Sufficient Conditions For Local Stability Research Articles

Bifurcation Analysis of a Delayed Worm Propagation Model with Saturated Incidence

This paper is concerned with a delayed SVEIR worm propagation model with saturated incidence. The main objective is to investigate the effect of the time delay on the model. Sufficient conditions for local stability of the positive equilibrium and existence of a Hopf bifurcation are obtained by choosing the time delay as the bifurcation parameter. Particularly, explicit formulas determining direction of the Hopf bifurcation and stability of the bifurcating periodic solutions are derived by using the normal form theory and the center manifold theorem. Numerical simulations for a set of parameter values are carried out to illustrate the analytical results.

A queueing system with on-demand servers: local stability of fluid limits

We study a system, where a random flow of customers is served by servers (called agents) invited on-demand. Each invited agent arrives into the system after a random time; after each service completion, an agent returns to the system or leaves it with some fixed probabilities. Customers and/or agents may be impatient, that is, while waiting in queue, they leave the system at a certain rate (which may be zero). We consider the queue-length-based feedback scheme, which controls the number of pending agent invitations, depending on the customer and agent queue lengths and their changes. The basic objective is to minimize both customer and agent waiting times. We establish the system process fluid limits in the asymptotic regime where the customer arrival rate goes to infinity. We use the machinery of switched linear systems and common quadratic Lyapunov functions to approach the stability of fluid limits at the desired equilibrium point, and derive a variety of sufficient local stability conditions. For our model, we conjecture that local stability is in fact sufficient for global stability of fluid limits; the validity of this conjecture is supported by numerical and simulation experiments. When local stability conditions do hold, simulations show good overall performance of the scheme.

Exponential Stability and Stabilization for Quadratic Discrete‐Time Systems with Time Delay

AbstractIn this note, we deal with the exponential stability and stabilization problems for quadratic discrete‐time systems with time delay. By using the quadratic Lyapunov function and a so called ‘Finsler's lemma', delay‐independent sufficient conditions for local stability and stabilization for quadratic discrete‐time systems with time delay are derived in terms of linear matrix inequalities (LMIs). Based on these sufficient conditions, iterative linear matrix inequality algorithms are proposed for maximizing the stability regions of the systems. Finally, two examples are given to illustrate the effectiveness of the methods presented in this paper.

Stability and instability of a neuron network with excitatory and inhibitory small-world connections

This study considers a delayed neural network with excitatory and inhibitory shortcuts. The global stability of the trivial equilibrium is investigated based on Lyapunov’s direct method and the delay-dependent criteria are obtained. It is shown that both the excitatory and inhibitory shortcuts decrease the stability interval, but a time delay can be employed as a global stabilizer. In addition, we analyze the bounds of the eigenvalues of the adjacent matrix using matrix perturbation theory and then obtain the generalized sufficient conditions for local stability. The possibility of small inhibitory shortcuts is helpful for maintaining stability. The mechanisms of instability, bifurcation modes, and chaos are also investigated. Compared with methods based on mean-field theory, the proposed method can guarantee the stability of the system in most cases with random events. The proposed method is effective for cases where excitatory and inhibitory shortcuts exist simultaneously in the network.

Stability, convergence, limit cycles and chaos in some models of population dynamics

In this paper, we study three models of population dynamics: (i) the classical delay logistic equation, (ii) its variant which incorporates a harvesting rate and (iii) the Perez–Malta–Coutinho (PMC) equation. For all these models, we first conduct a local stability analysis around their respective equilibria. In particular, we outline the necessary and sufficient condition for local stability. In the case of the PMC equation, we also outline a sufficient condition for local stability, which may help guide design considerations. We also characterise the rate of convergence, about the locally stable equilibria, for all these models. We then conduct a detailed local bifurcation analysis. We first show, by using a suitably motivated bifurcation parameter, that the models undergo a Hopf bifurcation when the necessary and sufficient condition gets violated. Then, we use Poincare normal forms and centre manifold theory to study the dynamics of the systems just beyond the region of local stability. We outline an analytical basis to establish the type of the Hopf and determine the stability of the limit cycles. In some cases, we are able to derive explicit analytic expressions for the amplitude and period of the bifurcating limit cycles. We also highlight that in the PMC model, variations in one of the model parameters can readily induce chaotic dynamics. Finally, we construct an equivalent electronic circuit for the PMC model and demonstrate the existence of multiple bifurcations in a hardware-based circuit implementation.

