Local Stability Behavior Research Articles

The impact of radio-chemotherapy on tumour cells interaction with optimal control and sensitivity analysis

Oncologists and applied mathematicians are interested in understanding the dynamics of cancer-immune interactions, mainly due to the unpredictable nature of tumour cell proliferation. In this regard, mathematical modelling offers a promising approach to comprehend this potentially harmful aspect of cancer biology. This paper presents a novel dynamical model that incorporates the interactions between tumour cells, healthy tissue cells, and immune-stimulated cells when subjected to simultaneous chemotherapy and radiotherapy for treatment. We analysed the equilibria and investigated their local stability behaviour. We also study transcritical, saddle–node, and Hopf bifurcations analytically and numerically. We derive the stability and direction conditions for periodic solutions. We identify conditions that lead to chaotic dynamics and rigorously demonstrate the existence of chaos. Furthermore, we formulated an optimal control problem that describes the dynamics of tumour-immune interactions, considering treatments such as radiotherapy and chemotherapy as control parameters. Our goal is to utilize optimal control theory to reduce the cost of radiotherapy and chemotherapy, minimize the harmful effects of medications on the body, and mitigate the burden of cancer cells by maintaining a sufficient population of healthy cells. Cost-effectiveness analysis is employed to identify the most economical strategy for reducing the disease burden. Additionally, we conduct a Latin hypercube sampling-based uncertainty analysis to observe the impact of parameter uncertainties on tumour growth, followed by a sensitivity analysis. Numerical simulations are presented to elucidate how dynamic behaviour of model is influenced by changes in system parameters. The numerical results validate the analytical findings and illustrate that a multi-therapeutic treatment plan can effectively reduce tumour burden within a given time frame of therapeutic intervention.

Resonant Actuation Based on Dynamic Characteristics of Bistable Laminates

Bistable or multi-stable structures have found broad applications in the fields of adaptive structures, flow control, and energy harvesting devices due to their unique nonlinear characteristics and strong local stability behavior. In this paper, a theoretical model based on the principle of minimum potential energy and the Rayleigh–Ritz method is established to study the dynamic characteristics of a bistable unsymmetric laminate with a fixed center. Numerical results of this theoretical model were obtained and verified by an FEA model using ABAQUS. The nonlinear dynamic characteristics and the structural response under different levels of external excitation were investigated and verified by experiments. The realization conditions of single-well vibration and cross-well vibration of bistable laminates were determined, with which the actuation strategies can be optimized for targeting modal frequencies of bistable laminates.

Effects of state‐of‐the‐art residual stress models on the member and local stability behaviour

StahlbauVolume 92, Issue 1 p. 38-38 Empfehlungen der Redaktion Effects of state-of-the-art residual stress models on the member and local stability behaviour First published: 03 January 2023 https://doi.org/10.1002/stab.202370105Read the full textAboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat No abstract is available for this article. Volume92, Issue1January 2023Pages 38-38 RelatedInformation

Dynamics of a predator-prey model with fear effects and gestation delays

<abstract><p>Recent studies have shown that, in addition to direct predation, fear of predators alters the physiological behavior of prey. Based on this fact, this paper investigates a three-species food chain based on ratio-dependent and Beddington-DeAngelis type functional responses, which incorporates fear effects and two gestation delays. The positivity, boundedness and existence of equilibrium points of the system are investigated, and the local stability behavior of the equilibrium points and the occurrence of Hopf-bifurcation when the time lag parameters exceed the critical values are studied by analyzing the corresponding characteristic equations. The main results show that Hopf-bifurcation occurs when the time delay parameters attain the thresholds. Finally, numerical simulations are performed to verify our main results.</p></abstract>

Effects of state‐of‐the‐art residual stress models on the member and local stability behaviour

