Global Stability Criteria Research Articles

The active role of co-evolving haloes in stellar bar formation

ABSTRACT We use idealized N-body simulations of equilibrium discs in live and static haloes to study how dark matter co-evolution impacts the assembly of stellar particles into a bar and the halo response. Initial conditions correspond to a marginally unstable disc according to commonly used disc stability criteria, and are evolved for the equivalent of about 150 disc dynamical times (10 Gyr). An extensive convergence study ensures accurate modelling of the bar formation process. Live haloes lead to the formation of a strong bar, but the same disc remains unbarred when evolved in a static halo. Neither seeded disc instabilities nor longer (60 Gyr) simulations result in the formation of a bar when the halo is static. When the live halo is replaced with a static analogue at later times, the previously robust bar slowly dissipates, suggesting that (1) the co-evolution of the disc and halo is critical for the assembly and long-term survival of bars in marginally unstable discs and (2) global disc stability criteria must be modified for discs in the presence of live haloes. In our live halo runs, a ‘dark bar’ grows synchronously with the stellar bar. Processes that inhibit the transfer of angular momentum between the halo and disc may stabilize a galaxy against bar formation, and can lead to the dissolution of the bar itself. This raises further questions about the puzzling stability of observed discs that are marginally unstable, but unbarred.

A particle swarm optimization inspired global and local stability driven predictive load balancing strategy

In distributed systems and parallel computing, optimal load balancing is difficult. These abstract addresses load balancing in distributed situations, highlighting current solutions' flaws and emphasizing the need for new ones. Load balancing research includes centralized and distributed algorithms, heuristics, and predictive models. Despite various successful methods, workload adaptability, overhead reduction, and scaling to large systems remain unresolved. This study proposes a particle swarm optimization (PSO) load balancing method that considers global and local stability considerations. The proposed method uses PSO principles to balance exploration and exploitation and allocate resources among distributed nodes. Predictive components improve preventative load management by predicting workload changes. Global and local load balancing stability criteria distinguish this study. The recommended method considers global system-wide performance indicators, local node-level characteristics, and micro-level stability to maximize system efficiency. A dual-focus technique distinguishes the proposed load balancing strategy from others, solving dynamic distributed system challenges. The study examines load balancing system advances and suggests improvements and further research. More accurate prediction modeling, stability measures, and application-specific enhancements may be studied in the future. Experimental validation and real-world implementation of the recommended approach are necessary to determine its practicality and ability to handle modern distributed computing systems.

Stability and Hopf-bifurcation analysis of diffusive Leslie–Gower prey–predator model with the Allee effect and carry-over effects

Identification of the various dynamical kinetics relating to prey–predator interconnection is the main goal of theoretical ecologists. In addition to direct killing (the lethal effect) in prey–predator interactions, a number of studies have suggested that there are some indirect effects (non-lethal), such as fear of predation. This non-lethal effect slows down the growth rate of prey and this phenomenon can be carried over to upcoming seasons or generations. The Allee effect also has an impact not only on prey populations but also on predator populations. Considering these, we first propose a modified Leslie–Gower predator–prey model that takes into account the fear-induced carry-over effect (COE), the predator’s Allee effect, and intraspecific competition among predators. The ecologically viable equilibrium points and corresponding local and global stability criteria are derived for the model. Moreover, we establish the conditions for the occurrence of Hopf-bifurcation around coexistence equilibrium and compute the first Lyapunov coefficient to verify the stability of the limit cycle. Secondly, we extend the proposed model to include the spatial variable by considering the spatial movements of the species. For the spatial model with zero flux boundary conditions, the criteria for Turing instability and Hopf-bifurcation are derived. Numerical simulations aim to verify the theoretical results. Our study suggests that the parameters related to fear-induced COE, the predator’s Allee effect, and intraspecific competition have a greater impact on system dynamics.

Non-separation Method-Based Global Stability Criteria for Takagi–Sugeno Fuzzy Quaternion-Valued BAM Delayed Neural Networks Using Quaternion-valued Auxiliary Function-Based Integral Inequality

This paper focuses on the global asymptotic stability (GAS) problem for Takagi–Sugeno (T-S) fuzzy quaternion-valued bidirectional associative memory neural networks (QVBAMNNs) with discrete, distributed and leakage delays by using non-separation method. By applying T-S fuzzy model, we first consider a general form of T-S fuzzy QVBAMNNs with time delays. Then, by constructing appropriate Lyapunov–Krasovskii functionals and employing quaternion-valued integral inequalities and homeomorphism theory, several delay-dependent sufficient conditions are obtained to guarantee the existence and GAS of the considered neural networks (NNs). In addition, these theoretical results are presented in the form of quaternion-valued linear matrix inequalities (LMIs), which can be verified numerically using the effective YALMIP toolbox in MATLAB. Finally, two numerical illustrations are presented along with their simulations to demonstrate the validity of the theoretical analysis.

