Local Stability Analysis Research Articles

Tornado-induced instability assessment of super large hyperbolic reinforced concrete cooling towers

Cooling towers are typical high-rise thin-walled structures which are of hyperbolic rotating shell structure type and significantly sensitive to wind loads. Wind-related local instability is the key control factor in the design of structural geometric shape and tower shell thickness optimization. In fact, this kind of stability analysis has only conducted aiming at normal climate (such as monsoon, etc.) under the guideline of series of national Codes. In order to deal with the issues under disaster climate, physical and numerical investigation to evaluate local stability performance of a super large cooling tower (SLCT) under tornado environments based on Buckling Stress State (BSS) method is implemented. Besides, whole process elastic–plastic collapse simulation of the SLCT exposed to the tornados was also carried out to study the failure mechanism and re-evaluated the reasonability of the well-accepted BSS method for structural wind-resistant design. The results revealed that the tornados would have obvious adverse effects on the structural local stability when the SLCT is located at the tornado vortex core radius compared with those under traditional synoptic winds. The swirl ratios and the central distances between the SLCT and the tornado vortex core have significant effects on the local instability of the SLCT. The extended elastic–plastic collapse simulation discovered that the position of the first crack during SLCT collapse process is basically consistent with the critical position from the elastic local stability analysis of the SLCT, indicating that strengthening measures can be carried out in advance at the critical position for structural local stability improvement under tornado conditions. Besides, the extended elastic–plastic collapse numerical simulation of the SLCT could be as the effective supplement to the structural elastic stability assessment (such as BSS method) in the wind-resistant design of the SLCT.

Impact of Yoga Awareness on Transmission Dynamics of Communicable Diseases: SIR Model Analysis

Communicable diseases are major health problems that affect the whole economy of the na-tion. So it becomes the prime agenda of developed and developing countries to educate peo-ple about disease dynamics and control strategies. Due to changes in people’s living styles, disease treatment modality has been changed. So, we have introduced Yoga awareness as control strategy which includes Ahar, Vihar, Achar and Vichar. They are means for achiev-ing the physical, mental, social and spiritual well-being. Our main aim is to model the role of Yoga awareness in controlling the disease dynamics. It maintains the physical fitness of practitioners and improve the whole metabolism system of the human body. In this work, the previous SIR model is modified by incorporating a new awareness transmission rate β 1 to the existing model. Real life situation data of Yoga aware and aware infected individuals were col-lected from different Yoga centers of Sudurpashchim province, Nepal and analyzed by using mathematical techniques. Stability analysis of governing by ordinary differential equations showed that model is stable. Yoga awareness reproduction number Ra was calculated by next-generation matrix method. Sensitivity analysis of Ra and concerning parameters indicate that Ra decreases with an increase in Yoga awareness coverage level. Also, the recovery rate has the opposite relation with Ra which indicates that the recovery period also decreases with an increase in awareness coverage. Local and global stability analysis showed that disease-free equilibrium exists when Ra < 1 and endemic equilibrium exists when Ra > 1. Numerical simulations also support the analytical results and suggest that Yoga awareness has a positive influence on controlling disease d ynamics. The increase in the coverage of awareness leads to reduced susceptibility and infectivity. So propagation of disease can be controlled by Yoga awareness.

Stability of Stuart vortices in rotating stratified fluids

The linear stability of the Stuart vortices, which is a model of arrays of vortices often observed in the atmosphere and the oceans, in rotating stratified fluids is investigated by local and modal stability analysis. As in the case of the two-dimensional (2-D) Taylor–Green vortices, five types of instability appear in general: the pure-hyperbolic instability, the strato-hyperbolic instability, the rotational-hyperbolic instability, the centrifugal instability and the elliptic instability. The condition for each instability and the estimate of the growth rate derived by Hattori & Hirota (J. Fluid Mech., vol. 967, 2023, A32) are shown to also be useful for the Stuart vortices, which supports their applicability to general flows. The properties of each instability depend on stratification and rotation in a way similar to the case of the 2-D Taylor–Green vortices. For the Stuart vortices, however, the centrifugal instability and the elliptic instability become more dominant than the three hyperbolic instabilities in comparison to the 2-D Taylor–Green vortices; this is explained by the larger ratios of the maximum vorticity and the strain rate at the elliptic stagnation points to the strain rate at the hyperbolic stagnation points. Direct correspondence between the modal and local stability results is further established by comparing unstable modes to solutions to the local stability equations; this is useful for identifying the types of modes since the mechanism of instability is readily known in the local stability analysis. This helps us to discover the modes of the ring-type elliptic instability, which have been predicted only theoretically.