AbstractResidual stresses have considerable effect on both the local and the member stability behaviour of steel structures. Previous studies have shown that the structural stability behaviour depends on both the distribution and the amplitude of the residual stresses. The residual stress distribution is affected by the cross‐section geometry, the steel grade and the manufacturing process, e.g. flame cutting and the welding procedure. A realistic consideration of residual stresses is necessary for an efficient and safe design of steel structures.This article presents an experimental and numerical study on the influence of residual stresses on the stability behaviour of structural members, i.e. steel columns and beams considering a variety of cross‐sectional slenderness, i.e. plastic, compact and slender cross‐sections are taken into account. A novel residual stress model for welded I‐shaped sections was developed and evaluated using test data. This model takes into account the main influencing parameters, i.e. the cross‐sections geometry, the steel grade as well as manufacturing process with thermal cuts or non‐thermal cuts of the steel plates. The novel residual stress model is then used to investigate the stability behaviour in terms of a numerical simulation study. The result of the study allows to propose an improvement of the buckling curve selection for welded high strength steel columns and beams.

Flexural bearing capacity and stiffness of stiffened hollow glulam beams: Experiments, finite element analysis and calculation theory

• Twenty-seven specimens were designed and four-point bending experiments were carried out. • The failure modes, elastic stiffness, bearing capacity, and ductility coefficients of the solid beam, hollow beam, stiffened hollow beam were compared and analyzed. • Numerical models were developed to analyze the mechanical performance of stiffened hollow beams and were validated by the experimental results. • The calculation equations for ultimate bearing capacity and bending stiffness of glulam beam with three different section forms were established. Inspired by the section structure of natural bamboo, a rectangular hollow glulam beam with stiffeners is proposed. In this paper, experimental study and finite element simulation were carried out for such glulam beams. The results showed that compared with hollow beam, the initial bending stiffness, ultimate bearing capacity and ductility of the stiffened hollow glulam beams improved significantly, indicating that stiffeners enhanced the local stability behavior of the webs and the bottom plates. Meanwhile, the calculation equations for ultimate bearing capacity and bending stiffness of glulam beam with three different section forms were established based on the material mechanics and constitutive model. The accuracy was verified by experiments and finite element analysis. This paper shows that the proposed stiffened hollow glulam beam can be used as a replacement for solid timber beam, which can save material while maintain a desirable mechanical performance.

Optical and thermodynamic behaviors of Ayón–Beato–García black holes for 4D Einstein Gauss–Bonnet gravity

We investigate a family of four dimensional (4D) Ayón–Beato–García (ABG) black holes in Einstein Gauss–Bonnet (EGB) gravity. We approach and examine the associated thermodynamic and the optical aspects by varying the involved parameters. We first compute and analyze the corresponding thermodynamic quantities. Among others, we inspect the global and the local stability behaviors. Then, we study the optical behaviors. Using Hamilton–Jacobi method, we study the shadow geometrical configurations in terms of one dimensional real closed curves. Using Gauss–Bonnet theorem, we calculate and examine the deflection angle of light rays by such black holes. In specific regions of a reduced moduli space obtained by fixing the mass and varying the remaining parameters, the present work recovers certain previous findings. Finally, we provide a possible speculative connection with observations from Event Horizon Telescope by imposing certain constraints on the involved b parameters in the light of the M87∗ image.

LOCAL STABILITY ANALYSIS OF A PREDATOR-PREY DYNAMICS INCORPORATING BOTH SPECIES DENSITY INCREASING FUNCTIONAL RESPONSE

Most of the functional responses which have been incorporated to formulate mathematical biology consider individual contact or predator cooperation. In this study, we have introduced a different functional response that describes the prey-predator system when predators form a line and cooperatively attack a group of predators. We have also described the effect of prey on this system. Additionally, we find all the equilibrium points and their local stability behaviour.

