The stochastic $\mathrm{\Theta }$-method is extended to solve nonlinear stochastic Volterra integro-differential equations. The mean-square convergence and asymptotic stability of the method are studied. First, we prove that the stochastic $\mathrm{\Theta }$-method is convergent of order $1/2$ in mean-square sense for such equations. Then, a sufficient condition for mean-square exponential stability of the true solution is given. Under this condition, it is shown that the stochastic $\mathrm{\Theta }$-method is mean-square asymptotically stable for every stepsize if $1/2\le \theta \le 1$ and when , the stochastic $\mathrm{\Theta }$-method is mean-square asymptotically stable for some small stepsizes. Finally, we validate our conclusions by numerical experiments.

This paper is concerned with the problem for the maximal number of limit cycles for a quadratic piecewise near-Hamiltonian system. By using the method of the first order Melnikov function, we find that it can have 8 limit cycles.

This paper is concerned with the computation of the normal form and its application to a viscoelastic moving belt. First, a new computation method is proposed for significantly refining the normal forms for high-dimensional nonlinear systems. The improved method is described in detail by analyzing the four-dimensional nonlinear dynamical systems whose Jacobian matrices evaluated at an equilibrium point contain three different cases, that are, (i) two pairs of pure imaginary eigenvalues, (ii) one nonsemisimple double zero and a pair of pure imaginary eigenvalues, and (iii) two nonsemisimple double zero eigenvalues. Then, three explicit formulae are derived, herein, which can be used to compute the coefficients of the normal form and the associated nonlinear transformation. Finally, employing the present method, we study the nonlinear oscillation of the viscoelastic moving belt under parametric excitations. The stability and bifurcation of the nonlinear vibration system are studied. Through the illustrative example, the feasibility and merit of this novel method are also demonstrated and discussed.

We studied the delayed periodic mutualism system with Gilpin-Ayala effect. Some new and interesting sufficient conditions are obtained to guarantee the existence of periodic solution for the multispecies mutualism system with infinite delays. Our method is based on Mawhin's coincidence degree. To the best knowledge of the authors, there is no paper considering the existence of periodic solutions for n-species mutualism system with infinite delays.

We discuss the global and blow-up solutions of the following nonlinear parabolic problems with a gradient term under Robin boundary conditions: $(b(u){)}_t=\nabla ·(h(t)k(x)a(u)\nabla u)+f(x,u,|\nabla u{|}^2,t)$, in $D\times (0,T)$, $(\partial u/\partial n)+\gamma u=0$, on $\partial D\times (0,T)$, $u(x,0)=u_0(x)>0$, in $\overline{D}$, where $D\subset {\mathbb{R}}^N(N\ge 2)$ is a bounded domain with smooth boundary $\partial D$. Under some appropriate assumption on the functions $f$, $h$, $k$, $b$, and $a$ and initial value $u_0$, we obtain the sufficient conditions for the existence of a global solution, an upper estimate of the global solution, the sufficient conditions for the existence of a blow-up solution, an upper bound for “blow-up time,” and an upper estimate of “blow-up rate.” Our approach depends heavily on the maximum principles.

A class of Lotka-Volterra mutualistic system with time delays of benefit and feedback delays is introduced. By analyzing the associated characteristic equation, the local stability of the positive equilibrium and existence of Hopf bifurcation are obtained under all possible combinations of two or three delays selecting from multiple delays. Not only explicit formulas to determine the properties of the Hopf bifurcation are shown by using the normal form method and center manifold theorem, but also the global continuation of Hopf bifurcation is investigated by applying a global Hopf bifurcation result due to Wu (1998). Numerical simulations are given to support the theoretical results.

An ecoepidemiological predator-prey model with stage structure for the predator and time delay due to the gestation of the predator is investigated. The effects of a prey refuge with disease in the prey population are concerned. By analyzing the corresponding characteristic equations, the local stability of each of the feasible equilibria of the model is discussed. Further, it is proved that the model undergoes a Hopf bifurcation at the positive equilibrium. By means of appropriate Lyapunov functions and LaSalle’s invariance principle, sufficient conditions are obtained for the global stability of the semitrivial boundary equilibria. By using an iteration technique, sufficient conditions are derived for the global attractiveness of the positive equilibrium.

