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In this paper, we propose a delayed viral infection model to incorporate a logistic proliferation, a cell pyroptosis effect and the three intracellular time delays. We present the basic reproduction number and investigate the existence and the global stability of equilibria: infection-free equilibrium $p_0$ and infection equilibrium $p^{\ast }$, respectively. By considering different combinations of the time delays, we investigate the existence and the properties of Hopf bifurcation from $p^{\ast }$ when it is unstable. We also numerically explore the viral dynamics beyond stability. Bifurcation diagrams are used to show the stability switches, multiperiod solutions and irregular sustained oscillations with the variation of time delays. The results reveal that both the logistic proliferation term and time delays are responsible for the rich dynamics, but the logistic proliferation term may be the main factor for the occurrence of the Hopf bifurcation. Moreover, we show that ignoring the cell pyroptosis effect may underevaluate the viral infection risk and the sensitivity analysis implies that taking effective strategies for reducing the impact of cell pyroptosis is beneficial for decreasing the viral infection risk.

The aim of this work is to develop a model of alcohol that takes drug features into account. We assess the model feasibility and explain its formulation in terms of a nonlinear differential equation. Using the subsequent matrix generation technique, we ascertain the reproductive number in order to assess the dynamics of the model. We also examine the system equilibrium points, namely the positive and free alcohol equilibrium points. To gain insights into the stability properties of the model, we utilize the Lyapunov function and the Routh–Hurwitz criterion. Through these methods, we investigate both the local stability and global stability of the considered model. Furthermore, we employ numerical simulations to complement and illustrate the theoretical results obtained. These simulations provide visual representations that enhance the understanding of the model dynamics and behavior.

In this paper, the problem of global robust exponential stabilization for a class of neural networks with reaction-diffusion terms and time-varying delays which covers the Hopfield neural networks and cellular neural networks is investigated. A feedback control gain matrix is derived to achieve the global robust exponential stabilization of the neural networks by using the Lyapunov stability theory, and the stabilization condition can be verified if a certain Hamiltonian matrix with no eigenvalues on the imaginary axis. This condition can avoid solving an algebraic Riccati equation. Finally, a numerical simulation illustrates the effectiveness of the results.

This paper presents new theoretical results on global exponential stability of bi-directional associative memory neural networks with distributed delays and reaction-diffusion terms based on the inequality technique, Lyapunov functional, and analysis technique. The results remove the usual assumption that the activation functions are of monotonous or differential character. Exponential converging velocity index is estimated, which depends on the delay kernel functions and system parameters. Finally, two numerical examples are given to show the validity and feasibility of our results.

In this paper, the robust synchronization problem is investigated for a new class of continuous-time complex networks that involve parameter uncertainties, time-varying delays, constant and delayed couplings, as well as multiple stochastic disturbances. The norm-bounded uncertainties exist in all the network parameters after decoupling, and the stochastic disturbances are assumed to be Brownian motions that act on the constant coupling term, the delayed coupling term as well as the overall network dynamics. Such multiple stochastic disturbances could reflect more realistic dynamical behaviors of the coupled complex network presented within a noisy environment. By using a combination of the Lyapunov functional method, the robust analysis tool, the stochastic analysis techniques and the properties of Kronecker product, we derive several delay-dependent sufficient conditions that ensure the coupled complex network to be globally robustly synchronized in the mean square for all admissible parameter uncertainties. The criteria obtained in this paper are in the form of linear matrix inequalities whose solution can be easily calculated by using the standard numerical software. The main results are shown to be general enough to cover many existing ones reported in the literature. Simulation examples are presented to demonstrate the feasibility and applicability of the proposed results.

In this paper, several simple and new conditions are presented to ensure existence, uniqueness and global exponential robust stability of the equilibrium point of neural networks with delays by using contraction mapping principle, homeomorphism property and Lyapunov method. The conditions given in this paper are easy to verify and apply, and the obtained results improve and extend to those in the literature. In addition, two examples are given to show the effectiveness of the proposed results.

In this paper, some global exponential stability criteria for the equilibrium point of discrete-time recurrent neural networks with variable delay are presented by using the linear matrix inequality (LMI) approach. The neural networks considered are assumed to have asymmetric weighting matrices throughout this paper. On the other hand, by applying matrix decomposition, the model is embedded into a cooperative one, the latter possesses important order-preserving properties which are basic to our analysis. A sufficient condition is obtained ensuring the componentwise exponential stability of the system with specific performances such as decay rate and trajectory bounds.

