On the Dynamics of the Classical Transverse Field Ising Chain

We investigate stationary and travelling wave solutions of a special lattice differential equation in one space dimension. Depending on a parameter λ, results are given on the existence, shape and stability for these kind of solutions. The analysis of travelling wave solutions leads us to a functional differential equation with both forward and backward shifts. The existence of solutions of this equation will be proved by use of the implicit function theorem. In particular, we consider kink solutions and periodic solutions.     

中立型时滞微分方程的渐近稳定性

考虑具有正负系数的中立型时滞微分方程ｄｄ狋［狓（狋）－犘（狋）狓（狋－τ）］＋犙（狋）狓（狋－δ）－犚（狋）狓（狋－σ）＝０，　狋≥狋０， 其中Ｐ（ｔ）∈Ｃ（［ｔ０，∞），Ｒ），Ｑ（ｔ），Ｒ（ｔ）∈Ｃ（［ｔ０，∞），Ｒ＋ ），τ，δ，σ∈（０，∞）．获得了该方程零解 一致稳定及渐近稳定的充分条件，它推广并改进了现有文献中的结论．     

数值求解NDDEs系统的单支方法的非线性稳定性

该文探讨了单支方法关于一类中立型延迟微分方程（ＮＤＤＥｓ）系统的整体稳定性和渐近稳定性．在适当的条件下，获得了单支方法关于ＮＤＤＥｓ系统的一些新的非线性稳定性判据．     

Periodic Solutions of the Duffing DifferentialEquation Revisited via the Averaging Theory

We use three different results of the averaging theory of first order for studying the existence of new periodic solutions in the two Duffing differential equations $ddot y+ a sin y= b sin t$ and $ddot y+a y-c y^3=bsin t$, where $a$, $b$ and $c$ are real parameters.     

Periodic Solutions of a Class of Duffing Differential Equations

In this work we study the existence of new periodic solutions for the well knwon class of Duffing differential equation of the form $x^{\prime\prime}+ c x^{\prime}+ a(t) x +b(t) x^3 = h(t)$, where $c$ is a real parameter, $a(t)$, $b(t)$ and $h(t)$ are continuous $T$--periodic functions. Our results are proved using three different results on the averaging theory of first order.     

Stability Analysis for the Numerical Simulation of Hybrid Stochastic Differential Equations

This paper is mainly concerned with the exponential stability of a class of hybrid stochastic differential equations–stochastic differential equations with Markovian switching (SDEwMSs). It first devotes to reveal that under the global Lipschitz condition, a SDEwMS is pth (p ∈ (0,1)) moment exponentially stable if and only if its corresponding improved Euler-Maruyama(IEM) method is pth moment exponentially stable for a suitable step size. It then shows that the SDEwMS is pth(p ∈ (0,1)) moment exponentially stable or its corresponding IEM method with small enough step sizes implies the equation is almost surely exponentially stable or the corresponding IEM method, respectively. In that sense, one can infer that the SDEwMS is almost surely exponentially stable and the IEM method, no matter whether the SDEwMS is pth moment exponentially stable or the IEM method. An example is demonstrated to illustrate the obtained results.     

Modeling of Lone Star Ticks with Deer Migration to Canada

Due to climate change and an increase of favourable habitat, ticks and tick-borne diseases are reported to expand to northern areas in north America. One main factor for lone star ticks to be established in Canada is due to the migration of white-tailed deers from US. In this work, we formulate a compartmental model to study the dynamics of lone star ticks and white-tailed deers, with a focus on migration effect of white-tailed deers. The tickhost interaction and the effect of deer migration are explored analytically and numerically. The positivity of the populations in the model is proved, and the unique positive equilibrium is proved to be asymptotically stable. We conduct sensitivity analysis on a set of parameters, revealing the correlation between the parameters and equilibrium populations. Numerical results show that migration rate of white-tailed deer is one crucial parameter that increases the populations of (infected) ticks and (infected) hosts.     

A HIV Infection Model with Periodic Multidrug Therapy

This paper investigates the effects of periodic drug treatment on a HIV infection model with two co-circulation populations of target cells. We first introduce the basic reproduction ratio for the model, and then show that the infection free equilibrium is globally asymptotically stable if R0 < 1, while the infection persists and there exists at least one positive periodic state when R0 > 1. Therefore, R0 serves as a threshold parameter for the infection. We then consider an optimization problem by shifting the phase of drug efficacy functions, which corresponds to change the dosage time of drugs in each time interval. It turns out that shifting the phase affect critically on the stability of the infection free steady state. Finally, exhaustive numerical simulations are carried out to support our theoretical analysis and explore the optimal phase shift.     

Multiplicity and Stability of Equilibrium States of Three-Dimensional Nonlinear System

The multiplicity and stability of the equilibrium states of a three-dimensional differential system with initial conditions and three cross terms are studied in this paper. The existence and multiplicity of equilibrium states are given under the different qualifications of parameters. Besides, the local stability of the equilibrium state is shown by analyzing the eigenfunction of the Jacobi matrix. The global stability of the equilibrium state is obtained by constructing the Lyapunov function. Furthermore, the numerical simulation intuitively reflected the relationship of variables and verified the correctness of theoretical analysis.