A mathematical model of HIV-1 infection within host cell to cell viral transmissions with RTI and discrete delays

A within host viral infection model with both virus to cell and cell to cell transmissions and three discrete delays using reverse transcriptase inhibitors is investigated. The effect of time delay on stability of the equilibria of the infection model has been studied. By choosing the immune delay as the bifurcation parameter, we establish the sufficient condition for local stability of chronic infection equilibrium, Hopf bifurcation occurs when the immune delay passes a critical value. The global stability analysis of the model is carried out in terms of the basic reproduction number $$R_0$$ . If $$R_0 < 1$$ , the infection-free (semi-trivial) equilibrium is globally stable for both ODE and DDE model; if $$R_0>1$$ , the chronic infection (positive) equilibrium exists and is globally stable for both ODE and DDE model. Numerical simulations are obtained to validate our analytical findings by varying the system parameter.

Stability and stabilisation for time-varying polytopic quadratic systems

ABSTRACTThis paper develops the problems of local stability and stabilisation for time-varying polytopic quadratic systems. The state-space data is assumed to be dependent on parameters that are measurable in real time and vary in a compact set with bounded variation rates. These problems are solved by utilising the parameter-dependent quadratic Lyapunov function and S-procedure approach. Sufficient conditions for local stability and stabilisation are first formulated as optimisation problems with ‘quasi’ linear matrix inequalities. Simulation examples are then provided to confirm the effectiveness of the given approach.

Chaos and synchronisation of a new fractional order system with only two stable equilibria

In this paper, a new fractional order three-dimensional autonomous quadratic generalised Sprott C chaotic system is proposed. It is found that it has only two stable equilibria. That means that it is different from general fraction order chaotic systems, which often have unstable equilibrium points. First, some sufficient conditions for local stability of equilibria are given. Note that in the parameter space where the equilibria of the system are both asymptotically stable, it is confirmed by numerical simulation that chaotic attractors coexist with two stable equilibria. Then synchronisation between such systems is analysed. Unlike the active control scheme, which will cause the error system to be linear, here the error system is still kept to be non-linear. By Lyapunov stability theory, synchronisation is realised. Numerical simulation is performed to verify the theoretical results.

Dynamical Analysis of a Computer Virus Propagation Model with Delay and Infectivity in Latent Period

A delayed SLB computer virus propagation model with infectivity in latent period is proposed in this paper. We establish sufficient conditions for local stability of the positive equilibrium and existence of Hopf bifurcation by analyzing distribution of the roots of the associated characteristic equation and applying the Hopf bifurcation theorem. Furthermore, properties of the Hopf bifurcation are determined by using the normal form theory and the center manifold theorem. Finally, numerical simulations supporting the theoretical analysis are carried out.

Hopf bifurcation analysis for an SIRS epidemic model with logistic growth and delays

A delayed SIRS epidemic model with logistic growth is investigated in this paper. The main results are given in terms of local stability and Hopf bifurcation. Sufficient conditions for local stability of the positive equilibrium and existence of the local Hopf bifurcation are obtained by regarding the possible combination of the two delays as a bifurcation parameter and analyzing distribution of roots of the corresponding characteristic equations. Particularly, the direction and stability of the local Hopf bifurcation are determined by using the normal form theory and center manifold theorem. Finally, some numerical simulations are provided in order to illustrate the theoretical results.

A dynamic model for infectious diseases: The role of vaccination and treatment

Understanding dynamics of an infectious disease helps in designing appropriate strategies for containing its spread in a population. Recent mathematical models are aimed at studying dynamics of some specific types of infectious diseases. In this paper we propose a new model for infectious diseases spread having susceptible, infected, and recovered populations and study its dynamics in presence of incubation delays and relapse of the disease. The influence of treatment and vaccination efforts on the spread of infection in presence of time delays are studied. Sufficient conditions for local stability of the equilibria and change of stability are derived in various cases. The problem of global stability is studied for an important special case of the model. Simulations carried out in this study brought out the importance of treatment rate in controlling the disease spread. It is observed that incubation delays have influence on the system even under enhanced vaccination. The present study has clearly brought out the fact that treatment rate even in presence of time delays would contain the disease as compared to popular belief that eradication can only be done through vaccination.

Predictive Control of Power Converters: Designs With Guaranteed Performance

In this work, a cost function design based on Lyapunov stability concepts for finite control set model predictive control is proposed. This predictive controller design allows one to characterize the performance of the controlled converter, while providing sufficient conditions for local stability for a class of power converters. Simulation and experimental results on a buck dc-dc converter and a two-level dc-ac inverter are conducted to validate the effectiveness of our proposal.