We address the question of the long-term coexistence of three competing species whose dynamics are governed by the partial differential equations. We obtain criteria for permanent coexistence in a Lotka-Volterra system modeling the interaction of three competing species in a bounded habitat whose exterior is lethal to each species. It is also proved that if the intercompeting strength is very weak, the system is always permanent, provided that each single one of the three species can survive in the absence of the two other species.

A mathematical model of human T-cell lymphotropic virus type 1 in vivo with cell-to-cell infection and mitosis is formulated and studied. The basic reproductive number $R_0$ is derived. It is proved that the dynamics of the model can be determined completely by the magnitude of $R_0$. The infection-free equilibrium is globally asymptotically stable (unstable) if $(R_0>1)$. There exists a chronic infection equilibrium and it is globally asymptotically stable if $R_0>1$.

The dual Ito equation can be seen as a two-component generalization of the well-known Camassa-Holm equation. By using the theory of planar dynamical system, we study the existence of its traveling wave solutions. We find that the dual Ito equation has smooth solitary wave solutions, smooth periodic wave solutions, and periodic cusp solutions. Parameter conditions are given to guarantee the existence.

The delay-dependent stochastic stability problem of Markovian jump systems with time-varying delays is investigated in this paper. Though the Lyapunov-Krasovskii functional is general and simple, less conservative results are derived by using the convex combination method, improved Wirtinger’s integral inequality, and a slack condition on Lyapunov matrix. The obtained results are formulated in terms of linear matrix inequalities (LMIs). Numerical examples are provided to verify the effectiveness and superiority of the presented results.

In order to obtain a more accurate description of the ecological system perturbed by human exploitation activities such as planting and harvesting, we need to consider the impulsive differential equations. Therefore, by applying the comparison theorem and the Lyapunov method of the impulsive differential equations, this paper gives some new sufficient conditions for the permanence and existence of a unique uniformly asymptotically stable positive almost periodic solution in a food chain system with almost periodic impulsive perturbations. The method used in this paper provides a possible method to study the permanence and existence of a unique uniformly asymptotically stable positive almost periodic solution of the models with impulsive perturbations in biological populations. Finally, an example and numerical simulations are given to illustrate that our results are feasible.

A class of planar cubic Kolmogorov systems with harvest and two positive equilibrium points is investigated. With the help of computer algebra system MATHEMATICA, we prove that five limit cycles can be bifurcated simultaneously from the two critical points (1, 1) and (2, 2), respectively, in the first quadrant. Moreover, the necessary conditions of centers are obtained.

We study the expansions of the first order Melnikov functions for general near-Hamiltonian systems near a compound loop with a cusp and a nilpotent saddle. We also obtain formulas for the first coefficients appearing in the expansions and then establish a bifurcation theorem on the number of limit cycles. As an application example, we give a lower bound of the maximal number of limit cycles for a polynomial system of Liénard type.

We establish the regularity criteria for the two-dimensional generalized liquid crystal model. It turns out that the global existence results satisfy our regularity criteria naturally.

A FitzHugh-Nagumo (FHN) neural system with multiple delays has been proposed. The number of equilibrium point is analyzed. It implies that the neural system exhibits a unique equilibrium and three ones for the different values of coupling weight by employing the saddle-node bifurcation of nontrivial equilibrium point and transcritical bifurcation of trivial one. Further, the stability of equilibrium point is studied by analyzing the corresponding characteristic equation. Some stability criteria involving the multiple delays and coupling weight are obtained. The results show that the neural system exhibits the delay-independence and delay-dependence stability. Increasing delay induces the stability switching between resting state and periodic activity in some parameter regions of coupling weight. Finally, numerical simulations are taken to support the theoretical results.

This paper deals with a Neumann boundary value problem for a volume-filling chemotaxis model with logistic growth in a $d$-dimensional box ${\mathbb{T}}^d=(0,\pi {)}^d(d=\mathrm{1,2},3)$. It is proved that given any general perturbation of magnitude $\delta$, its nonlinear evolution is dominated by the corresponding linear dynamics along a finite number of fixed fastest growing modes, over a time period of the order $\ln (\mathrm{1}/\delta )$. Each initial perturbation certainly can behave drastically different from another, which gives rise to the richness of patterns.

