Sufficient Conditions For Global Stability Research Articles

On viscous stratified Darcy–Forchheimer flow in a horizontal porous layer with thermal anisotropy and variable permeability

This study analyzes the effect of anisotropy and the internal heat source in a Darcy–Forchheimer porous layer. It is well known that the variations in viscosity can be attributed to the temperature. Therefore, in the present problem, we consider a linear variation in viscosity with temperature for simplicity. We first derived the linear instability theory and then established global stability using the energy functional approach. In the global stability analysis, we show that working with the L2 norm fails to give a sufficient condition for global stability by exhibiting that the associated maximization problem is unbounded in the underlying stability measure space. Then, we show that a conditional stability bound can be achieved by restricting the internal heat source parameter Q with higher-order norms. The eigenvalue problems obtained in linear and nonlinear theories were integrated numerically. The linear and nonlinear instability thresholds are then compared to identify the potential regions of sub-critical instabilities. It is observed that the system is stabilized when the horizontal component of thermal diffusivity dominates and is unstable when the vertical component of thermal diffusivity dominates. We also found that increasing the variable permeability parameter λ destabilized the system. It is observed that increasing viscosity stabilizes the system, and decreasing viscosity encourages the start of convection. It is also interesting that, in the presence of an internal heat source, the region of subcritical instability increases with increasing viscosity effect but reduces with increasing vertical permeability λ.

Air-Jet Levitation of Continuum Robots: Stable Head Floating by Passive Thrust Vectoring for Enhancing Mobility

Flexible, continuum-type robots can access narrow spaces in debris areas during search-and-rescue missions. However, their current problem is the lifting capability of the head, which is necessary to surmount the rubble. This study proposes a passive-thrust vectoring method to stabilize head levitation for air-jet-actuated, long, continuum robots, and thus enhance their step-climbing abilities. An air jet can generate a thrust force that is sufficient for head levitation. A critical issue in levitation involves the backward bending of the head without using any air-jet control; however, thrust control is inappropriate because of the delay caused by the long channel. Therefore, the proposed method maintains the thrust direction constant. Sufficient conditions for global stability are derived and confirmed via dynamic simulations. The proposed method is simple and can be mechanically realized with a passive head-bending mechanism, thus contributing to a lightweight design. Experiments demonstrated that the developed air-floating-type, 7-m long robot, can achieve stabilized head levitation, and that the robot can climb a step with a height of 250 mm. Robot demonstrations in rubble justify the robot’s capacity to surmount the rubble. This passive-thrust vectoring method is expected to contribute to the future enhancement of the mobility of continuum robots owing to its simplicity and practicality.

Qualitative Study of a Discrete-Time Harvested Fishery Model in the Presence of Toxicity

This paper analyses a discrete-time Michaelis-Menten type harvested fishery model in the presence of toxicity. Boundary and interior (positive) fixed points are examined. Using an iteration scheme and the comparison principle of difference equations, we determined the sufficient condition for global stability of the interior fixed point. It is shown that the sufficient criterion for Neimark-Sacker bifurcation and flip bifurcation can be established. It is observed that the system behaves in a chaotic manner when a specific set of system parameters is selected, which are controlled by a hybrid control method. Examples are cited to illustrate our conclusions.

Global Stability and Bifurcation Analysis in a Discrete-Time Two Predator-One Prey Model with Michaelis-Menten Type Prey Harvesting

This article studies a discrete-time Leslie-Gower two predator-one prey system with Michaelis-Menten type prey harvesting. Positivity and boundedness of the model solution are investigated. Existence and stability of fixed points are examined. Using an iteration scheme and the comparison principle of difference equations, we find out the sufficient condition for global stability of the positive fixed point. It is shown that the sufficient criterion for Neimark-Sacker bifurcation can be developed. It is observed that the system behaves in a chaotic manner when a specific set of system parameters is chosen, which are regulated by a hybrid control method. Examples are provided to illustrate our conclusions.

Interlayer antisynchronization in degree-biased duplex networks.

