ISSN:

1531-3492

eISSN:

1553-524X

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## Discrete & Continuous Dynamical Systems - B

January 2020 , Volume 25 , Issue 1

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**Abstract:**

Scaling transformations involving a small parameter (*degenerate scalings*) are frequently used for ordinary differential equations that model chemical reaction networks. They are motivated by quasi-steady state (QSS) of certain chemical species, and ideally lead to slow-fast systems for singular perturbation reductions, in the sense of Tikhonov and Fenichel. In the present paper we discuss properties of such scaling transformations, with regard to their applicability as well as to their determination. Transformations of this type are admissible only when certain consistency conditions are satisfied, and they lead to singular perturbation scenarios only if additional conditions hold, including a further consistency condition on initial values. Given these consistency conditions, two scenarios occur. The first (which we call standard) is well known and corresponds to a classical quasi-steady state (QSS) reduction. Here, scaling may actually be omitted because there exists a singular perturbation reduction for the unscaled system, with a coordinate subspace as critical manifold. For the second (nonstandard) scenario scaling is crucial. Here one may obtain a singular perturbation reduction with the slow manifold having dimension greater than expected from the scaling. For parameter dependent systems we consider the problem to find all possible scalings, and we show that requiring the consistency conditions allows their determination. This lays the groundwork for algorithmic approaches, to be taken up in future work. In the final section we consider some applications. In particular we discuss relevant nonstandard reductions of certain reaction-transport systems.

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**Abstract:**

In this paper, we establish an abstract infinite dimensional KAM theorem. As an application, we use the theorem to study the higher dimensional beam equation system

under periodic boundary conditions, where

*+*[Abstract](694)

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**Abstract:**

The main objective of this paper is to study the long-time behavior of solutions for the three dimensional non-autonomous Navier-Stokes equations with nonlinear damping for

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**Abstract:**

In the recent paper [

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**Abstract:**

The global behaviour of the compact pairwise approximation of SIS epidemic propagation on networks is studied. It is shown that the system can be reduced to two equations enabling us to carry out a detailed study of the dynamic properties of the solutions. It is proved that transcritical bifurcation occurs in the system at

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**Abstract:**

Stemming from the stochastic Lotka-Volterra or predator-prey equations, this work aims to model the spatial inhomogeneity by using stochastic partial differential equations (SPDEs). Compared to the classical models, the SPDE models are more versatile. To incorporate more qualitative features of the ratio-dependent models, the Beddington-DeAngelis functional response is also used. To analyze the systems under consideration, first existence and uniqueness of solutions of the SPDEs are obtained using the notion of mild solutions. Then sufficient conditions for permanence and extinction are derived.

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**Abstract:**

The systems of reaction-diffusion equations coupled with moving boundaries defined by Stefan condition have been widely used to describe the dynamics of spreading population. There are several numerical difficulties to efficiently handle such systems. Firstly extremely small time steps are usually demanded due to the stiffness of the system. Secondly it is always difficult to efficiently and accurately handle the moving boundaries. To overcome these difficulties, we first transform the one-dimensional problem with a moving boundary into a system with a fixed computational domain, and then introduce four different temporal schemes: Runge-Kutta, Crank-Nicolson, implicit integration factor (IIF) and Krylov IIF for handling such stiff systems. Numerical examples are examined to illustrate the efficiency, accuracy and consistency for different approaches, and it can be shown that Krylov IIF is superior to other three approaches in terms of stability and efficiency by direct comparison.

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**Abstract:**

In this paper, we present some controllability results for the heat equation in the framework of hierarchic control. We present a Stackelberg strategy combining the concept of controllability with robustness: the main control (the leader) is in charge of a null-controllability objective while a secondary control (the follower) solves a robust control problem, this is, we look for an optimal control in the presence of the worst disturbance. We improve previous results by considering that either the leader or follower control acts on a small part of the boundary. We also present a discussion about the possibility and limitations of placing all the involved controls on the boundary.