This paper gives a new version of Gronwall’s inequality on time scales. The method used in the proof is much different from that in the literature. Finally, an application is presented to show the feasibility of the obtained Gronwall’s inequality.

The impulsive control and synchronization of unified chaotic system are proposed. By applying impulsive control theory and introducing a piecewise continuous auxiliary function, some novel and useful conditions are provided to guarantee the globally asymptotical stabilization and synchronization of unified chaotic system under impulsive control. Compared with some previous results, our criteria are superior and less conservative. Finally, the effectiveness of theoretical results is shown through numerical simulations.

Vaccination is one of the most effective measures for suppressing the spread of computer virus, and the bilinear incidence rate assumption for the majority of previous models, which is a good first approximation of the general incidence rate, is in disagreement with the reality. In this paper, a new dynamical model with two kinds of generic nonlinear probabilities (incidence rate and vaccination probability) is established. An exhaustive mathematical analysis of this model shows that (a) there are two equilibria, virus-free equilibrium and viral equilibrium, and (b) the virus-free (or viral) equilibrium is globally asymptotically stable when the basic reproduction number is less (or greater) than unity. The analysis of the basic reproduction number is also included. Additionally, some numerical examples are given to illustrate the main results, from which it can be seen that the generic nonlinear vaccination is helpful to strengthen computer security.

We use bifurcation method of dynamical systems to study exact traveling wave solutions of a nonlinear evolution equation. We obtain exact explicit expressions of bell-shaped solitary wave solutions involving more free parameters, and some existing results are corrected and improved. Also, we get some new exact periodic wave solutions in parameter forms of the Jacobian elliptic function. Further, we find that the bell-shaped waves are limits of the periodic waves in some sense. The results imply that we can deduce bell-shaped waves from periodic waves for some nonlinear evolution equations.

Rumor has no basis in fact and flies around. And in general, it is propagated for a certain motivation, either for business, economy, or pleasure. It is found that the web does expose us to more rumor and increase the speed of the rumors spread. Corresponding to these new ways of spreading, the government should carry out some measures, such as issuing message by media, punishing the principal spreader, and enhancing management of the internet. In order to assess these measures, dynamical models without and with control measures are established. Firstly, for two models, equilibria and the basic reproduction number of models are discussed. More importantly, numerical simulation is implemented to assess control measures of rumor spread between individuals-to-individuals and medium-to-individuals. Finally, it is found that the amount of message released by government has the greatest influence on the rumor spread. The reliability of government and the cognizance ability of the public are more important. Besides that, monitoring the internet to prevent the spread of rumor is more important than deleting messages in media which already existed. Moreover, when the minority of people are punished, the control effect is obvious.

We consider the sufficient conditions for asymptotic stability and instability of certain higher order nonlinear difference equations with infinite delays in finite-dimensional spaces. With the aid of the general comparison condition on the right-hand side function $f_k(·)$, we generalize the stability and instability result.

We are concerned with the following nonlinear three-point fractional boundary value problem: $D_{0+}^{\alpha }u\left(t\right)+\lambda a\left(t\right)f\left(t,u\left(t\right)\right)=0$, , $u\left(0\right)=0$, and $u\left(1\right)=\beta u\left(\eta \right)$, where , , , $D_{0+}^{\alpha }$ is the standard Riemann-Liouville fractional derivative, $a\left(t\right)>0$ is continuous for $0\le t\le 1$, and $f\ge 0$ is continuous on $\left[\mathrm{0,1}\right]\times \left[0,\left.\infty \right)\right.$. By using Krasnoesel'skii's fixed-point theorem and the corresponding Green function, we obtain some results for the existence of positive solutions. At the end of this paper, we give an example to illustrate our main results.

We investigate an n-species food chain model with spatial diffusion and time delays. By using Schauder’s fixed point theorem, we obtain the result about the existence of the travelling wave solutions of the food chain model with reaction term satisfying the partial quasimonotonicity conditions.