With synchronization being one of nature's most ubiquitous collective behaviors, the field of network synchronization has experienced tremendous growth, leading to significant theoretical developments. However, most previous studies consider uniform connection weights and undirected networks with positive coupling. In the present article, we incorporate the asymmetry in a two-layer multiplex network by assigning the ratio of the adjacent nodes' degrees as the weights to the intralayer edges. Despite the presence of degree-biased weighting mechanism and attractive-repulsive coupling strengths, we are able to find the necessary conditions for intralayer synchronization and interlayer antisynchronization and test whether these two macroscopic states can withstand demultiplexing in a network. During the occurrence of these two states, we analytically calculate the oscillator's amplitude. In addition to deriving the local stability conditions for interlayer antisynchronization via the master stability function approach, we also construct a suitable Lyapunov function to determine a sufficient condition for global stability. We provide numerical evidence to show the necessity of negative interlayer coupling strength for the occurrence of antisynchronization, and such repulsive interlayer coupling coefficients cannot destroy intralayer synchronization.

Global stability and bifurcation analysis in a discrete-time two prey-one predator model with help

ABSTRACT This article examines a discrete-time predator-prey model where prey teams cooperate. Positivity and boundedness of the model solution are investigated. Positive fixed points are examined. Using an iteration scheme and the comparison principle of difference equations, we determined the sufficient condition for global stability of the positive fixed point. It is shown that the sufficient criterion for Neimark-Sacker bifurcation and flip bifurcation can be established. It is observed that the system behaves in a chaotic manner when a specific set of system parameters is selected, which are controlled by a hybrid control method. Examples are provided to illustrate our conclusions.

STABILITY ANALYSIS AND NUMERICAL SIMULATIONS OF IVGTT GLUCOSE-INSULIN INTERACTION MODELS WITH TWO TIME DELAYS

This paper presents a systematic study of a mathematical model of glucose and insulin interaction with two time delays, with a focus on analytical studies, bifurcation analysis, and very well numerical simulations. This model based on the Intra-Venous Glucose Tolerance Test (IVGTT) and is presented with two time delays. One delay is the insulin response time to an increase in glucose concentration, and the hepatic glucose production time delay is the other. Then, we establish results on positivity, boundedness, and persistence. We also provide sufficient stability analysis conditions for both local and global asymptotic stability of the proposed models. For the latter, two different strategies are used: stability bifurcation analysis and Lyapunov-Krasovskii functionals. We investigate different regions of parameter space using two approaches, that yield different sets of sufficient conditions for global stability. The bifurcation graphs generated from our extensive and carefully designed simulations complement and confirm these analytical results. The insulin concentration level peaks after the glucose concentration level, according to the numerical simulations.

Stability Analysis of a Delayed Immune Response Model to Viral Infection

The purpose of this work is the study of the qualitative behavior of the homogeneous in space solution of a delay differential equation arising from a model of infection dynamics. This study is mainly based on the monotone dynamical systems theory. Existence and smoothness of solutions are proved, and conditions of asymptotic stability of equilibriums in the sense of monotone dynamical systems are formulated. Then, sufficient conditions of global stability of the nonzero steady state are derived, for the two typical forms of the function f, specifying the efficiency of immune response-mediated virus elimination. Numerical simulations illustrate the analytical results. The obtained theoretical results have been applied, in a context of COVID-19 data calibration, to forecast the immunological behaviour of a real patient.

A careful study of the effect of the infectious diseases and refuge on the dynamical behavior of prey-scavenger modeling

In this paper, the dynamics of scavenger species predation of both susceptible and infected prey at different rates with prey refuge is mathematically proposed and studied. It is supposed that the disease was spread by direct contact between susceptible prey with infected prey described by Holling type-II infection function. The existence, uniqueness, and boundedness of the solution are investigated. The stability constraints of all equilibrium points are determined. In addition to establishing some sufficient conditions for global stability of them by using suitable Lyapunov functions. Finally, these theoretical results are shown and verified with numerical simulations.

Behavioural analysis of two prey-two predator model

To deal with real-life diversity of our ecosystem, this paper analyzes two prey-two predator model including both Type-I and Type-II functional responses. The interior equilibrium point of the proposed model is calculated; and behaviour of the model around that point is studied. Local stability at an interior equilibrium point is discussed; and possibility of Hopf-bifurcation with probable direction is studied. A generalized form of the Poincaré-Bendixon criterion is applied to establish the sufficient conditions for global stability of the proposed model surrounding at an interior equilibrium point. Numerical simulations are also conducted in support of our work. Conclusions of our findings and some probable future directions are also included at the end.

