Boundary Equilibrium Research Articles

Development of a tokamak magnetohydrodynamic code with the discontinuous Galerkin and Weighted Essentially Non-Oscillatory methods

In this study, we present the development of a new initial-value magnetohydrodynamic (MHD) code for toroidal geometry using discontinuous Galerkin (DG) and Weighted Essentially Non-Oscillatory (WENO) methods. The code utilizes a triangular mesh based on the flux of the fixed boundary equilibrium in the poloidal plane, which is uniformly divided in the toroidal direction. By solving the conservative perturbed generalized Lagrange multiplier (GLM) MHD model, the code can simulate both ideal and resistive toroidal plasmas. We validate the code by performing the linear calculations of the internal kink mode and tearing mode, and our results show good agreement with calculations conducted using an eigen-value MHD code. Furthermore, nonlinear simulations of the resistive internal kink mode and tearing mode are also carried out. We discuss the challenges and limitations of the code, as well as the future efforts to improve its performance. The development of this new initial-value MHD code opens up new possibilities for studying and understanding the dynamics of toroidal plasmas, contributing to the advancement of fusion research.

Slope Stability Evaluation and Geometrical Recommendation Using The Morgenstern Price Method

The study is located on PT Indocement Tunggal Prakarsa Tbk., which is still actively engaged in mining operations. The study aimed to determine the slope Safety Factor (SF) and offer suggestions for safe slope geometry for mining operations. Primary data collection through observation of lithology conditions, sample testing in the laboratory, scanline mapping, slope geometry measurements, and secondary data obtained from company inventories and related sources. The kinematic approach is used to assess the risk of landslides. This slope stability study uses boundary equilibrium based on the Morgenstern Price and Mohr-Coulomb failure criteria. Limestone is the majority of the rocks in the location. The kinematic analysis demonstrated that direct toppling and wedge toppling are the landslide potential at the research location. Using the non-circular boundary equilibrium approach, slope stability analysis was carried out on four Regions with dry and saturated groundwater conditions i.e, Regions A', B140, C120, and C135. The four Regions have stable slopes and are in good condition (SF values over 1.25). For the excavation to be carried out optimally, the recommendations for optimization of the overall slope geometry are given, namely, the height of the bench is 10 m, and the width of the ladder is 4 m. The slope angle is 80o, with SF value of 3.035 in dry conditions and an SF value of 2.021 in saturated conditions.

Stability, bifurcation, and control: Modeling interaction of the predator-prey system with Alles effect

In this study, we investigate the resilience and bifurcation properties of a discrete-time predator-prey model incorporating an Allee effect, the transfer of prey to predator energy, and a necessary threshold for prey consumption prior to predator reproduction. The discrete-time formulation involves iteratively adjusting the predator and prey populations at distinct time steps. The model encompasses equations that depict the fluctuations in population sizes, considering processes like breeding, predation, death, and the influence of the Allee effects. The investigation results indicate that the system's boundary equilibrium goes through a period-doubling bifurcation. Moreover, when the system is in close proximity to the exclusive positive equilibrium influenced by the prey-predator population, it exhibits Neimark-Sacker bifurcations. It's worth noting that our observations suggest that the predator population might approach extinction or achieve a stable equilibrium when chaotic dynamics appear in the prey population. The validity of this phenomenon is confirmed using the Lyapunov exponents calculation, which verifies the range of diverse dynamic behaviors. The outcomes and analysis of this research offer intriguing insights in both mathematical and biological realms. Implementing methods for chaos control could be beneficial in stabilizing or mitigating the occurrence of chaotic dynamics. Additionally, the theoretical results are supported by numerical simulations, phase portraits, bifurcation diagrams, graphs depicting chaos control, and maximum Lyapunov exponent graphs.

