ISSN:

1531-3492

eISSN:

1553-524X

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### Volume 10, 2008

## Discrete & Continuous Dynamical Systems - B

March 2017 , Volume 22 , Issue 2

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*+*[Abstract](317)

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**Abstract:**

The synchronization of stochastic differential equations (SDEs) with additive noise is investigated in pathwise sense, moreover convergence rate of synchronization is obtained. The optimality of the convergence rate is illustrated through examples.

*+*[Abstract](514)

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**Abstract:**

This paper is concerned with the spreading or vanishing of a epidemic disease which is characterized by a diffusion SIS model with nonlocal incidence rate and double free boundaries. We get the full information about the sufficient conditions that ensure the disease spreading or vanishing, which exhibits a detailed description of the communicable mechanism of the disease. Our results imply that the nonlocal interaction may enhance the spread of the disease.

*+*[Abstract](342)

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**Abstract:**

The paper concerns the study and applications of a new class of optimal control problems governed by a perturbed sweeping process of the hysteresis type with control functions acting in both play-and-stop operator and additive perturbations. Such control problems can be reduced to optimization of discontinuous and unbounded differential inclusions with pointwise state constraints, which are immensely challenging in control theory and prevent employing conventional variation techniques to derive necessary optimality conditions. We develop the method of discrete approximations married with appropriate generalized differential tools of modern variational analysis to overcome principal difficulties in passing to the limit from optimality conditions for finite-difference systems. This approach leads us to nondegenerate necessary conditions for local minimizers of the controlled sweeping process expressed entirely via the problem data. Besides illustrative examples, we apply the obtained results to an optimal control problem associated with of the crowd motion model of traffic flow in a corridor, which is formulated in this paper. The derived optimality conditions allow us to develop an effective procedure to solve this problem in a general setting and completely calculate optimal solutions in particular situations.

*+*[Abstract](353)

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**Abstract:**

This paper investigates the existence of a uniform in time

*+*[Abstract](612)

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**Abstract:**

In this paper, we consider an initial boundary value problem for nonlocal-in-time parabolic equations involving a nonlocal in time derivative. We first show the uniqueness and existence of the weak solution of the nonlocal-in-time parabolic equation, and also the spatial smoothing properties. Moreover, we develop a new framework to study the local limit of the nonlocal model as the horizon parameter *δ* approaches 0. Exploiting the spatial smoothing properties, we develop a semi-discrete scheme using standard Galerkin finite element method for the spatial discretization, and derive error estimates dependent on data smoothness. Finally, extensive numerical results are presented to illustrate our theoretical findings.

*+*[Abstract](333)

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**Abstract:**

We derive a sufficient condition for stability in probability of an equilibrium of a randomly perturbed map in

*+*[Abstract](307)

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**Abstract:**

We introduce a new model to predict the spread of an epidemic, focusing on the contamination process and simulating the disease propagation by the means of a unique function viewed as a measure of the local infective energy. The model is intended to illustrate a map of the epidemic spread and not to compute the densities of various populations related to an epidemic, as in the classical models. First, the model is constructed as a cellular automaton exhibiting a self-organizing-type criticality process with two thresholds. This induces the consideration of an associate continuous model described by a nonlinear equation with two singularities, for whose solution we prove existence, uniqueness and certain properties. We provide numerical simulations to put into evidence the effect of some model parameters in various scenarios of the epidemic spread.

*+*[Abstract](351)

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**Abstract:**

In this paper, we consider a particular type of nonlinear McKend-rick-von Foerster equation with a diffusion term and Robin boundary condition. We prove the existence of a global solution to this equation. The steady state solutions to the equations that we consider have a very important role to play in the study of long time behavior of the solution. Therefore we address the issues pertaining to the existence of solution to the corresponding state equation. Furthermore, we establish that the solution of McKendrick-von Foerster equation with diffusion converges pointwise to the solution of its steady state equations as time tends to infinity.