Fixed time steps discrete-time sliding mode consensus protocols for two degree of freedom helicopter systems

This paper presents analysis of number of steps required for the consensus in a leader-following discrete multi-agent system (DMAS) with discrete-time sliding mode protocols designed by Gao’s reaching law and power rate reaching laws. The DMAS is configured for communication with a fixed, undirected graph topology having one leader and other agents as followers. The sufficient condition for global stability is established using the Lyapunov function in both the cases. The efficacy of both the protocols is compared in simulation for number of steps required for the consensus of a homogeneous multiple two degree of freedom helicopter systems where the pitch angle and its velocity and yaw angle and its velocity are used for consensus. The simulation results reveal that the consensus performance due to protocol with power rate reaching law outperforms the protocol with Gao’s reaching law.

New sufficient conditions of global stability of nonlinear positive electrical circuits

The global stability of electrical circuits composed of positive linear parts and nonlinear static element with given characteristic and positive gain feedbacks is investigated. New sufficient conditions for the global stability of this class of nonlinear positive electrical circuits are established. The effectiveness of these new stability conditions are demonstrated on simples examples of positive nonlinear electrical circuits.

An epidemic model with multiple delays for the propagation of worms in wireless sensor networks

A delayed SEIRS worm propagation model with the inclusion of a finite communication radius and node density for wireless sensor networks is investigated in this paper. By using different combinations of the three delays as bifurcating parameter and analyzing distribution of roots of the corresponding characteristic equation, sufficient conditions are derived for local stability of the endemic equilibrium and the existence of a Hopf bifurcation at the endemic equilibrium are addressed. By constructing a suitable Lyapunov function, sufficient conditions for global stability of the endemic equilibrium are determined. Finally, numerical simulations for a set of parameter values are performed to illustrate the analytical findings.

Theory of Hidden Oscillations and Stability of Control Systems

The development of the theory of absolute stability, the theory of bifurcations, the theory of chaos, theory of robust control, and new computing technologies has made it possible to take a fresh look at a number of well-known theoretical and practical problems in the analysis of multidimensional control systems, which led to the emergence of the theory of hidden oscillations, which represents the genesis of the modern era of Andronov’s theory of oscillations. The theory of hidden oscillations is based on a new classification of oscillations as self-excited or hidden. While the self-excitation of oscillations can be effectively investigated analytically and numerically, revealing a hidden oscillation requires the development of special analytical and numerical methods and also it is necessary to determine the exact boundaries of global stability, to analyze and reduce the gap between the necessary and sufficient conditions for global stability, and distinguish classes of control systems for which these conditions coincide. This survey discusses well-known theoretical and engineering problems in which hidden oscillations (their absence or presence and location) play an important role.

Dynamics of a delayed SIR model for the transmission of PRRSV among a swine population

The objective of this paper is to propose a delayed susceptible-infectious-recovered (SIR) model for the transmission of porcine reproductive respiratory syndrome virus (PRRSV) among a swine population, including the latent period delay of the virus and the time delay due to the period the infectious swines need to recover. By taking different combinations of the two delays as the bifurcation parameter, local stability of the disease-present equilibrium and the existence of Hopf bifurcation are analyzed. Sufficient conditions for global stability of the disease-present equilibrium are derived by constructing a suitable Lyapunov function. Directly afterwards, properties of the Hopf bifurcation such as direction and stability are studied with the aid of the normal form theory and center manifold theorem. Finally, numerical simulations are presented to justify the validity of the derived theoretical results.

Discrete-time event-triggered higher order sliding mode control for consensus of 2-DOF robotic arms

This paper proposes a discrete-time event-triggered higher-order sliding mode protocol for the leader-following consensus of Discrete Homogeneous Multi-Agent Systems (DHMAS). The discrete version of the Super-Twisting Algorithm (DSTA) which is one of the most popular algorithm of higher order sliding mode control blended with event-triggered control approach is used to design the protocol. The DSTA algorithm is robust against the uncertainties present in the DHMAS and improve the consensus performance. The event-triggered control approach reduces the usage of power and communication resources due to the intermittent communication with the neighboring agents to achieve consensus. For implementing the proposed consensus protocol, each follower agent of DHMAS monitor for the event-triggering condition as defined in this paper. The sufficient condition for global stability with proposed consensus protocol is also derived to ensure the stable operation of DHMAS. Further, the proposed protocol is implemented on a discrete homogeneous multi-agent system set up comprise of 2-DOF flexible joint robotic arm considering one robotic arm as a leader and others as followers to show the efficacy of the proposed protocol.