A DELAY NONAUTONOMOUS PREDATOR–PREY MODEL FOR THE EFFECTS OF FEAR, REFUGE AND HUNTING COOPERATION

Fear of predation may assert privilege to prey species by restricting their exposure to potential predators, meanwhile it can also impose costs by constraining the exploration of optimal resources. A predator–prey model with the effect of fear, refuge, and hunting cooperation has been investigated in this paper. The system’s equilibria are obtained and their local stability behavior is discussed. The existence of Hopf-bifurcation is analytically shown by taking refuge as a bifurcation parameter. There are many ecological factors which are not instantaneous processes, and so, to make the system more realistic, we incorporate three discrete time delays: in the effect of fear, refuge and hunting cooperation, and analyze the delayed system for stability and bifurcation. Moreover, for environmental fluctuations, we further modify the delayed system by incorporating seasonality in the fear, refuge and cooperation. We have analyzed the seasonally forced delayed system for the existence of a positive periodic solution. In the support of analytical results, some numerical simulations are carried out. Sensitivity analysis is used to identify parameters having crucial impacts on the ecological balance of predator–prey interactions. We find that the rate of predation, fear, and hunting cooperation destabilizes the system, whereas prey refuge stabilizes the system. Time delay in the cooperation behavior generates irregular oscillations whereas delay in refuge stabilizes an otherwise unstable system. Seasonal variations in the level of fear and refuge generate higher periodic solutions and bursting patterns, respectively, which can be replaced by simple 1-periodic solution if the cooperation and fear are also allowed to vary with time in the former and latter situations. Higher periodicity and bursting patterns are also observed due to synergistic effects of delay and seasonality. Our results indicate that the combined effects of fear, refuge and hunting cooperation play a major role in maintaining a healthy ecological environment.

Design analysis of torsichrone centrifugal pendulum vibration absorbers

The number of applications of centrifugal pendulum vibration absorber (CPVA) in the automotive industry is growing steadily. The request of low fuel consumption and torque increase at low speed stimulated a large variety of drive concepts, such as downsizing, cylinder deactivation and hybrid powertrains. These design solutions increased the severity of torsional vibrations as well as the need of more effective devices to damp them out. For this reason, in recent years new types of CPVAs have appeared on the market. In particular, a pendulum with trapezoidal architecture, called torsichrone, has been developed. Its distinctive feature consists in the exploit of absorbers rotational kinetic energy for additional vibration attenuation. This paper investigates the kinematics and the nonlinear dynamics of trapezoidal dampers by means of intrinsic geometry theoretical tools and perturbation methods, respectively. It also offers insights about the influence of absorber center of mass path on the overall performance. Furthermore, optimization criteria for the rotational component of absorbers motion are proposed. On the basis of such analyses, design guidelines are recommended. Finally, to assess vibration isolation effectiveness and local stability behavior of trapezoidal CPVAs, a numerical comparison with the tautochrone parallel pendulum and results of multibody dynamics simulation are discussed.

A study on the stability behavior of an epidemic model with ratio-dependent incidence and saturated treatment.

In the present article, the dynamics of a novel combination of ratio-dependent incidence rate and saturated treatment rate in susceptible-infected-recovered disease compartmental model has been presented. The ratio-dependent incidence rate has been incorporated into the model to monitor the situation when ratio of the number of infectives to that of the susceptibles is getting higher. The saturated treatment rate of the infected population has been considered as Holling type II functional, which explains the limitation in treatment availability. From the mathematical analysis of the model, two types of equilibria of the model have been obtained, which are named as disease-free equilibrium (DFE) and endemic equilibrium (EE). The local stability behavior of equilibria has been investigated by the basic reproduction number [Formula: see text], center manifold theory and Routh-Hurwitz criterion. It has been investigated that the DFE is locally asymptotically stable when [Formula: see text], and when [Formula: see text], the DFE exhibits either a forward bifurcation or a backward bifurcation under some conditions. The local stability behavior of the EE has also been analyzed, and some conditions are obtained for the same. Finally, some numerical computations have been performed in support of our theoretical results.

Conditioned Lyapunov exponents for random dynamical systems

We introduce the notion of Lyapunov exponents for random dynamical systems, conditioned to trajectories that stay within a bounded domain for asymptotically long times. This is motivated by the desire to characterize local dynamical properties in the presence of unbounded noise (when almost all trajectories are unbounded). We illustrate its use in the analysis of local bifurcations in this context. The theory of conditioned Lyapunov exponents of stochastic differential equations builds on the stochastic analysis of quasi-stationary distributions for killed processes and associated quasi-ergodic distributions. We show that conditioned Lyapunov exponents describe the local stability behaviour of trajectories that remain within a bounded domain and - in particular - that negative conditioned Lyapunov exponents imply local synchronisation. Furthermore, a conditioned dichotomy spectrum is introduced and its main characteristics are established.