A nonautonomous model for the effect of rarity value on the dynamics of a predator–prey system with variable harvesting

The human inclination to value rarity contributes to the exploitation of many wildlife species by driving up their market value when their numbers are scarce. This research investigates a model to explore how the value attached to rarity impacts the harvesting of predator species and subsequently influences the dynamics of predator–prey interactions in an ecological system. To broaden the model's scope, it's extended to its nonautonomous form by allowing the harvesting rate of predator species and the degree of neutrality in consumers to respond for the rarity of predator species to change over time. Both the autonomous and nonautonomous systems are explored through analytical and numerical investigation. Numerical findings reveal that as the prey population's growth rate gradually increases, the unforced system shifts from stable boundary equilibrium (harvesting absent) to stable interior equilibrium, then to oscillatory interior equilibrium, followed by period doubling oscillations leading to chaotic behavior. The autonomous system exhibits limit cycle oscillations around the boundary equilibrium when predator species are minimally harvested. Moderate harvesting stabilizes the system, while excessive harvesting induces instability. However, the chaotic oscillations can be eliminated through period‐halving route upon decreasing the consumers' response to predator rarity. Furthermore, the seasonally forced model demonstrates periodic solutions, higher order periodic solutions, and chaotic dynamics contingent upon different settings of prey growth rate, the strength of seasonality in predator harvesting, and the consumers' response to predator rarity.

Positive steady-state solutions for a vegetation–water model with saturated water absorption

In this paper, a vegetation–water model with saturated water absorption is considered. The global stability of boundary equilibrium is first obtained. Then the stability and Turing instability of positive equilibria are discussed and we find that the equilibrium with vegetation small is always unstable and if the vegetation diffusion is small or the water diffusion is large, the other positive equilibrium loses its stability and Turing instability occurs. In addition, a priori estimates of nonnegative steady-state solutions is obtained by the maximum principle. Moreover, some detailed qualitative analyses are carried out on the steady-state bifurcations at both simple and double eigenvalues. In particular, we establish the criterion to determine the bifurcation direction. Finally, some dynamics near the bifurcation point are showed numerically and we depict the evolution processes of vegetation patterns under different parameters. Our results show that the parameter 1/p, which represents the conversion of water absorption, has a great impact on the vegetation patterns formation: with the increase of p, the vegetation biomass decreases, and meanwhile it can induce the transition of different pattern structures.

Dynamics of a plant-herbivore system with Ricker plant growth and the strong Allee effects on plant population

We examine a discrete-time model of plant-herbivore interaction, in which the population of plants follows the Ricker law and is affected by strong Allee effects. We discuss the equilibrium points, including their characteristics and number. Additionally, we analyze in detail the local behavior of the solutions around the equilibrium points. Under certain initial conditions, both populations may go extinct. We conduct research demonstrating the occurrence of transcritical and period-doubling bifurcations, resulting in a stable two-cycle at the boundary equilibrium. However, the interior equilibrium becomes unstable due to the emergence of the Neimark-Sacker bifurcation and period-doubling. We confirm the latter's existence through analytical methods and illustrate the period-doubling bifurcation through numerical simulations.

Piecewise immunosuppressive infection model with viral logistic growth and effector cell-guided therapy

<abstract><p>This work investigated a piecewise immunosuppressive infection model that assessed the effectiveness of implementing this therapeutic regimen once the effector cell count falls below a specific threshold level by introducing a threshold strategy. The sliding mode dynamics, global dynamics, and boundary equilibrium bifurcations of the Filippov system were examined based on the global dynamics of the two subsystems. Our primary findings indicate that the HIV viral loads and effector cell counts can be stabilized within the required predetermined level. This outcome depends on the threshold level, immune intensity, and the initial values of the system. Therefore, properly combining these key factors makes it possible to effectively curb the abnormal increase of virus and keep the effector cells at a reliable level. This approach maximizes the controllable range of the HIV. The proposed switching system incorporating pseudo-equilibrium exhibits three types of equilibriums that could be bistable or tristable. It means there is a possibility of controlling the virus after administering therapy if the immune intensity $ c $ is limited within the range of the post-treatment control threshold and the elite control threshold when $ {R_0} > {R_{{c_1}}} > {R_{{c_2}}} > 1 $.</p></abstract>