Stochastic effects on convergence dynamics of reaction–diffusion recurrent neural networks (RNNs) with constant transmission delays are studied. Without assuming the boundedness, monotonicity and differentiability of the activation functions, nor symmetry of synaptic interconnection weights, by skillfully constructing suitable Lyapunov functionals and employing the method of variational parameters, M-matrix properties, inequality technique, stochastic analysis and non-negative semimartingale convergence theorem, delay independent and easily verifiable sufficient conditions to guarantee the almost sure exponential stability, mean value exponential stability and mean square exponential stability of an equilibrium solution associated with temporally uniform external inputs to the networks are obtained, respectively. The results are compared with the previous results derived in the literature for discrete delayed RNNs without diffusion or stochastic perturbation. Two examples are also given to demonstrate our results.

For a large class of reaction–diffusion bidirectional associative memory (RDBAM) neural networks with periodic coefficients and general delays, several new delay-dependent or delay-independent sufficient conditions ensuring the existence and global exponential stability of a unique periodic solution are given, by constructing suitable Lyapunov functionals and employing some analytic techniques such as Poincaré mapping. The presented conditions are easily verifiable and useful in the design and applications of RDBAM neural networks. Moreover, the employed analytic techniques do not require the symmetry of the bidirectional connection weight matrix, the boundedness, monotonicity and differentiability of activation functions of the network. In several ways, the results generalize and improve those established in the current literature.

We present a variational formulation for the Kardar–Parisi–Zhang (KPZ) equation that leads to a thermodynamic-like potential for the KPZ as well as for other related kinetic equations. For the KPZ case, with the knowledge of such a potential we prove some global shift invariance properties previously conjectured by other authors. We also show a few results about the form of the stationary probability distribution function for arbitrary dimensions. The procedure used for KPZ was extended in order to derive more general forms of such a functional leading to other nonlinear kinetic equations, as well as cases with density dependent surface tension.

In the current paper, a class of stochastic cellular neural networks with reaction–diffusion effects, both discrete and distributed time delays, is studied. Several sufficient conditions guaranteeing the almost sure and pth moment exponential stability of its equilibrium solution are respectively obtained through analytic methods such as employing Lyapunov functional, applying Itô's formula, inequality techniques, embedding in Banach space, Matrix analysis and semimartingale convergence theorem. The yielded conclusions, which are independent of diffusion terms and delays, assume much less restrictions on activation functions and interconnection weights, and can be applied within a broader range of neural networks. Moreover, through the obtained results, it could be noted that noise will affect the exponential stability of the system. For illustration, two examples are given to show the feasibility of results.

A functional version of the LaSalle invariance principle is introduced. Rather than the usual pointwise Lyapunov-like functions, this extended version of the principle uses specially constructed functionals along system trajectories. This modification enables the original principle to handle not only autonomous, but also some nonautonomous systems. The new theoretical result is used to study robust synchronization of general Liénard-type nonlinear systems. The new technique is finally applied to coupled chaotic van der Pol oscillators to achieve synchronization. Numerical simulation is included to demonstrate the effectiveness of the proposed methodology.

The global exponential stability is studied for a class of high-order bi-directional associative memory (BAM) neural networks with time delays and reaction–diffusion terms. By constructing suitable Lyapunov functional, using differential mean value theorem and homeomorphism, several sufficient conditions guaranteeing the existence, uniqueness and global exponential stability of high-order BAM neural networks with time delays and reaction–diffusion terms are given. Two illustrative examples are also given in the end to show the effectiveness of our results.

We devote to studying the problem for the existence of homoclinic and heteroclinic orbits of Unified Lorenz-Type System (ULTS). Other than the known results that the ULTS has two homoclinic orbits to $E_0=(0,0,0)$ for $b=-2a_1$, $d=-a_1$, $a_1^2+a_{2}c>0$, $e<0$ and two heteroclinic orbits to $E_{1,2}=(\pm \sqrt{-\frac{2(a_1^2+a_{2}c)}{e}},\mp \frac{a_1}{a_2}\sqrt{-\frac{2(a_1^2+a_{2}c)}{e}},-\frac{a_1^2+a_{2}c}{a_{2}e})$ for $b=-2a_1$, $d=-a_1$, $a_1^2+a_{2}c<0$, $e>0$ on its invariant algebraic surface $Q(x,y,z)=z-\frac{x^2}{2a_2}=0$, formulated in the literature by Yang and Chen [2014], we seize two new heteroclinic orbits of this Unified Lorenz-Type System. Namely, we rigorously prove that this system has another two heteroclinic orbits to $E_0$ and $E_{\pm }=(\pm \sqrt{\frac{b(a_{2}c-a_{1}d)}{a_{1}e}},\mp \frac{a_1}{a_2}\sqrt{\frac{b(a_{2}c-a_{1}d)}{a_{1}e}},\frac{a_{1}d-a_{2}c}{a_{2}e})$ while no homoclinic orbit when $a_1<0$, $e<0$, $a_1+d<0$, $a_2\ne 0$, $a_{2}c-a_{1}d>0$, $b+2a_1\ge 0$.