This paper is concerned with the existence of quasiperiodic solutions with two frequencies of completely resonant, quasiperiodically forced nonlinear wave equations subject to periodic spatial boundary conditions. The solutions turn out to be, at the first order, the superposition of traveling waves, traveling in the opposite or the same directions. The proofs are based on the variational Lyapunov-Schmidt reduction and the linking theorem, while the bifurcation equations are solved by variational methods.

The purpose of this paper is to obtain some sufficient conditions for the global existence of multiple positive periodic solutions of a delayed stage-structured plant-hare model with a toxin-determined functional response. Some novel estimation techniques to construct two open subsets for a priori bounds are employed.

We investigate integrodifferential functional differential equations $\ddot{x}+f(t,x,\dot{x})\dot{x}+{\int }_{t-r(t)}^t\mathrm{‍}a(t,s)g(x(s))ds=0$ with variable delay. By using the fixed point theory, we obtain conditions which ensure that the zero solution of this equation is stable under an exponentially weighted metric. Then we establish necessary and sufficient conditions ensuring that the zero solution is asymptotically stable. We will give an example to apply our results.

This work is concerned with the quadratic-mean asymptotically almost periodic mild solutions for a class of stochastic functional differential equations $\text{d}x\left(t\right)=\left[A\left(t\right)x\left(t\right)+F\left(t,x\left(t\right),x_t\right)\right]\text{d}t+H(t,x\left(t\right),x_t)\circ \text{d}W(t)$. A new criterion ensuring the existence and uniqueness of the quadratic-mean asymptotically almost periodic mild solutions for the system is presented. The condition of being uniformly exponentially stable of the strongly continuous semigroup ${\{T\left(t\right)\}}_{t\ge 0}$ is essentially removed, which is generated by the linear densely defined operator $A:D(A)\subset L^2(\mathbb{P},ℍ)\to L^2(\mathbb{P},ℍ)$, only using the exponential trichotomy of the system, which reflects a deeper analysis of the behavior of solutions of the system. In this case the asymptotic behavior is described through the splitting of the main space into stable, unstable, and central subspaces at each point from the flow’s domain. An example is also given to illustrate our results.

We investigate the long time behavior of the damped, forced KdV-BO equation driven by white noise. We first show that the global solution generates a random dynamical system. By energy type estimates and dispersive properties, we then prove that this system possesses a weak random attractor in the space $H^1(ℝ)$.

A virus dynamics model with logistic function, general incidence function, and cure rate is considered. By carrying out mathematical analysis, we show that the infection-free equilibrium is globally asymptotically stable if the basic reproduction number ${ℛ}_0\le 1$. If ${ℛ}_0>1$, then the infection equilibrium is globally asymptotically stable under some assumptions. Furthermore, we also obtain the conditions for which the model exists an orbitally asymptotically stable periodic solution. Examples are provided to support our analytical conclusions.

We study a class of quasi symmetric seventh degree systems and obtain the conditions that its two singular points can be two centers at the same step by careful computing and strict proof. In addition, the condition of an isochronous center is also given. In terms of quasi symmetric systems, our work is interesting and obtained conclusions about bicenters are new.

We drive a scalar delay differential system to model the congestion of a wireless access network setting. The Hopf bifurcation of this system is investigated using the control and bifurcation theory; it is proved that there exists a critical value of delay for the stability. When the delay value passes through the critical value, the system loses its stability and a Hopf bifurcation occurs. Furthermore, the direction and stability of the bifurcating periodic solutions are derived by applying the normal form theory and the center manifold theorem. Finally, some examples and numerical simulations are presented to show the feasibility of the theoretical results.

Information plays an important role in modern society. In this paper, we presented a mathematical model of information spreading with isolation. It was found that such a model has rich dynamics including Hopf bifurcation. The results showed that, for a wide range of parameters, there is a bistable phenomenon in the process of information spreading and thus the information cannot be well controlled. Moreover, the model has a limit cycle which implies that the information exhibits periodic outbreak which is consistent with the observations in the real world.

We develop a nonlinear mathematical model with the effect of awareness programs on the binge drinking. Due to the fact that awareness programs are capable of inducing behavioral changes in nondrinkers, we introduce a separate class by avoiding contacts with the heavy drinkers. Furthermore we assume that cumulative density of awareness programs increases at a rate proportional to the number of heavy drinkers. We establish some sufficient conditions for the stability of the alcohol free and the alcohol present equilibria and give some numerical simulations to explain our main result. Our results show that awareness programs is an effective measure in reducing alcohol problems.