Global Stability Analysis of a General Scalar Difference Equation

We consider a general first-order scalar difference equation with and without Allee effect. The model without Allee effect represents asexual reproduction of a species while the model including Allee effect represents sexual reproduction. We analyze global stabilities of both models analytically and compare the results obtained. Numerical simulations are included to support the analytical results. We conclude that Allee effect has a destabilizing effect on the global stability of the model. This result is different from the local stability behaviour of the same fixed point of the model.

Deterministic Seirs Epidemic Model for Modeling Vital Dynamics, Vaccinations, and Temporary Immunity

In this paper, the author proposes a new SEIRS model that generalizes several classical deterministic epidemic models (e.g., SIR and SIS and SEIR and SEIRS) involving the relationships between the susceptible S, exposed E, infected I, and recovered R individuals for understanding the proliferation of infectious diseases. As a way to incorporate the most important features of the previous models under the assumption of homogeneous mixing (mass-action principle) of the individuals in the population N, the SEIRS model utilizes vital dynamics with unequal birth and death rates, vaccinations for newborns and non-newborns, and temporary immunity. In order to determine the equilibrium points, namely the disease-free and endemic equilibrium points, and study their local stability behaviors, the SEIRS model is rescaled with the total time-varying population and analyzed according to its epidemic condition R0 for two cases of no epidemic (R0 ≤ 1) and epidemic (R0 > 1) using the time-series and phase portraits of the susceptible s, exposed e, infected i, and recovered r individuals. Based on the experimental results using a set of arbitrarily-defined parameters for horizontal transmission of the infectious diseases, the proportional population of the SEIRS model consisted primarily of the recovered r (0.7–0.9) individuals and susceptible s (0.0–0.1) individuals (epidemic) and recovered r (0.9) individuals with only a small proportional population for the susceptible s (0.1) individuals (no epidemic). Overall, the initial conditions for the susceptible s, exposed e, infected i, and recovered r individuals reached the corresponding equilibrium point for local stability: no epidemic (DFE X ¯ D F E ) and epidemic (EE X ¯ E E ).

Mathematical analysis of the effect of cuckoo bird’s incubation period in population dynamics

Although caring and raising children is the duty of all parents, some animals relieve themselves of this labored work by relying on others to raise their young. After the host has made a nest and laid eggs, the parasite locates the nest and lays a few eggs in the host’s nest. Several possible interactions occur between the host and parasite in brood parasitism: the parasite does not destroy the host’s eggs and the host does not abandon her nest with eggs. Here, we establish a mathematical framework for studying an ecological model on brood parasitism with stage structure of the parasite, where time delay represents the incubation period of the parasite. For the proposed model, we studied the impact of incubation time delay on the coexistence equilibrium. By using corresponding characteristic equations, the local stability behavior of each of the feasible equilibria of the model and existence of Hopf bifurcations at the coexistence equilibrium are established. Further, the global properties of some equilibria are established using Lyapunov–LaSalle invariance principle. The incubation time delay was found to be crucial for establishing stability behavior. When the incubation time delay increased and crossed some critical values, Hopf bifurcation may occur at the coexistence equilibrium. Further, by using normal form and center manifold theories, formulas were established to determine the stability and direction of the Hopf bifurcation. Numerical simulations were conducted to support the theoretical analysis. Our mathematical model could help lay a foundation for further studies on brood parasitism by the cuckoo bird.

Space independent xenon oscillations control in VVER reactor: A bifurcation analysis approach

A comprehensive parametric bifurcation analysis has been carried out in order to control space independent xenon instabilities in a reduced order model of VVER 1000 reactor. A lumped parameter kinetic model which takes into account the fundamental mode of reactor power is developed to describe nonlinear dynamic behavior of the reactor. The usual state space controller design techniques for such a model often require canonical representations which impose certain challenges in practical implementation of a rather easy and feasible control task. Bifurcation control methods are therefore resorted which rely basically on a physical intuition into the reactor nonlinear behavior. Reactor local stability behavior is essentially related to thermal feedback effects represented by the power coefficient of reactivity, a variant parameter during system operation causing Hopf bifurcation induced oscillations. The plant is shown to experience instability as the bifurcation parameter takes value in the working region of the reactor. Control procedure is thereby carried out accordingly via manipulation of power coefficient of the reactivity as the Hopf parameter. Taking into account the power (temperature) dependent origin of the selected Hopf parameter, a Washout filter is designed and implemented to relocate this parameter outside the instability region through application of reactor power as the state feedback variable. Numerical simulations exhibit quite reasonable performance of the proposed controller to maintain reactor stability.