Complex Dynamics and Fractional-Order Optimal Control of an Epidemic Model with Saturated Treatment and Incidence

In this study, we have developed a novel SIR epidemic model by incorporating fractional-order differential equations and utilizing saturated-type functions to describe both disease incidence and treatment. The intricate dynamical characteristics of the proposed model, encompassing the determination of the conditions for the existence of all possible feasible equilibria with their local and global stability criteria, are investigated thoroughly. The model undergoes backward bifurcation with respect to the parameter representing the side effects due to treatment. This phenomenon emphasizes the critical role of treatment control parameters in shaping epidemic outcomes. In addition, to understand the optimal role of the treatment in mitigating the disease prevalence and minimizing the associated cost, we investigated a fractional-order optimal control problem. To further visualize the analytical results, we have conducted simulation works considering feasible parameter values for the model. Finally, we have employed local and global sensitivity analysis techniques to identify the factors that have the greatest potential to reduce the impact of the disease.

A sampled-data stabilisation analysis on memristive BAM neural networks for memductance function

In this paper, we explore a stabilisation problem for two different class of memductance functions for memristive bidirectional associative memory neural networks (MBAMNNs) with time-varying delay. A new Lyapunov-Krasovskii function (LKF) with double and triple integral terms is derived with some sufficient conditions to guarantee the global asymptotic stability criteria using the Lyapunov stability theory. The controller gain matrix for sampled-data controller is obtained for the state-dependent switched system and state-dependent continuous system in terms of linear matrix inequalities (LMIs). Finally, we stabilise the considered model by visualising the simulation part using standard MATLAB software which demonstrates the validness of the theoretical results.

Global robust exponential stability of interval BAM neural networks with multiple time-varying delays: A direct method based on system solutions

This paper analyzes global robust exponential stability of interval bidirectional associative memory (BAM) neural networks with multiple time-varying delays, proposes a direct method based on system solutions, and gives sufficient conditions under which interval BAM neural networks have a unique and globally robustly exponentially stable equilibrium point. This method not only avoids the difficult to set up any Lyapunov–Krasovskii functional, but also derives simpler global robust exponential stability criteria. Compared with the data from other literature, the robust exponential stability criteria obtained in this paper have been presented to have more merits theoretically and numerically.

Qualitative analysis of TB transmission dynamics considering both the age since latency and relapse

Since the beginning of time, tuberculosis (TB) has been a fatal illness that predominantly affects the human lungs before spreading to other organs including the brain, spine, etc. The main elements of TB mitigation are age-dependent heterogeneity, identifying those who are latently infected, and treating them using the right diagnostic strategy. In this present work, the complex transmission mechanism of this disease in a population is described by a coupled system of integro-partial differential equations (IDE-PDE). The system’s well-posedness requirement is confirmed. The proposed system’s basic reproduction number (R0) is obtained. This work provides a complete analysis of the qualitative properties of the model, including steady state existence, asymptotic smoothness of the solution semi-flow, uniform persistence of the endemic equilibrium, and the global asymptotic stability criterion. It is observed that in assessing the severity of the pandemic, the value of R0 is crucial. Additionally, the stability results are visually illustrated by solving the model equations numerically while assuming two hypothetical cases. The current work also suggests several methods for reducing the value of the basic reproductive number (R0) by manipulating a few parameter values, which may help to lessen the prevalence of TB in a community.

Overflow Oscillation Elimination in Fixed-Point 2D Digital Filters Based on the Roesser Model

This paper investigates the overflow oscillation elimination problem in the fixed-point two-dimensional (2D) digital filter (DF) based on Roesser model subjected to two’s complement overflow (TCO) nonlinearities. By utilizing the system information and behavior of TCO nonlinearities more effectively, a new global asymptotic stability (GAS) criterion for 2D DFs is presented. A comparison of the obtained criterion with existing criteria is made with the help of examples.