Optimal Congestion Control and Routing for Multipath Networks with Random Losses

In this paper we consider optimal congestion control and routing schemes for multipath networks with non-congestion related packet losses which can be caused by, for example, errors on links on the routes, and develop a relaxed multipath network utility maximization problem. In order to obtain the optimum, we present a primal algorithm which is shown to be globally stable in the absence of round-trip delays. When round-trip delays are considered, decentralized sufficient conditions for local stability of the algorithm are proposed, in both continuous-time and discrete- time forms. Finally, a window-flow control mechanism is presented which can approximate the optimum of the multipath network utility maximization model.

Stability and Hopf Bifurcation in Delayed Predator-Prey System with Ratio Dependent

A delayed predator-prey model with stage structure for predator and ratio dependent response function is considered. By calculating characteristic equations and analyzing characteristic roots, the sufficient conditions for local stability of all the equilibria and Hopf bifurcation are obtained. Moreover, We use an iteration technique and comparison arguments to derive the sufficient conditions of the global stability of the boundary and positive equilibrium.

Stability and bifurcation analysis of delay induced tumor immune interaction model

A modified mathematical model of Kuznetsov et al. (1994) representing tumor-immune interaction with discrete time delay is proposed in this paper. By choosing the discrete time delay as the bifurcation parameter, we establish the sufficient condition for local stability of interior steady state, Hopf bifurcation occurs when the time delay passes a critical value. The estimation of the length of delay to preserve stability has been derived. Moreover, based on the normal form theory and center manifold theorem, we establish explicit expressions to determine the direction of Hopf bifurcation and the stability of bifurcating periodic solutions. Numerical simulations are obtained to validate our analytical findings by varying the system parameter. A two parameters bifurcation diagrams are also shown by varying the system parameters, which bring out the rich biological dynamics of the system.

Stability of Synchronized Motions of Inverter–Based Microgrids Under Droop Control

We consider the problems of frequency stability, voltage stability and power sharing in droop–controlled inverter–based microgrids with meshed topologies and dominantly inductive power lines. Assuming that the conductances in the microgrid can be neglected, a port–Hamiltonian description of a droop–controlled microgrid is derived. The model is used to establish sufficient conditions for local stability. Furthermore, we propose a condition for the controller parameters such that a desired steady–state active power distribution is achieved. The robustness of the stability condition with respect to the presence of conductances is analyzed via a simulation example based on the CIGRE benchmark medium voltage distribution network.

Hopf Bifurcation Analysis for a Computer Virus Model with Two Delays

This paper is concerned with a computer virus model with two delays. Its dynamics are studied in terms of local stability and Hopf bifurcation. Sufficient conditions for local stability of the positive equilibrium and existence of the local Hopf bifurcation are obtained by regarding the possible combinations of the two delays as a bifurcation parameter. Furthermore, explicit formulae for determining direction of the Hopf bifurcation and stability of the bifurcating periodic solutions are obtained by using the normal form method and center manifold theory. Finally, some numerical simulations are presented to support the theoretical results.

Hopf Bifurcation Analysis for a Four-Dimensional Recurrent Neural Network with Two Delays

A four-dimensional recurrent neural network with two delays is considered. The main result is given in terms of local stability and Hopf bifurcation. Sufficient conditions for local stability of the zero equilibrium and existence of the Hopf bifurcation with respect to both delays are obtained by analyzing the distribution of the roots of the associated characteristic equation. In particular, explicit formulae for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are established by using the normal form theory and center manifold theory. Some numerical examples are also presented to verify the theoretical analysis.

Stability and Hopf Bifurcation in a Delayed SEIRS Worm Model in Computer Network

A delayed SEIRS epidemic model with vertical transmission in computer network is considered. Sufficient conditions for local stability of the positive equilibrium and existence of local Hopf bifurcation are obtained by analyzing distribution of the roots of the associated characteristic equation. Furthermore, the direction of the local Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by using the normal form theory and center manifold theorem. Finally, a numerical example is presented to verify the theoretical analysis.

Stability analysis of diffusive predator–prey model with modified Leslie–Gower and Holling-type III schemes

The stability of a diffusive predator–prey model with modified Leslie–Gower and Holling-type III schemes is investigated. A threshold property of the local stability is obtained for a boundary steady state, and sufficient conditions of local stability and un-stability for the positive steady state are also obtained. Furthermore, the global asymptotic stability of these two steady states are discussed. Our results reveal the dynamics of this model system.