Wiener and Lévy processes to prevent disease outbreaks: Predictable vs stochastic analysis

The study considers stochastic behavior along with modeling, mathematical analysis, theory development, and numerical simulation of the COVID-19 virus. To evaluate current trends and make future projections regarding the basic reproduction number and infection case, we have taken the modified five-compartment SEIRD mathematical model. Since the basic reproduction number R0 is not sufficient to predict the outbreak, we applied the Itô stochastic differential equations (SDEs) along with the Weiner process and Lévy jump to investigate the nature of the disease outbreak. Since COVID-19 is an infectious disease caused by the severe acute respiratory syndrome coronavirus 2 “(SARS-CoV-2)”, that has common symptoms including fever, cough, and difficulty breathing, we have illustrated the boundedness and positivity of solutions and investigated the stability of endemic and disease-free equilibrium points. The findings show that the dynamics of disease epidemics are influenced by contact patterns. For all the models, the threshold quantity, R0 is calculated which is the key reason to prove the global and local stability analysis of disease-free equilibrium and endemic equilibrium points. Using the least squares method for data fitting, we have performed a case study on COVID-19 data in Italy for this article. A sensitivity analysis is part of our investigation to find important factors. This paper investigates a stochastic epidemic model (SDE) using Lévy jumps and Weiner processes. Local and global stability in a stochastic setting are examined in the analysis. To view the analytical study, a multitude of numerical results are achieved. The study is important to comprehend the stochastic behavior of the virus in the contagious model in order to prevent similar disease outbreaks in the future.

Sediment DNA Records the Critical Transition of Bacterial Communities in the Arid Lake

It is necessary to predict the critical transition of lake ecosystems due to their abrupt, non-linear effects on social-economic systems. Given the promising application of paleolimnological archives to tracking the historical changes of lake ecosystems, it is speculated that they can also record the lake’s critical transition. We studied Lake Dali-Nor in the arid region of Inner Mongolia because of the profound shrinking the lake experienced between the 1300 s and the 1600 s. We reconstructed the succession of bacterial communities from a 140-cm-long sediment core at 4-cm intervals and detected the critical transition. Our results showed that the historical trajectory of bacterial communities from the 1200 s to the 2010s was divided into two alternative states: state1 from 1200 to 1300 s and state2 from 1400 to 2010s. Furthermore, in the late 1300 s, the appearance of a tipping point and critical slowing down implied the existence of a critical transition. By using a multi-decadal time series from the sedimentary core, with general Lotka-Volterra model simulations, local stability analysis found that bacterial communities were the most unstable as they approached the critical transition, suggesting that the collapse of stability triggers the community shift from an equilibrium state to another state. Furthermore, the most unstable community harbored the strongest antagonistic and mutualistic interactions, which may imply the detrimental role of interaction strength on community stability. Collectively, our study showed that sediment DNA can be used to detect the critical transition of lake ecosystems.

Spatiotemporal dynamics of prey-predator model incorporating Holling-type II functional response with fear and its carryover effects.