A tumor–immune interaction model with the effect of impulse therapy

A mathematical model describing the growth of tumor cells and the effect of impulsive therapy is studied in this paper. The governing equations and parameters related to the proposed model are based on the experimental and clinical outcome. Employing the quasi-steady-state approximation in delineating the dynamics of tumor–immune under different cytokines, a four-dimensional deterministic system is obtained. We explore the steady states, local and global stability criteria, transcritical bifurcation and boundary separation for the given model. The tumor cells are characterized by their growth rate and the tumor control region is determined in the parametric space. An impulsive differential equation is considered to determine the interval for perfect dosing of dendritic cell vaccination to boost the immune response. The effect of dendritic cell vaccination is determined by taking into account both perfect and imperfect therapeutics and the optimal number for therapeutics is computed. Our model demonstrates that the dendritic cell vaccination boosts the immune system and eradicates the tumor cell population.

Existence and stability of a periodic solution of a general difference equation with applications to neural networks with a delay in the leakage terms

In this paper, a new global exponential stability criterion is obtained for a general multidimensional delay difference equation using induction arguments. In the cases that the difference equation is periodic, we prove the existence of a periodic solution by constructing a type of Poincaré map. The results are used to obtain stability criteria for a general discrete-time neural network model with a delay in the leakage terms. As particular cases, we obtain new stability criteria for neural network models recently studied in the literature, in particular for low-order and high-order Hopfield and Bidirectional Associative Memory (BAM).

Input-to-state stability of positive delayed neural networks via impulsive control

This paper is concerned with the positivity and impulsive stabilization of equilibrium points of delayed neural networks (DNNs) subject to bounded disturbances. With the aid of the continuous dependence theorem for impulsive delay differential equations, a relaxed positivity condition is derived, which allows the neuron interconnection matrix to be Metzler if the activation functions satisfy a certain condition. The notion of input-to-state stability (ISS) is introduced to characterize internal global stability and disturbance attenuation performance for impulsively controlled DNNs. The ISS property is analyzed by employing a time-dependent max-separable Lyapunov function which is able to capture the positivity characterization and hybrid structure of the considered DNNs. A ranged dwell-time-dependent ISS condition is obtained, which allows to design an impulsive control law via partial state variables. As a byproduct, an improved global exponential stability criterion for impulse-free positive DNNs is obtained. The applicability of the achieved results is illustrated through three numerical examples.

Global Stabilization of Delayed Feedback Financial System Involved in Advertisement under Impulsive Disturbance

Diffusion is an inevitable important factor in advertising dynamic systems. However, previous literature did not involve this important diffusion factor, and only involved the local stability of the advertising model. This paper develops a global stability criterion for the impulsive advertising dynamic model with a feedback term under the influence of diffusion. Since global stability requires the unique existence of equilibrium points, variational methods are employed to solve it in the infinite dimensional function space, and then a global stability criterion of the system is derived by way of the impulse inequality lemma and orthogonal decomposition of a class of Sobolev spaces. Numerical simulations verify the effectiveness of the proposed method.

A high-order compact difference method and its Richardson extrapolation for semi-linear reaction–diffusion equations with piecewise continuous argument in diffusion term

In this paper, for the initial–boundary value problems of semi-linear reaction–diffusion equations with piecewise continuous argument in spatial derivative, we suggest Crank–Nicolson method, high-order compact difference (HOCD) method and HOCD-based Richardson extrapolation (RHOCD) method. Under the appropriate conditions, it is proved that HOCD (resp. RHOCD) method has the computational accuracy O(τ2+h4) (resp. O(τ4+h4)). This shows that RHOCD method improves the calculation accuracy of HOCD method in temporal direction. Moreover, we also analyze the stability of HOCD method and thus derive a global stability criterion of this method. Finally, with a series of numerical experiments, we further confirm the computational effectiveness and theoretical accuracy of the concerned methods.

Stability Criteria and Λ-Fractional Mechanics

Global stability criteria for Λ-fractional Mechanics are established. The fractional extension of a bar under axial loading is discussed. Globally minimizing the total energy function, non-smooth deformations are introduced. The co-existence of phases phenomenon is established in Λ-fractional elasticity. It is pointed out that global minimization only is valid in fractional analysis.