Local buckling of shear-deformable laminated composite beams with arbitrary cross-sections using discrete plate analysis

The present paper deals with the onset of local buckling of compressively loaded thin-walled beams made of orthotropically laminated composite materials using discrete plate analysis. The analysis model focusses on the buckling of webs and flanges of composite beams with arbitrary cross-sections under uniform longitudinal compressive load. In order to account for transverse shear deformations as they are typical for moderately thick to thick laminated composite materials made of e.g. carbon fibre reinforced plastics, the present analysis is based on first-order shear deformation theory, thus employing the classical Reissner–Mindlin plate theory for the analysis of laminated composite structures. The idealisation consists of modelling the webs as being simply supported at all four edges, while at the longitudinal unloaded edges an elastic clamping is assumed which is represented by a clamping stiffness that takes material, geometry and layups of the adjacent flanges of the beams into account. Accordingly, the flanges are treated as plates with three simply supported edges and one free edge, wherein the unloaded simply supported edge is elastically clamped in order to represent the rotational support by the adjacent web. The analysis of the web and flange buckling loads is performed using the Rayleigh–Ritz-method employing specifically chosen shape functions for the out-of-plane displacements and the rotations of the cross-sections. The accuracy of the employed approaches is established by comparison with accompanying finite element simulations of actual thin-walled composite beams. It is revealed that the presented methodology is highly efficient in terms of computational effort and yet performs with satisfying accuracy which makes it very attractive for actual practical applications whenever the local stability behaviour of composite beams is to be considered.

Interconnected jumping time‐delay systems: Mode‐dependent decentralized stability and stabilization

SUMMARYIn this paper, we deal with the problems of mode‐dependent decentralized stability and stabilization with ℋ︁∞ performance for a class of continuous‐time interconnected jumping time‐delay systems. The jumping parameters are governed by a finite state Markov process and the delays are unknown time‐varying and mode‐dependent within interval. The interactions among subsystems satisfy quadratic bounding constraints. To characterize mode‐dependent local stability behavior, we employ an improved Lyapunov–Krasovskii functional at the subsystem level and express the stability conditions in terms of linear matrix inequalities (LMIs). A class of local decentralized state‐feedback controllers is developed to render the closed‐loop interconnected jumping system stochastically stable. Then, we extend the feedback strategy to dynamic observer‐based control and establish the stochastic stabilization via LMIs. It has been established that the developed results encompass several existing results as special cases which are illustrated by simulation of examples. Copyright © 2011 John Wiley & Sons, Ltd.

Local buckling of wide-flange thin-walled anisotropic composite beams

The present contribution deals with the onset of local buckling of compressively loaded thin-walled beams with open I, C, Z, T and L-cross-sections made of laminated composite materials. The method employs a discrete plate analysis approach in the course of which each structural subelement of interest—which presently is the flange—of the thin-walled cross-section is considered as a separate composite plate with elastic rotational restraints at those edges where an adjacent substructural element is located. While in many investigations the lamination schemes of webs and flanges are considered to be purely orthotropic, in the present paper the laminate layups are allowed to be of an arbitrary non-orthotropic nature, which also allows for the analysis of laminates with inherent bending–torsion coupling. The analysis of the buckling loads of the flanges of thin-walled composite beams is performed using the Ritz-method for which some especially adjusted displacement shape functions are employed. For the case of pure orthotropy, a novel closed-form solution is described. The accuracy of the employed approaches is established by comparison with accompanying finite element simulations of thin-walled composite beams. It is revealed that the presented methodology is highly efficient in terms of computational effort and yet performs with satisfying accuracy, which makes it very attractive for actual practical applications whenever the local stability behaviour of wide-flange thin-walled composite beams is to be considered.