The recent focus in the fields of biology and ecology has centered on the significant attention given to the mathematical modeling and analyzing the spatiotemporal population distribution among species engaged in interactions. This paper explores the dynamics of the temporal and spatiotemporal delayed Bazykin-type prey-predator model, incorporating fear and its carryover effect. In our model, we incorporated a functional response of the Holling-type II. In the temporal model, a detailed dynamic analysis was carried out, investigating the positivity and boundedness of solutions, establishing the uniqueness and existence of positive interior equilibria, and examining both local and global stability. Additionally, we explored the presence of saddle-node, transcritical, and Hopf bifurcations varying attack rate parameter. The delayed system shows highly periodic behavior. Additionally, for the spatiotemporal model, we provide a complete analysis of local and global stability, and we derive the conditions for the existence of Turing instability for both self-diffusion and cross-diffusion, respectively. The two-dimensional diffusive model is further discussed, highlighting various Turing patterns, including holes, stripes, and hot and cold spots, along with their biological significance. Numerical simulations are executed to validate the analytical findings in both temporal and spatiotemporal models.

Delay-induced nutrient recycling in plankton system: Application to Sundarban mangrove wetland

The paper discusses the nutrient–plankton system with effect of time delay in nutrient recycling and toxin determined function response (TDFR). The designed model system explores the delay-induced system dynamics. We present the local stability analysis of interior equilibrium points in absence as well as in presence of time delay. Further, the direction of Hopf bifurcation is obtained. We perform the numerical computation and observe that time delay in nutrient recycling can generate the periodic solution in a stable nutrient–plankton system. Some other essential parameters, such as input concentration of nutrients and natural removal rate of nutrients, also regulate the dynamical system. The system shows Hopf and double-Hopf bifurcation in the presence of time delay. Our study shows that the delay in the nutrient recycling causes instability transition phenomenon. The delay-induced nutrient recycling and different input concentrations of nutrients can regulate the estuarine system. Finally, the stability switching is observed for delayed system.

Analisis Kestabilan Lokal Model Transmisi Demam Berdarah Dengue

Dengue fever transmission in Indonesia has an advanced amount. In this article, dynamic model of interaction between human and Aedes aegypti mosquitos is learned. The SEIRRD (Susceptible, Exposed, Infected, Recovered, Deceased) model is used in this article. The prurpose in this model is to describe the stability of dengue transmission, so that we can analyze the developed of epidemic model in mathemtic field. Using NGM method to analyze basic reproduction number and applying Routh-Hurwitz criteria method to show the local stability of model. Then, two equilibrium points, called endemic and non-endemic equilibrium points, are obtained. The result of basic reproduction number is described the stability analysis. If basic reproduction number less then one, the endemic equilibrium point is locally asymptotically stable and otherwise. Local stability analysis at the equilibrium point is determined through parameter analysis. Furthermore, numerical simulations are carried out by fitting the data to obtaine the result of the parameters. The results of numerical simulations explaine the spread of dengue transmission Keywords: Dynamic Model, Epidemic Model, Equilibrium Point, Local Stability, Routh Hurwitz

Controlling flat-foot limit cycle walkers with compliant joints based on local stability variation.

This study investigates local stability of a four-link limit cycle walking biped with flat feet and compliant ankle joints. Local stability represents the behavior along the solution trajectory between Poincare sections, which can provide detailed information about the evolution of disturbances. The effects of ankle stiffness and foot structure on local stability are studied. In addition, we apply a control strategy based on local stability analysis to the limit cycle walker. Control is applied only in the phases with poor local stability. Simulation results show that the energy consumption is reduced without sacrificing disturbance rejection ability. This study may be helpful in motion control of limit cycle bipedal walking robots with flat feet and ankle stiffness and understanding of human walking principles.

Bifurcations and complex dynamics in a banking duopoly model with macroprudential policy

We consider a banking duopoly model with a macroprudential policy in Indonesia called loan-to-deposit ratio-based reserve requirement (LDR-RR). The objective of the policy is to control the banking loans growth using a LDR-based penalty scheme that requires banks to save more money in the central bank to maintain their liquidity ratio. Following recent studies on banking models, we analyze a piecewise discrete-time model with two banks. We assume that the dynamics of the deposits follows the discrete logistic growth. Moreover, our model has two borders, hence we examine the resulting border-collision bifurcations. From the local stability analysis we find that, according to the parameter values, only the border-collision bifurcation or the flip bifurcation occurs. Finally, we perform several numerical simulations to confirm the stability analysis’ results. Our discrete dynamical system offers various possibilities of development for future research perspectives.