The main purpose of this paper is to study the periodicity and global asymptotic stability of a generalized Lotka-Volterra’s competition system with delays. Some sufficient conditions are established for the existence and stability of periodic solution of such nonlinear differential equations. The approaches are based on Mawhin’s coincidence degree theory, matrix spectral theory, and Lyapunov functional.

A special predator-prey system is investigated in which the prey population exhibits herd behavior in order to provide a self-defense against predators, while the predator is intermediate and its population shows individualistic behavior. Considering the fact that there always exists a time delay in the conversion of the biomass of prey to that of predator in this system, we obtain a delayed predator-prey model with square root functional response and quadratic mortality. For this model, we mainly investigate the stability of positive equilibrium and the existence of Hopf bifurcation by choosing the time delay as a bifurcation parameter.

An integral-differential model equation arising from neuronal networks with very general kernel functions is considered in this paper. The kernel functions we study here include pure excitations, lateral inhibition, lateral excitations, and more general synaptic couplings (e.g., oscillating kernel functions). The main goal of this paper is to prove the existence and uniqueness of the traveling wave front solutions. The main idea we apply here is to reduce the nonlinear integral-differential equation into a solvable differential equation and test whether the solution we get is really a wave front solution of the model equation.

We consider the nonlinear dynamics in a double-chain model of DNA which consists of two long elastic homogeneous strands connected with each other by an elastic membrane. By using the method of dynamical systems, the bounded traveling wave solutions such as bell-shaped solitary waves and periodic waves for the coupled nonlinear dynamical equations of DNA model are obtained and simulated numerically. For the same wave speed, bell-shaped solitary waves of different heights are found to coexist.

Sufficient conditions are obtained for the global attractivity of the following integrodifferential model of mutualism: $d{N}_{\mathrm{1}}\mathrm{}(t)/dt=r_{\mathrm{1}}{N}_{\mathrm{1}}(t)[((K_{\mathrm{1}}+{\alpha }_{\mathrm{1}}\mathrm{}{\int }_{\mathrm{0}}^{\mathrm{\infty }}{J}_{\mathrm{2}}\mathrm{}(s)N_{\mathrm{2}}\mathrm{}(t-s)ds)\mathrm{‍}/(\mathrm{1}+{\int }_{\mathrm{0}}^{\mathrm{\infty }}{J}_{\mathrm{2}}\mathrm{}(s)N_{\mathrm{2}}\mathrm{}(t-s)ds))\mathrm{‍}-N_{\mathrm{1}}(t)]$, $d{N}_{\mathrm{2}}\mathrm{}(t)/dt=r_{\mathrm{2}}{N}_{\mathrm{2}}(t)[((K_{\mathrm{2}}+{\alpha }_{\mathrm{2}}\mathrm{}{\int }_{\mathrm{0}}^{\mathrm{\infty }}\mathrm{}{J}_{\mathrm{1}}(s)N_{\mathrm{1}}\mathrm{}(t-s)ds)\mathrm{‍}/(\mathrm{1}+{\int }_{\mathrm{0}}^{\mathrm{\infty }}{J}_{\mathrm{1}}\mathrm{}(s)N_{\mathrm{1}}\mathrm{}(t-s)ds\mathrm{‍}))-N_{\mathrm{2}}(t)]$, where $r_i,K_i,$ and ${\alpha }_i$, $i=\mathrm{1,2}$, are all positive constants. Consider ${\alpha }_i>K_i$, $i=\mathrm{1,2}.$ Consider $J_i\in C([\mathrm{0},+\mathrm{\infty }),[\mathrm{0},+\mathrm{\infty }))$ and ${\int }_{\mathrm{0}}^{\mathrm{\infty }}\mathrm{‍}{J}_i(s)ds=\mathrm{1}$, $i=\mathrm{1,2}.$ Our result shows that conditions which ensure the permanence of the system are enough to ensure the global stability of the system. The result not only improves but also complements some existing ones.