tail characterized by a spectral index κ. Here we consider how kappa distributions influence the whistler mode instability. In a relativistic regime, we analyze the effects of the bi‐kappa and kappa loss‐cone distributions on the linear and nonlinear growth of whistler mode waves. We find that for κ = 2, the linear growth rate corresponding to the bi‐kappa distribution exceeds that for a bi‐Maxwellian distribution (κ = ∞) for electron anisotropies less than a critical value. The threshold wave amplitude for nonlinear growth corresponding to a hot injected bi‐kappa distribution can be sensitively dependent on the value of κ, but also depends crucially on the electron anisotropy, parallel hot electron temperature, and wave frequency. We plot time profiles of the wave magnetic field, frequency, total nonlinear growth rate and local nonlinear growth rate, and examine how these depend on κ. The sweep (chirp) rates of the whistler mode chorus waves are found to be in the range of observed values, for the adopted plasma parameters. For a given value of the loss‐cone parameter σ, realistic whistler mode wave profiles may only exist for a restricted range of κ. For example, for σ ≥ 4, realistic wave profiles require κ ≥ 4 as well as a sufficiently large anisotropy. In summary, this new study finds that the influence of kappa distributions on the whistler mode instability is complex, not least because the instability depends on several system parameters as well as κ.

Mutual inhibition in presence of a virus in continuous culture.

In this paper, we consider two species competing for a limiting substrate such that each species impedes the growth of the other one (Mutual inhibition) in presence of a virus inhibiting one bacterial species. A system of ordinary differential equations is proposed as a mathematical model for this competition. A detailed local qualitative analysis of the system is carried out. We proved that for a general nonlinear growth rates, the Competitive Exclusion Principle still valid, that at least one species goes extinct. For some cases where we have two locally stable equilibrium points, initial species concentrations are important in determining which is the winning species. Obtained results were confirmed by some numerical simulations using Matlab software.

Study of Unsteady Flow Through and Around an Array of Isolated Square Cylinders

Abstract Flow past a porous square cylinder, which is constituted by an array of small square cylinders, was studied by simplified and highly stable lattice Boltzmann method (LBM) at Re = 4000. The effects of solid volume fractions (SVF) on the flow structure and important aerodynamic parameters were investigated and the internal fluid was described by phase-averaged method and its interaction with the near region. Several energy-contained Strouhal numbers come up in cylinder arrays, and mass flux through arrays is calculated to estimate the effects of blocking. The average total force on the array is found to decrease monotonically with decreasing SVF, and it has a dramatic drop as SVF decreasing from 0.062 to 0.036. The mass flux of array for SVF = 0.062 is smaller than that of 0.073 due to the stagger arrangement of isolated cylinders. The underlying Reynolds stresses reveal the increase in mean wake length as SVF decreases, and it behaves like the solid one for SVF = 0.132. Comparing with the other SVFs, the shear layer region at SVF = 0.132 indicates a significant decrease of curvature and a linear increase of growth rate, but a nonlinear growth rate for the other SVFs can be observed. The local maximum turbulent kinetic energy (TKE) and vorticity in shear layer region reveal the variation during the initial stages following separation. Moreover, detailed investigations of instantaneous wake dynamics provide an assistance to understand the flow physics of cylinder arrays for different SVF.

Global exponential stabilization for a class of nonlinear systems by output feedback control with three gain‐scaling factors

SummaryIn this article, we consider a problem of global exponential stabilization for a class of nonlinear systems with the perturbed nonlinearity by output feedback. We provide a newly designed observer‐based output feedback controller with three gain‐scaling factors to deal with extended structure of nonlinearity and larger growth rate of nonlinearity over the existing results. Utilizing three gain‐scaling factors, we have more flexibility in selecting control parameters. Moreover, we present two procedures for selecting gain‐scaling parameters which are associated with stabilizing the system and improving the convergence rate of the system states and observer errors, respectively. Via numerical and practical examples, we illustrate the improved features of the proposed control scheme over the existing schemes.

Adaptive control of feedforward nonlinear systems with uncertain time‐varying parameters and an unknown time‐varying delay in the input

SummaryIn this paper, we consider a global regulation problem for a class of feedforward nonlinear systems. The key features of our considered system are identified by the presence of uncertain time‐varying parameters associated with main diagonal states and input and an unknown time‐varying delay in the input. Moreover, the growth rates of nonlinearity and the input‐delay are only known to be finite. To solve our considered problem, we give a process to obtain a compact set that contains the allowed time‐varying parameters. Then, we propose an adaptive controller which handles both unknown growth rate of nonlinearity and input‐delay for system regulation. We carry out the rigorous system analysis and show the effectiveness of our proposed control scheme via an application example.