*+*[Abstract](279)

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**Abstract:**

We extend the taming techniques developed in [

*+*[Abstract](310)

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**Abstract:**

We consider the chemotaxis-growth system

under no-flux boundary conditions, in a convex bounded domain

It is shown that under the assumption that

for any given nonnegative

any bounded classical solution constructed above stabilizes to the constant stationary solution

*+*[Abstract](611)

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**Abstract:**

We study the periodic solutions of the second-order differential equations of the form

*+*[Abstract](277)

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**Abstract:**

Recently, Gu et al. [

*+*[Abstract](441)

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**Abstract:**

In this paper, we investigate the nonlinear wave equation in a bounded domain with a time-varying delay term in the weakly nonlinear internal feedback

The asymptotic behavior of solutions is studied by using an appropriate Lyapunov functional. Moreover, we extend and improve the previous results in the literature.

*+*[Abstract](286)

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**Abstract:**

We construct Lyapunov functionals for delay differential equation models of infectious diseases in vivo to analyze the stability of the equilibria. The Lyapunov functionals contain the terms that integrate over all previous states. An appropriate evaluation of the logarithm functions in those terms guarantees the existence of the integrals. We apply the rigorous analysis for the one-strain models to multistrain models by using mathematical induction.

*+*[Abstract](298)

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**Abstract:**

The free boundary problem of planar full compressible magnetohydrodynamic equations with large initial data is studied in this paper, when the initial density connects to vacuum smoothly. The global existence and uniqueness of classical solutions are established, and the expanding rate of the free interface is shown. Using the method of Lagrangian particle path, we derive some *L*^{∞} estimates and weighted energy estimates, which lead to the global existence of classical solutions. The main difficulty of this problem is the degeneracy of the system near the free boundary, while previous results (cf. [

*+*[Abstract](272)

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**Abstract:**

In this paper, we study the effect of small Brownian noise on a switching dynamical system which models a first-order DC/DC buck converter. The state vector of this system comprises a continuous component whose dynamics switch, based on the ON/OFF configuration of the circuit, between two ordinary differential equations (ODE), and a discrete component which keeps track of the ON/OFF configurations. Assuming that the parameters and initial conditions of the unperturbed system have been tuned to yield a stable periodic orbit, we study the stochastic dynamics of this system when the forcing input in the ON state is subject to small white noise fluctuations of size

*+*[Abstract](302)

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**Abstract:**

We offer two improvements to prior results concerning global stability of the 2D Ricker Equation. We also offer some new methods of approach for the more pathological cases and prove some miscellaneous results including a special nontrivial case in which the mapping is conjugate to the 1D Ricker map along an invariant line and a proof of the non-existence of period-2 points.

*+*[Abstract](390)

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**Abstract:**

The current paper is devoted to the ergodicity of stochastic coupled fractional Ginzburg-Landau equations driven by

*+*[Abstract](319)

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**Abstract:**

This paper looks into the stability of equilibria, existence and non-existence of traveling wave solutions in a diffusive producer-scrounger model. We find that the existence and non-existence of traveling wave solutions are determined by a minimum wave speed

*+*[Abstract](911)

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**Abstract:**

A delayed reaction-diffusion Schnakenberg system with Neumann boundary conditions is considered in the context of long range biological self-organisation dynamics incorporating gene expression delays. We perform a detailed stability and Hopf bifurcation analysis and derive conditions for determining the direction of bifurcation and the stability of the bifurcating periodic solution. The delay-diffusion driven instability of the unique spatially homogeneous steady state solution and the diffusion-driven instability of the spatially homogeneous periodic solution are investigated, with limited simulations to support our theoretical analysis. These studies analytically demonstrate that the modelling of gene expression time delays in Turing systems can eliminate or disrupt the formation of a stationary heterogeneous pattern in the Schnakenberg system.

*+*[Abstract](271)

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**Abstract:**

The Neumann boundary value problem for the chemotaxis system generalizing the prototype

is considered in a smooth bounded convex domain

If

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