FORMATION AND DISSOCIATION OF CO2 HYDRATES WITHIN A NATURAL BASALTIC-BASED POROUS MEDIUM FROM ICELANDIC MARGINS

Carbon dioxide hydrates were formed and dissociated in a lab-scale apparatus, filled with demineralized water and porous sediments. Two different porous media were tested; the first consists of glass beads, commonly used to reproduce marine environments during lab-scale hydrate formation experiments. Natural basaltic sand, typical of the Icelandic margins, was chosen as second sediment. The role played by the two sediments was analyzed during both the hydrate formation and dissociation processes and the results were compared with each other. In particular, the pressure-temperature values measured during the dissociation phase were compared with the phase boundary equilibrium conditions for pure carbon dioxide hydrates, carried out in the absence of any porous sediment, collected from the scientific literature. The presence of porous sand favored the preservation of hydrates during the dissociation phase; in detail, hydrates melted at pressures 1.1 bar higher (on average) in the presence of glass beads, and 1.0 bar higher with the basaltic sand. The formation was more abundant in the presence of glass beads: 0.227-0.232 mol of hydrates were produced, against 0.141-0.21 mol achieved when the basaltic sediment was used.

Stability analysis of solutions of certain May's host-parasitoid model by using KAM theory

<abstract><p>We use the Kolmogorov-Arnold-Moser (KAM) theory to investigate the stability of solutions of a system of difference equations, a certain class of a generalized May's host-parasitoid model. We show the existence of the extinction, interior, and boundary equilibrium points and examine their stability. When the rate of increase of hosts is less than one, the zero equilibrium is globally asymptotically stable, which means that both populations are extinct. We thoroughly describe the dynamics of 1:1 non-isolated resonance fixed points and have used the KAM theory to determine the stability of interior equilibrium point. Also, we have conducted several numerical simulations to support our findings by using the software package Mathematica.</p></abstract>

A double time-delay Holling Ⅱ predation model with weak Allee effect and age-structure

<abstract><p>A double-time-delay Holling Ⅱ predator model with weak Allee effect and age structure was studied in this paper. First, the model was converted into an abstract Cauchy problem. We also discussed the well-posedness of the model and the existence of the equilibrium solution. We analyzed the global stability of boundary equilibrium points, the local stability of positive equilibrium points, and the conditions of the Hopf bifurcation for the system. The conclusion was verified by numerical simulation.</p></abstract>

Reconstruction of Natural Slope Stability by Limit Equilibrium Methods and Finite Element Methods

Slope stability analysis is particularly important for natural slopes that are relatively undesigned to be technically safe. Natural slopes are prone to collapse depending on the condition of the slope material and rainfall. A comprehensive stability analysis using numerical methods to calculate the factor of safety and probability of collapse can be used as a reference to assess the safety of a slope. This study discusses boundary equilibrium and finite element in the analysis of slope safety factor and landslide probability by reconstructing the slope with landslide history. The reconstruction of the slope is based on the actual slope condition with saturated material due to rain that has a safety factor of 0.5 using mathematical methods which are then analyzed numerically in this study. The results of the factor of safety on the saturated condition slope have a value that is not much different from the actual condition with a 100% probability of landslide. The results of low shear strength values in each slice of the limit equilibrium analysis for saturated conditions also indicate that the slope is in an unsafe condition which is supported by the development of shear and tensile strains in the finite element method analysis in the slope slide plane area which causes the stress distribution in the actual landslide area to be unstable.

The Impact of Predator-dependent Prey Refuge on the Dynamics of a Leslie-Gower Predator-prey Model

In this paper, we propose a new Leslie-Gower predator-prey model with predator-dependent prey refuge. Firstly, we obtain the positivity and boundedness of the system solution. Secondly, we prove that the origin is unstable using blow-up method, analyze the existence and local stability of the boundary equilibrium point and positive equilibrium point, and prove that the unique positive equilibrium point of the system is globally asymptotically stable by constructing a suitable Dulac function. Finally, mathematic analysis and numerical simulation show that: (1) when the strength of the predator-dependent prey refuge k = 0 , the dynamics of the predator-prey system without predator-dependent prey refuge are consistent with the results obtained from the traditional Leslie-Gower predator-prey system; (2) when k tends to positive infinity, the predator-dependent refuge lead to prey population densities fall somewhere between without prey refuge and with proportional refuge. However, the predator densities within this new form of the predator-dependent prey refuge is greater than the densities of predators without prey refuge and with proportional refuge; (3) increasing the strength k of the predator-dependent prey refuge can increase the densities of predator and prey populations respectively.