*+*[Abstract](653)

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**Abstract:**

This paper deals with the two-species chemotaxis system with two chemicals

under homogeneous Neumann boundary conditions in a bounded domain

Moreover, by constructing suitable energy functions, it is shown that:

*+*[Abstract](602)

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**Abstract:**

The paper investigates the existence of global attractors for a few classes of multi-valued operators. We establish some criteria and give their applications to a strongly damped wave equation with fully supercritical nonlinearities and without the uniqueness of solutions. Moreover, the geometrical structure of the global attractors of the corresponding multi-valued operators is shown.

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**Abstract:**

The stochasticity of transcription can be quantified by mRNA distribution

*+*[Abstract](643)

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This paper is devoted to the mathematical analysis of the diffraction of an electromagnetic plane wave by a biperiodic structure. The wave propagation is governed by the time-domain Maxwell equations in three dimensions. The method of a compressed coordinate transformation is proposed to reduce equivalently the diffraction problem into an initial-boundary value problem formulated in a bounded domain over a finite time interval. The reduced problem is shown to have a unique weak solution by using the constructive Galerkin method. The stability and a priori estimates with explicit time dependence are established for the weak solution.

*+*[Abstract](561)

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**Abstract:**

This paper considers stochastic functional differential equations (SFDEs) with infinite delay. The main aim is to establish the LaSalle-type theorems to locate limit sets for this class of SFDEs. In comparison with the existing results, this paper gives more general results under the weaker conditions imposed on the Lyapunov function. These results can be used to discuss the asymptotic stability and asymptotic boundedness for SFDEs with infinite delay. In the end, two examples will be given to illustrate applications of our new results established.

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**Abstract:**

We show how recent existence results for pullback exponential attractors can be applied to non-autonomous delay differential equations with time-varying delays. Moreover, we derive explicit estimates for the fractal dimension of the attractors.

As a special case, autonomous delay differential equations are also discussed, where our results improve previously obtained bounds for the fractal dimension of exponential attractors.

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**Abstract:**

In this work, the time fractional KdV equation with Caputo time derivative of order

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**Abstract:**

We focus on the asymptotic behavior of strongly anisotropic parabolic problems. We concentrate on heat equations, whose diffusion matrix fields have disparate eigenvalues. We establish strong convergence results toward a profile. Under suitable smoothness hypotheses, by introducing an appropriate corrector term, we estimate the convergence rate. The arguments rely on two-scale analysis, based on average operators with respect to unitary groups.

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**Abstract:**

In this paper we investigate the bi-center problem and the total Hopf cyclicity of two center-foci for the general cubic Liénard system which has three distinct equilibria and is equivalent to the general Liénard equation with cubic damping and restoring force. The location of these three equilibria is arbitrary, specially without any kind of symmetry. We find the necessary and sufficient condition for the existence of bi-centers and prove that there is no case of a unique center. On the Hopf cyclicity we prove that there are totally

*+*[Abstract](1782)

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**Abstract:**

The multiframe super-resolution (SR) techniques are considered as one of the active research fields. More precisely, the construction of the desired high resolution (HR) image with less artifacts in the SR models, which are always ill-posed problems, requires a great care. In this paper, we propose a new fourth-order equation based on a diffusive tensor that takes the benefit from the diffusion model of Perona-Malik in the flat regions and the Weickert model near boundaries with a high diffusion order. As a result, the proposed SR approach can efficiently preserve image features such as corner and texture much better with less blur near edges. The existence and uniqueness of the proposed partial differential equation (PDE) are also demonstrated in an appropriate functional space. Finally, the given experimental results show the effectiveness of the proposed PDE compared to some competitive methods in both visually and quantitatively.

*+*[Abstract](413)

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**Abstract:**

In this paper we propose and study a hybrid *discrete–continuous* mathematical model of collective motion under alignment and chemotaxis effect. Starting from paper [*t* → +∞, the migrating aggregate exponentially converges to a state in which all the particles have a same position with zero velocity. Finally, we present a comparison between the analytical findings and some numerical results, concerning the behaviour of the full nonlinear system.

*+*[Abstract](439)

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**Abstract:**

This paper is concerned with the forward dynamical behavior of nonautonomous systems. Under some general conditions, it is shown that in an arbitrary small neighborhood of a pullback attractor of a nonautonomous system, there exists a family of sets

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