Invariance and stability conditions of interlayer synchronization manifold.

We investigate interlayer synchronization in a stochastic multiplex hypernetwork which is defined by the two types of connections, one is the intralayer connection in each layer with hypernetwork structure and theother is the interlayer connection between the layers. Here all types of interactions within and between the layers are allowed to vary with a certain rewiring probability. We address the question about the invariance and stability of the interlayer synchronization state in this stochastic multiplex hypernetwork. For the invariance of interlayer synchronization manifold, the adjacency matrices corresponding to each tier in each layer should be equal and the interlayer connection should be either bidirectional or the interlayer coupling function should vanish after achieving the interlayer synchronization state. We analytically derive a necessary-sufficient condition for local stability of the interlayer synchronization state using master stability function approach and a sufficient condition for global stability by constructing a suitable Lyapunov function. Moreover, we analytically derive that intralayer synchronization is unattainable for this network architecture due to stochastic interlayer connections. Remarkably, our derived invariance and stability conditions (both local and global) are valid for any rewiring probabilities, whereas most of the previous stability conditions are only based on a fast switching approximation.

Harmonic balance analysis of pull-in range and oscillatory behavior of third-order type 2 analog PLLs

The most important design parameters of each phase-locked loop (PLL) are the local and global stability properties, and the pull-in range. To extend the pull-in range, engineers often use type 2 PLLs. However, the engineering design relies on approximations which prevent a full exploitation of the benefits of type 2 PLLs. Using an exact mathematical model and relying on a rigorous mathematical thinking this problem is revisited here and the stability and pull-in properties of the third-order type 2 analog PLLs are determined. Both the local and global stability conditions are derived. As a new idea, the harmonic balance method is used to derive the global stability conditions. That approach offers an extra advantage, the birth of unwanted oscillations can be also predicted. As a verification it is shown that the sufficient conditions of global stability derived by the harmonic balance method proposed here and the well-known direct Lyapunov approach coincide with each other, moreover, the harmonic balance predicts the birth of oscillations in the gap between the local and global stability conditions. Finally, an example when the conditions for local and global stability coincide, is considered.

Switching-based stabilization of aperiodic sampled-data Boolean control networks with all subsystems unstable

We aim to further study the global stability of Boolean control networks (BCNs) under aperiodic sampled-data control (ASDC). According to our previous work, it is known that a BCN under ASDC can be transformed into a switched Boolean network (SBN), and further global stability of the BCN under ASDC can be obtained by studying the global stability of the transformed SBN. Unfortunately, since the major idea of our previous work is to use stable subsystems to offset the state divergence caused by unstable subsystems, the SBN considered has at least one stable subsystem. The central thought in this paper is that switching behavior also has good stabilization; i.e., the SBN can also be stable with appropriate switching laws designed, even if all subsystems are unstable. This is completely different from that in our previous work. Specifically, for this case, the dwell time (DT) should be limited within a pair of upper and lower bounds. By means of the discretized Lyapunov function and DT, a sufficient condition for global stability is obtained. Finally, the above results are demonstrated by a biological example.

A Mathematical Model for the Transmission Dynamics of Lymphatic Filariasis with Intervention Strategies.

This manuscript considers the transmission dynamics of lymphatic filariasis with some intervention strategies in place. Unlike previously developed models, our model takes into account both the exposed and infected classes in both the human and mosquito populations, respectively. We also consider vaccinated, treated and recovered humans in the presented model. The global dynamics of the proposed model are completely determined by the basic ([Formula: see text]) and effective reproduction numbers ([Formula: see text]). We then use Lyapunov function theory to find the sufficient conditions for global stability of both the disease-free equilibrium and endemic equilibrium. The Lyapunov functions show that when the basic reproduction number is less than or equal to unity, the disease-free equilibrium is globally asymptotically stable, and when it is greater than unity then the endemic equilibrium is also globally asymptotically stable. Finally, numerical simulations are carried out to investigate the effects of the intervention strategies and key parameters to the spread of lymphatic filariasis. The numerical simulations support the analytical results and illustrate possible model behavioral scenarios.