A general framework for robust stability analysis of neural networks with discrete time delays

Robust stability of different types of dynamical neural network models including time delay parameters have been extensively studied, and many different sets of sufficient conditions ensuring robust stability of these types of dynamical neural network models have been presented in past decades. In conducting stability analysis of dynamical neural systems, some basic properties of the employed activation functions and the forms of delay terms included in the mathematical representations of dynamical neural networks are of crucial importance in obtaining global stability criteria for dynamical neural systems. Therefore, this research article will examine a class of neural networks expressed by a mathematical model that involves the discrete time delay terms, the Lipschitz activation functions and possesses the intervalized parameter uncertainties. This paper will first present a new and alternative upper bound value of the second norm of the class of interval matrices, which will have an important impact on obtaining the desired results for establishing robust stability of these neural network models. Then, by exploiting wellknown Homeomorphism mapping theory and basic Lyapunov stability theory, we will state a new general framework for determining some novel robust stability conditions for dynamical neural networks possessing discrete time delay terms. This paper will also make a comprehensive review of some previously published robust stability results and show that the existing robust stability results can be easily derived from the results given in this paper.

Mathematical modelling of COVID-19 pandemic in Pakistan with optimal control

We propose an innovative mathematical modeling to examine the previous coronavirus disease of 2019 pandemic or (COVID-19). This analysis has been performed qualitatively through differential equation stability theory, as well as the basic reproductive value, which indicates a pandemic index, then calculated from the maximum eigenvalue of the subsequent matrix. We establish the global asymptotic stability criteria for such disease-free state. The actual COVID-19 occurrence documents and data from 01 July, 2021 to 14 August, 2022 in Pakistan are analyzed for estimation methods, leading in fitted values for biological parameters. Sensitivity analyzation is used to identify the much more relevant attributes in the developed framework. This scientific research revealed a deterministic computational formula that assesses the impact of various mitigation measures on the propagation of COVID-19 in a worldwide population. The analysis will concentrate on Pakistan, and relevant data gathered out of that region. If the method were modified for the total number of COVID-19 reported patients as well as the total number of active patients in the Pakistan region, infection rates would be approximated fairly. This research will in some way give government authorities including local hospitals additional knowledge about how to enhance precautionary efforts to minimize infection propagation.

Novel Global Asymptotic Stability Conditions for Discrete-Time Systems With Time-Varying Delay and Generalized Overflow Arithmetic

Novel global asymptotic stability (GAS) criterion for discrete-time (DT) systems with generalized overflow arithmetic and time-varying delay is presented in this brief. The criterion is based on two unique lemmas that allow for a more precise characterization of nonlinearities by utilizing system information in greater detail. The approach leads to significantly relaxed stability conditions as compared to a recently reported criterion.

On the Control over the Distribution of Ticks Based on the Extensions of the KISS Model

Ticks and tick-borne diseases present a well-known threat to the health of people in many parts of the globe. The scientific literature devoted both to field observations and to modeling the propagation of ticks continues to grow. To date, the majority of the mathematical studies have been devoted to models based on ordinary differential equations, where spatial variability was taken into account by a discrete parameter. Only a few papers use spatially nontrivial diffusion models, and they are devoted mostly to spatially homogeneous equilibria. Here we develop diffusion models for the propagation of ticks stressing spatial heterogeneity. This allows us to assess the sizes of control zones that can be created (using various available techniques) to produce a patchy territory, on which ticks will be eventually eradicated. Using averaged parameters taken from various field observations we apply our theoretical results to the concrete cases of the lone star ticks of North America and of the taiga ticks of Russia. From the mathematical point of view, we give criteria for global stability of the vanishing solution to certain spatially heterogeneous birth and death processes with diffusion.

Stability Verification for Heterogeneous Complex Networks via Iterative SOS Programming

This letter is to investigate the stability verification for heterogeneous polynomial complex networks through iterative sum-of-squares programming approach. With polynomial Lyapunov functions, a global asymptotic stability criterion is established for the heterogeneous complex networks under the directed topology. Based on the proposed criterion, the stability verification problem is then reduced to a sum-of-squares optimization problem for solving polynomial matrix inequalities. To process non-convex terms of the polynomial matrix inequalities, we present an iterative sum-of-squares programming approach to turn them into the convex polynomial matrix inequalities, yielding polynomial Lyapunov functions effectively to realize the stability verification. Finally, a numerical example is given to show the fully automatic verification of global asymptotic stability for the heterogeneous polynomial complex networks, which demonstrates the theoretical and algorithmic advantages of the proposed method.