Modelling the Transmission Dynamics of Anthrax Disease with Simultaneous Vaccination for both Human and Animal Population Using the Saturated Incidence Rate

In this study, a mathematical model based on saturated incidence rate that explores the transmission dynamics of anthrax as a zoonotic disease with vaccination compartment introduced into the human and animal population was formulated using ordinary differential equations. The model’s flow diagram was presented with qualitative and quantitative analysis carried out on the model. The local and global stability analysis of the equilibrium points investigated using the Ruth-Hurwitz criterion and Castillo-Chaves method respectively were found to be locally asymptotically stable if the basic reproductive number is less than one and unstable if the basic reproductive number is greater than one. Test for backward bifurcation was done with the result showing no backward bifurcation in the system. The sensitivity analysis of the model's parameters was performed to determine the contribution of each parameter to the basic reproduction number and a plot of partial ranking correlation (PRCC) obtained. The analysis revealed that, by decreasing human and animal contact rate, it would cause a decrease in the basic reproduction number. Also, numerical simulation further showed that increase in vaccination for both human and animal population reduces the spread of anthrax disease in the population. Anthrax disease, Bifurcation, Mathematical modeling, Sensitivity index, Stability

Role of Allee and Fear for Controlling Chaos in a Predator–Prey System with Circulation of Disease in Predator

This paper explores a predator–prey system featuring fear and disease within the predator population,utilizing the Rosenzweig–MacArthur model with Holling type-II functional response. The primary focus lies in investigating the impact of a fear factor, wherein the prey’s growth rate is hindered due to predator-induced fear. Additionally, the model accounts for the spread of disease among predators,leading to a division between susceptible and infected predator subpopulations. The inclusion of an Allee effect in the susceptible predator further enriches the model. The study involves a thorough examination, encompassing local and global stability analysis as well as Hopf bifurcation analysis around the interior equilibrium point. Numerical simulations underscore a noteworthy observation: an escalation in interaction force propels the system into chaotic dynamics,marked by stable focus, limit cycles and period-doubling phenomena. A noteworthy finding pertains to the influence of the Allee parameter ([Formula: see text]) on chaotic dynamics. As the Allee parameter values increase, the system tends to stable focus through a sequence of chaotic states, period-doubling and limit cycles. Subsequently, the paper introduces the role of another pivotal parameter, the fear factor, into the chaotic dynamics. Intriguingly, chaos transforms into stable focus through diverse nonlinear phenomena, including period-doubling and limit cycles. This nuanced exploration of parameters sheds light on the intricate dynamics governing the predator–prey system, offering a comprehensive understanding of the interplay between fear, disease and ecological factors. So our observation throughout this paper that how chaos behaves here after one by one injection of our new features: fear factor and Allee parameter?

On the Local and String Stability Analysis of Traffic Collision Risk

Conventional traffic stability studies primarily concentrate on the evolution of disturbances in vehicle motion but seldom consider how collision risk changes spatially and temporally. This study bridges the gap by extending the principles of traffic stability analysis to the field of traffic safety, focusing specifically on the temporal and spatial dynamics of collision risk. Leveraging the concepts of local and string stability, we formulate conditions under which collision risk behaves in a stable manner over time and space through the transfer function approach. A comparative analysis between conventional traffic stability and the newly introduced concept of collision risk stability reveals that while conditions for local stability are largely aligned in both domains, the criteria for string stability differ. These theoretical insights are substantiated through microscopic simulations using a variety of car-following models. The simulations also indicate that the consistency between theoretical and simulation outcomes diminishes as the disturbance magnitude increases, which is attributed to the linearization errors inherent in applying the transfer function in the theoretical derivations.