Application of porous layer in enhancing stability of salt-gradient solar pond

The salt gradient solar pond (SGSP) can capture and store a large amount of solar energy for a long time, making it an efficient and economical solar energy facility. The long-term operational stability of SGSP can be damaged by interface erosion, which is caused by thermosolutal convection. Attaching porous layers is a promising approach to slow down interface erosion. In this study, we simulated the unsteady thermosolutal convection in SGSP with Lattice Boltzmann methods (LBM). The effects of a porous layer on salinity gradient stability are analyzed with boundary equilibrium and marginal criterion. The simulation results show that the SGSP becomes thermally unstable when the interface erosion begins. By attaching a porous layer to the bottom surface, the convective flow is weakened and the instability in SGSP is reduced. The stability analysis shows that the thermally unstable state in SGSP transits into a theoretically stable state by attaching a porous layer. As the stable state is prolonged, the fluid temperature in the storage layer increases. We also find that reducing the permeability of the porous layer or increasing the layer thickness prolongs the stable state. This study helps to improve future SGSP design and promote its engineering application.

Limit Cycles of a Planar Piecewise Linear System with an Improper Node

This paper is concerned with the number of limit cycles for a planar piecewise linear (PWL) system with two zones separated by a straight line. Assume that one of the subsystems of the PWL system has an improper node. The number of limit cycles for saddle-improper node type, focus-improper node type and center-improper node type (the focus or the center is a virtual or boundary equilibrium) are studied. First, we introduce displacement functions and study the number of zeros of displacement functions for different types. Then, we give the parameter regions where the exact number of limit cycles is one or two (at least two) for different types.

Dynamical analysis of a chemostat model for 4-chlorophenol and sodium salicylate mixture biodegradation

We consider a mathematical continuous-time model for biodegradation of 4-chlorophenol and sodium salicylate mixture by the microbial strain Pseudomonas putida in a chemostat. The model is described by a system of three nonlinear ordinary differential equations and is proposed for the first time in the paper [Y.-H. Lin, B.-H. Ho, Biodegradation kinetics of phenol and 4-chlorophenol in the presence of sodium salicylate in batch and chemostat systems, Processes, 10:694, 2022], where the model is only quantitatively verified. This paper provides a detailed analysis of the system dynamics. Some important basic properties of the model solutions like existence, uniqueness and uniform boundedness of positive solutions are established. Computation of equilibrium points and study of their local asymptotic stability and bifurcations in dependence of the dilution rate as a key model parameter are also presented. Thereby, particular intervals for the dilution rate are found, where one or three interior (with positive components) equilibrium points do exist and possess different types of local asymptotic stability/instability. Hopf bifurcations are detected leading to the occurrence of stable limit cycles around some interior equilibrium points. A transcritical bifurcation also exists and implies stability exchange between an interior and the boundary (washout) equilibrium. The results are illustrated by lots of numerical examples.

Limited agricultural spectral dataset expansion based on generative adversarial networks

With the rise of deep learning, the combination of spectroscopy analysis techniques and deep learning methods has been extensively utilized in the field of agriculture, such as the detection of crop diseases, soil analysis, and crop quality assessment. Unfortunately, it is often difficult to obtain enough spectral samples that can be used for deep learning because of environmental constraints, equipment limitations, and labour costs. To address this issue, we propose a spectral sample augmentation technique based on the K-condition boundary equilibrium generative adversarial networks (KC-BEGAN). First, a stable boundary equilibrium generative adversarial network (BEGAN) model is constructed, and the KC-BEGAN model is built by incorporating chemical property labels, multiscale gradient information, and the k-nearest neighbour algorithm. The goal is to enrich complete spectral samples with chemical properties. Second, we compare the differences between generated samples and real samples using methods such as t-distributed stochastic neighbour embedding, Mahalanobis distance, F test, and maximum mean discrepancy. Following data augmentation by the KC-BEGAN model, the R2 on the test set for traditional regression models (PLSR, SVR, RR, PCR) and deep regression models (Inception-ResNet, Inception, 1D-CNN) improved by 4.9%, 0.4%, 2.5%, 4.4%, 3.6%, 4.4%, and 6.8% respectively. Furthermore, this study replaces the GAN module in the KC-BEGAN model with Diffusion-GAN and conducts experiments following the same procedure to evaluate the feasibility of other GAN models in augmenting labelled near-infrared spectral samples. The research results indicate that the spectral samples generated by the KC-BEGAN model are reliable and can meet the expansion requirements of small-scale spectral sample sets. Simultaneously, replacing the GAN module in the KC-BEGAN model with other GAN models is also feasible, suggesting that the continuous label marking method proposed in this study is indeed effective.