One-point probability distribution function from spherical collapse: early dark energy versus ΛCDM

ABSTRACT We compute the one-point probability distribution function (PDF) of an initially Gaussian dark matter density field using spherical collapse (SC). We compare the results to other forms available in the literature and also compare the PDFs in the Λ-cold dark matter model with an early dark energy (EDE) model. We find that the skewed lognormal distribution provides the best fit to the non-linear PDF from SC for both cosmologies, from a = 0.1 to 1 and for scales characterized by the comoving width of the Gaussian: σG = 0.5, 1, and 2. To elucidate the effect of cosmology, we examine the linear and non-linear growth rates through test cases. For overdensities, when the two models have the same initial density contrast, the differences due to cosmology are amplified in the non-linear regime, whereas, if the two models have the same linear density contrast today, then the differences in cosmology are damped in the non-linear regime. This behaviour is in contrast with voids, where the non-linear growth becomes ‘self-regulatory’ and is less sensitive to cosmology and initial conditions. To compare the PDFs, we examine the difference of the PDFs and evolution of the width of the PDF. The trends with scale and redshift are as expected. A tertiary aim of this paper was to check if the fitting form for the non-linear density–velocity divergence relation, derived for constant equation of state (w) models by Nadkarni-Ghosh holds for the EDE model. We find that it does with an accuracy of 4 per cent, thus increasing its range of validity.

On the theory of phase transformation process in a binary supercooled melt

The processes of bulk crystallization of a supercooled binary melt where the nucleation and growth of solid particles occur are considered with allowance for different mechanisms of nucleation kinetics. A nonlinear set of integro-differential equations is analytically solved by means of the saddle-point technique. The crystal-radius distribution function, temperature, solute concentration, and supercooling of a binary liquid are found taking into account the nonlinear growth rates of spherical crystals. The supercooling represents a decreasing function of time while the distribution function increases up to a maximum size of crystals in a metastable melt.

Output feedback stabilization of time‐delay nonlinear systems with unknown continuous time‐varying output function and nonlinear growth rate

SummaryThis article addresses the problem of global output feedback stabilization for a class of time‐varying delay nonlinear systems with polynomial growth rate. The systems under investigation possess two remarkable features: the output is perturbed by an unknown sensitivity function that is not differentiable but continuous, and the nonlinearities are bounded by a polynomial function of the output multiplied by unmeasurable state variables. The new full‐order observer is established by introducing a dynamic gain and filtering unknown nonlinearities and time‐varying delay. With the help of the transformation skill and the reasonable combination of several systems, this article proposes a linear output feedback controller with the dynamic gain and completes the performance analysis based on the construction of two integral Lyapunov functions. Finally, a simulation example is presented to demonstrate the effectiveness of control strategy.