The seizure classification of focal epilepsy based on the network motif analysis

Due to the complexity of focal epilepsy and its risk for transiting to the generalized epilepsy, the development of reliable classification methods to accurately predict and classify focal and generalized seizures is critical for the clinical management of patients with epilepsy. In order to holistically understand the seizure propagation behavior of focal epilepsy, we propose a three-node motif reduced network by respectively simplifying the focal region, surrounding healthy region and their critical regions as the single node. Because three-node motif can richly characterize information evolutions, the motif analysis method could comprehensively investigate the seizure behavior of focal epilepsy. Firstly, we define a new seizure propagation marker value to capture the seizure onsets and intensity. Based on the three-node motif analysis, it is shown that the focal seizure and spreading can be categorized as inhibitory seizure, focal seizure, focal-critical seizure and generalized seizures, respectively. The four types of seizures correspond to specific modal types respectively, reflecting the strong correlation between seizure behavior and information flow evolution. In addition, it is found that the intensity difference of outflow and inflow information from the critical node (connection heterogeneity) and the excitability of the critical node significantly affected the distribution and transition of the four seizure types. In particular, the method of local linear stability analysis also verifies the effectiveness of four types of seizures classification. In sum, this paper computationally confirms the complex dynamic behavior of focal seizures, and the study of criticality is helpful to propose novel seizure control strategies.

Coexistence of different mechanisms underlying the dynamics of supersonic turbulent flow over a compression ramp

Supersonic turbulent flow over a compression ramp is studied using wall-resolved large eddy simulation with a freestream Mach number of 2.95 and a Reynolds number [based on δ0: the thickness of incoming turbulent boundary layer (TBL)] of 63 560. The unsteady dynamics of the present shock wave/turbulent boundary layer interaction (STBLI) flow are investigated by using dynamic mode decomposition techniques, linear and nonlinear disambiguation optimization, local stability analysis (LSA), and global stability analysis (GSA). By analyzing the dynamic system for the STBLI flow, three dynamically important modes with characteristic spanwise wavelengths of 2δ0, 3δ0, and 6δ0 are captured. The 2δ0 mode approximates the spanwise scale of the Görtler-like vortices and Görtler mode of LSA, suggesting the presence of Görtler instability, which is believed to be related to the unsteady motion of streaks downstream of reattachment in the flow. The features of the 3δ0 mode are also observed in large-scale motions of the incoming TBL, implying the existence of a convective mechanism that is excited and maintained by such motions. Additionally, the GSA results show the most unstable mode features a spanwise wavelength of around 6δ0, indicating the existence of global instability that is believed to be related to the oscillating motion of separation shock. The coexistence of these three mechanisms is confirmed. Discussions on the above findings provide an interpretation for low-frequency unsteadiness that the unsteadiness of surface streaks results from the combined effects of the Görtler instability near flow reattachment and the convection of large-scale motions in the incoming boundary layer, while the low-frequency shock motion may be related to a global mode driven by upstream disturbances.

Saturated lysing efficiency of CD8<sup>+</sup> cells induced monostable, bistable and oscillatory HIV kinetics

<p>Effector CD8<sup>+</sup> cells lyse human immunodeficiency viruses (HIV)-infected CD4<sup>+</sup> cells by recognizing a viral peptide presented by human leukocyte antigens (HLA) on the CD4<sup>+</sup> cell surface, which plays an irreplaceable role in within-host HIV clearance. Using a semi-saturated lysing efficiency of a CD8<sup>+</sup> cell, we discuss a model that captures HIV dynamics with different magnitudes of lysing rate induced by different HLA alleles. With the aid of local stability analysis and bifurcation plots, exponential interactions among CD4<sup>+</sup> cells, HIV, and CD8<sup>+</sup> cells were investigated. The system exhibited unexpectedly complex behaviors that were both mathematically and biologically interesting, for example monostability, periodic oscillations, and bistability. The CD8<sup>+</sup> cell lysing rate, the CD8<sup>+</sup> cell count, and the saturation effect were combined to determine the HIV kinetics. For a given CD8<sup>+</sup> cell count, a low CD8<sup>+</sup> cell lysing rate and a high saturation effect led to monostability to a high viral titre, and a low CD8<sup>+</sup> cell lysing rate and a low saturation effect triggered periodic oscillations; this explained why patients with a non-protective HLA allele were always associated with a high viral titer and exhibited bad infection control. On the other hand, a high CD8<sup>+</sup> cell lysing rate led to bistability and monostability to a low viral titer; this explained why protective HLA alleles are not always associated with good HIV infection outcomes. These mathematical results explain how differences in HLA alleles determine the variability in viral infection.</p>