In this paper, a delayed virus infection model with cell-to-cell transmission and CTL immune response is investigated. In the model, time delay is incorporated into the CTL response. By constructing Lyapunov functionals, global dynamical properties of the two boundary equilibria are established. Our results show that time delay in the CTL response process may lead to sustained oscillation. To further investigate the nature of the oscillation, we apply the method of multiple time scales to calculate the normal form on the center manifold of the model. At the end of the paper, numerical simulations are carried out, which support our theoretical results.

In this paper, we consider a viral infection dynamics model with immune impairment and time delay in immune expansion. By calculation, it is shown that the model has three equilibria: infection-free equilibrium, immunity-inactivated infection equilibrium, and immunity-activated infection equilibrium. By analyzing the distributions of roots of corresponding characteristic equations, the local stability of the infection-free equilibrium and the immunity-inactivated infection equilibrium is established. Furthermore, we discuss the existence of Hopf bifurcation at the immunity-activated infection equilibrium. Sufficient conditions are obtained for the global asymptotic stability of each feasible equilibria of the model by using LaSalle’s invariance principle and iteration technique, respectively. Numerical simulations are carried out to illustrate the main theoretical results.

Recent studies have demonstrated that immune impairment is an essential factor in viral infection for disease development and treatment. In this paper, we formulate an age-structured viral infection model with a nonmonotonic immune response and perform dynamical analysis to explore the effects of both immune impairment and virus control. The basic infection reproduction number is derived for a general viral production rate, which determines the global stability of the infection-free equilibrium. For the immune intensity, we get two important thresholds, the post-treatment control threshold and the elite control threshold. The interval between the two thresholds is a bistable interval, where there are two immune-present infected equilibria. When the immune intensity is greater than the elite control threshold, only one immune-present infected equilibrium exists and it is stable. By assuming the death rate and virus production rate of infected cells to be constants, with the immune intensity as a bifurcation parameter, the system exhibits saddle-node bifurcation, transcritical bifurcation, and backward/forward bifurcation.

We investigate rescaling transformations for the Vlasov–Poisson and Euler–Poisson systems and derive in the plasma physics case Lyapunov functionals which can be used to analyze dispersion effects. The method is also used for studying the long time behavior of the solutions and can be applied to other models in kinetic theory (two-dimensional symmetric Vlasov–Poisson system with an external magnetic field), in fluid dynamics (Euler system for gases) and in quantum physics (Schrödinger–Poisson system, nonlinear Schrödinger equation).

We study a uniform-in-time convergence from the discrete-time (in short, discrete) Cucker–Smale (CS) model to the continuous-time CS model, which is valid for the whole time interval, as time-step tends to zero. Classical theory yields the convergence results which are valid only in any finite-time interval. Our uniform convergence estimate relies on two quantitative estimates “asymptotic flocking estimate” and “uniform${\ell }_2$-stability estimate with respect to initial data”. In the previous literature, most studies on the CS flocking have been devoted to the continuous-time model with general communication weights, whereas flocking estimates have been done for the discrete-time model with special network topologies such as the complete network with algebraically decaying communication weights and rooted leaderships. For the discrete CS model with a regular and algebraically decaying communication weight, asymptotic flocking estimate has been extensively studied in the previous literature. In contrast, for a general decaying communication weight, corresponding flocking dynamics has not been addressed in the literature due to the difficulty of extending the Lyapunov functional approach to the discrete model. In this paper, we present asymptotic flocking estimate for the discrete model using the Lyapunov functional approach. Moreover, we present a uniform ${\ell }_2$-stability estimate of the solution for the discrete CS model with respect to initial data. We combine asymptotic flocking estimate and uniform stability to derive a uniform-in-time convergence from the discrete CS model to the continuous CS model, as time-step tends to zero.

In this paper, we have considered a nonautonomous stage-structured epidemic model having two stages of the period of infection according to the progressing process of some infectious diseases (e.g. Chagas' disease, hepatitis C, etc.) with varying total population size and distributed time delay to become infectious. The infected persons in the different stages have different ability of transmitting disease. We have established some sufficient conditions on the permanence and extinction of the disease by using inequality analytical technique. We have obtained the explicit formula of the eventual lower bounds of infected persons. We have introduced some new threshold values R₀ and R* and further obtained that the disease will be permanent when R₀ > 1 and the disease will be going to extinct when R* < 1. By Lyapunov functional method, we have also obtained some sufficient conditions for global asymptotic stability of this model. Computer simulations are carried out to explain the analytical findings. The aim of the analysis of this model is to identify the parameters of interest for further study, with a view to informing and assisting policy-maker in targeting prevention and treatment resources for maximum effectiveness.