Evolution of a thin film down an incline: A new perspective

A new model which accounts for energy balance while describing the evolution of a thin viscous, Newtonian film down an incline at high Reynolds numbers and moderate Weber numbers has been derived. With a goal to improve the predictions by the model in inertia dominated regimes, the study employs the Energy Integral Method with ellipse profile EIM(E) as a weight function and is motivated by the success of EIM in effectively and accurately predicting the squeeze film force in squeeze flow problems and in predicting the inertial effects on the performance of squeeze film dampers [Y. Han and R. J. Rogers, “Squeeze film force modeling for large amplitude motion using an elliptical velocity profile,” J. Tribol. 118(3), 687–697 (1996)]. The focus in the present study is to assess the performance of the model in predicting the instability threshold, the model successfully predicts the linear instability threshold accurately, and it agrees well with the classical result [T. Benjamin, “Wave formation in laminar flow down an inclined plane,” J. Fluid Mech. 2, 554–573 (1957)] and the experiments by Liu et al. [“Measurements of the primary instabilities of film flows,” J. Fluid Mech. 250, 69–101 (1993)]. The choice of the ellipse profile allows us to have a free parameter that is related to the eccentricity of the ellipse, which helps in refining the velocity profile, and the results indicate that as this parameter is increased, there is a significant improvement in the inertia dominated regimes. Furthermore, the full numerical solutions to the coupled nonlinear evolution equations are computed through approximations using the finite element method. Based on a measure {used by Tiwari and Davis [“Nonmodal and nonlinear dynamics of a volatile liquid film flowing over a locally heated surface,” Phys. Fluids 21, 102101 (2009)]} of the temporal growth rate of perturbations, a comparison of the slope of the nonlinear growth rate with the linear growth rate is obtained and the results show an excellent agreement. This confirms that the present model’s performance is as good as the other existing models, weighted residual integral boundary layer (WRIBL) by Ruyer-Quil and Manneville [“Improved modeling of flows down inclined planes,” Eur. Phys. J: B 15, 357–369 (2000)] and energy integral method with parabolic profile [EIM(P)] by Usha and Uma [“Modeling of stationary waves on a thin viscous film down an inclined plane at high Reynolds numbers and moderate Weber numbers using energy integral method,” Phys. Fluids 16, 2679–2696 (2004)]. Furthermore, for any fixed inclination θ of the substrate, 0 < θ < π/2, there is no significant difference between the EIM(E) and EIM(P) results for weaker inertial effects, but when the inertial effects become stronger, the EIM(E) results for energy contribution from inertial terms to the perturbation at any streamwise location is enhanced. More detailed investigation on the model’s performance due to this enhancement in energy contribution and the assessment of the model as compared to the other existing theoretical models, experimental observations, and numerical simulations, in the inertia dominated regimes, will be reported in a future study.

Assessment of differences between DNA content of cell-cultured and freely suspended oocysts of Cryptosporidium parvum and their suitability as DNA standards in qPCR

BackgroundAlthough more modern methods are available, quantitative PCR (qPCR) is reproducible, sensitive and specific with instruments and expertise readily available in many laboratories. As such, the use of qPCR in Cryptosporidium research is well established and still widely used by researchers globally. This method depends upon the generation of standards at different concentrations to generate standard curves subsequently used for the quantification of DNA.MethodsWe assessed four types of DNA template used to generate standard curves in drug screening studies involving Cryptosporidium spp.: (i) serially diluted Cryptosporidium parvum oocysts (106–1); (ii) diluted template DNA from pure oocysts (×10–×106 dilution of 106 oocyst DNA template); (iii) oocysts incubated in human ileocecal adenocarcinoma (HCT-8) cells (105–1 and 5 × 104–50); and (iv) diluted DNA template (5 × 104) from cell culture incubated parasites (×10–×1000).ResultsSerial dilutions of both cell culture and pure oocyst suspension DNA template yielded better linearity than cell culture derived standards, with dilutions of 106 oocysts exhibiting similar quantification cycle (Cq) values to those obtained from DNA template dilutions of 106 oocysts. In contrast, cell culture incubated oocysts demonstrated significantly higher DNA content than equivalent freely suspended oocysts and diluted DNA template from both cell culture derived and freely suspended oocysts across numerous concentrations.ConclusionsFor many studies involving Cryptosporidium, only relative DNA content is required and as such, the superior linearity afforded by freely suspended oocysts and diluted DNA template (from either cell culture derived standards or freely suspended oocysts) will allow for more accurate relative quantification in each assay. Parasite division in the cell culture standards likely explains the higher DNA content found. These standards, therefore, have the potential to more accurately reflect DNA content in cell culture assays, and despite more modern methods available for absolute quantification, i.e. droplet digital PCR (ddPCR), the ubiquity of qPCR for the foreseeable future encourages further investigation into the reduced linearity observed in these standards such as varying oocyst seeding density, non-linear growth rates and assay efficiency.

Instability and transition induced by wall roughness in a finite Stokes layer

Abstract The transition of a classic unsteady flow, namely, a Stokes layer, induced by wall roughness is investigated numerically. On the basis of numerical simulations using a semispectral method, a detailed analysis of the evolution characteristics of the disturbances is carried out. The results show that turbulence arises from the presence of large pre-existing two-dimensional waves accompanied by rapidly increasing three-dimensional waves. It is found that the 3D disturbances exhibit exponential growth prior to the onset of the transition, while the 2D disturbances reach a saturated state. The nonlinear growth rate of the 3D disturbances displays a linear relation to the spanwise wavenumber, with the fundamental spanwise wavenumber itself being connected to the appearance of the transition. The breakdown in the transition process is also analyzed, and similar to steady flows, an enhanced instability is found.