Mathematical Model on Distributed Denial of Service Attack in the Computer Network

In this paper, an electronic- epidemic two-folded mathematical model is formulated with help of non-linear ordinary differential equations. Distributed Denial of Service (DDoS) attacks in the computer network are studied. The modeling of both attacking nodes and targeting nodes is performed. Botnet based malicious devices and their threats on computer networks are addressed using appropriate parameters. The basic reproduction numbers for both the attacking and the targeting population are calculated and interpreted. Local and global stability analysis is carried out for the infection-free and endemic equilibrium points. Differential equations are solved with the help of the Runge-Kutta 4th order numerical method and graphs are analyzed using MATLAB software. Simulation shows that the success or failure depends on the number of initially infected computers in the attacking group. The proposed model exhibits the phenomenon of backward bifurcation for different values of transmission parameters. This model gives the theoretical base for controlling and predicting the DDoS attack. This shows the way to minimize the attack in the network. This study will be helpful to identify the botnet devices and run the latest version of antivirus in the network to protect against DDoS attacks from attacking sources. The application of this study is to ascertain online crime and locate the attacking nodes in the field of online transactions of real-life problems that involve the internet and computer networking systems. Moreover, our model can play an important role in policy-making against the distributed attack.

Numerical Solution for the Heat Equation Using Crank-Nicolson Difference Method

Abstract: In this paper, the Crank-Nicolson difference method is applied to a simple problem involving one dimensional heat equation. Numerical solution to the heat equation using Crank-Nicolson difference equation is obtained. The method of solving the problem is implemented by using Python Programming. We have discussed the local truncation error and stability analysis of Crank-Nicolson difference method difference method of heat equation.

A constant proportional caputo operator for modeling childhood disease epidemics

We propose a constant proportional-Caputo (CPC) operator for modeling childhood disease epidemics by treating positivity, boundedness, well-posedness, and biological viability and checking the reproductive and strength numbers for the biological system. For local and global stability analysis, we used the Volterra-type Lyapunov function with the first and second derivatives, providing symmetric results for fractional-order derivatives. We used different techniques to invert the proportional-Caputo (PC) and constant proportional-Caputo operators to evaluate the fractional integral operator. We also used our suggested model’s fractional differential equations to derive the eigenfunctions of the CPC operator. Modern mathematical control with the fractional operator has many applications, including the complex and crucial study of systems with symmetry. Symmetry analysis is a powerful tool that enables the user to construct numerical solutions to a given fractional differential equation fairly systematically. Therefore, we construct the comparative results with the Caputo derivative and constant proportional-Caputo operator to understand the dynamic behavior of the disease. We also present an additional analysis of the constant proportional-Caputo and Hilfer generalized proportional operators. Using the Laplace Adomian Decomposition Method, we numerically simulate a system of fractional differential equations through tables and graphs.

Roles of astrocytes and prions in Alzheimer's disease: insights from mathematical modeling.

We present a mathematical model that explores the progression of Alzheimer's disease, with a particular focus on the involvement of disease-related proteins and astrocytes. Our model consists of a coupled system of differential equations that delineates the dynamics of amyloid beta plaques, amyloid beta protein, tau protein, and astrocytes. Amyloid beta plaques can be considered fibrils that depend on both the plaque size and time. We change our mathematical model to a temporal system by applying an integration operation with respect to the plaque size. Theoretical analysis including existence, uniqueness, positivity, and boundedness is performed in our model. We extend our mathematical model by adding two populations, namely prion protein and amyloid beta-prion complex. We characterize the system dynamics by locating biologically feasible steady states and their local stability analysis for both models. The characterization of the proposed model can help inform in advancing our understanding of the development of Alzheimer's disease as well as its complicated dynamics. We investigate the global stability analysis around the interior equilibrium point by constructing a suitable Lyapunov function. We validate our theoretical analysis with the aid of extensive numerical illustrations.