On the undesired equilibria induced by control barrier function based quadratic programs

In this paper, we analyze the system behavior for general nonlinear control-affine systems when a control barrier function-induced quadratic program-based controller is employed for feedback. In particular, we characterize the existence and locations of possible equilibrium points of the closed-loop system and also provide analytical results on how design parameters affect them. Based on this analysis, a simple modification on the existing quadratic program-based controller is provided, which, without any assumptions other than those taken in the original program, inherits the safety set forward invariance property, and further guarantees the complete elimination of undesired equilibrium points in the interior of the safety set as well as one type of boundary equilibrium points, and local asymptotic stability of the origin. Numerical examples are given alongside the theoretical discussions.

An analytical approach of applying the Lyapunov direct method to polynomial differential systems with discrete time delays

The Lyapunov direct method is a common and effective tool for discussing the global stability of dynamical systems. In this paper, we propose an analytical approach to apply this method for the global stability of the positive equilibrium of a polynomial differential system with discrete time delays, which includes how to construct an applicable Lyapunov functional and to verify the negative (semi-)definiteness of its derivative. Here the linear combination of the Volterra-type functions/functionals and their integrals plays an important role. Moreover, when the delayed system also has a boundary equilibrium, we show that the Lyapunov functional for the global stability of the positive equilibrium can be reformulated to establish the global stability of the boundary equilibrium. As an illustration, we apply this approach to a virus infection model with two delays.

AE-BEGAN based Synthetic Data Augmentation for Sample-Limited High-Dimensional Problems with Application to NIR Spectral Data

Synthetic data augmentation holds substantial research and application value in scenarios characterized by limited samples and high dimensions. It enhances the analytical ability and efficiency of spectral analysis models. This paper proposes Autoencoder-Combined Boundary Equilibrium Generative Adversarial Networks (AE-BEGAN) as a new method for augmenting synthetic data in scenarios with limited samples and high dimensions, with a specific emphasis on near-infrared (NIR) spectral data. The spectral data first undergoes preprocessing procedures that encompass advanced noise reduction algorithms and techniques for removing abnormal samples, guaranteeing elimination of unwanted disturbances and outliers. Then, the pre-processed data is utilized to train the AE-BEGAN model, which generates augmented synthetic samples. Finally, real NIR spectral data obtained from lubricant samples exhibiting different water contents were employed to validate and test the performance of the model. The experimental results demonstrate that the AE-BEGAN model outperforms other GANs in generating synthetic data of high quality and diversity, as quantified by two evaluation metrics, α-Precision and β-Recall with scores of approximately 0.86 and 0.28, respectively. The application case study confirms that the AE-BEGAN model exhibits the capability to generate derived NIR spectra and expand the number of spectra in scenarios with limited samples and high dimensions.

Lyapunov functions for studying global asymptotic stability of two rumor spreading models

In a previous work (2018, Commun. Theor. Phys. 70, 795–802), a new compartment model for the spreading of rumors was introduced and analyzed. However, only the local asymptotic stability of this model was discussed. In the present work, we first provide a rigorous mathematical analysis for the global asymptotic stability (GAS) of the above-mentioned rumor spreading model. By constructing suitable Lyapunov candidate functions, we obtain the GAS of a rumor-free (boundary) equilibrium point and a unique rumor-spreading (positive) equilibrium point. After that, we utilize the approach based on the Lyapunov candidate functions to study the GAS of another rumor spreading model with control strategies, which was proposed in (2022, Physica A 606, 128157). As an important consequence, the GAS of the rumor spreading model with control strategies is determined fully without resorting to technical hypotheses used in the benchmark work. Lastly, the theoretical findings are supported by a set of illustrative numerical examples. The obtained results not only improve the ones constructed in the two above-mentioned benchmark papers but also can be extended to study the global dynamics of other rumor propagation models in the context of both integer-order and fractional-order derivatives.