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Discrete & Continuous Dynamical Systems - B

January 2020 , Volume 25 , Issue 1

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Singular perturbations and scaling
Christian Lax and Sebastian Walcher
2020, 25(1): 1-29 doi: 10.3934/dcdsb.2019170 +[Abstract](382) +[HTML](197) +[PDF](509.58KB)

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.

Quasi-periodic solutions for a class of beam equation system
Yanling Shi and Junxiang Xu
2020, 25(1): 31-53 doi: 10.3934/dcdsb.2019171 +[Abstract](439) +[HTML](231) +[PDF](537.72KB)

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 \begin{document}$ 0<\sigma \in [ \sigma_1,\sigma_2 ], $\end{document} \begin{document}$ 0<\mu\in [ \mu_1,\mu_2 ] $\end{document} are real parameters. By establishing a block-diagonal normal form, we obtain the existence of a Whitney smooth family of small amplitude quasi-periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional dynamic system.

Pullback exponential attractors for the three dimensional non-autonomous Navier-Stokes equations with nonlinear damping
Fang Li and Bo You
2020, 25(1): 55-80 doi: 10.3934/dcdsb.2019172 +[Abstract](345) +[HTML](163) +[PDF](502.7KB)

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 \begin{document}$ r>4. $\end{document} Inspired by the the methods of \begin{document}$ \ell $\end{document}-trajectories in [27], we will prove the existence of a finite dimensional pullback attractor and a pullback exponential attractor, which gives another way of considering the long-time behavior of the non-autonomous evolutionary equations.

Qualitative analysis on an SIS epidemic reaction-diffusion model with mass action infection mechanism and spontaneous infection in a heterogeneous environment
Chengxia Lei, Jie Xiong and Xinhui Zhou
2020, 25(1): 81-98 doi: 10.3934/dcdsb.2019173 +[Abstract](372) +[HTML](200) +[PDF](429.14KB)

In the recent paper [29], a susceptible-infected-susceptible (SIS) epidemic reaction-diffusion model with a mass action infection mechanism and linear birth-death growth with no flux boundary condition was studied. It has been recognized that spontaneous infection is an important factor in disease epidemics, in addition to disease transmission [43]. In this paper, we investigate the SIS model in [29] with spontaneous infection. We establish the global boundedness and uniform persistence in the general heterogeneous environment, and derive the global stability of the unique constant endemic equilibrium in the homogeneous environment case. Moreover, we analyze the asymptotic behavior of the endemic equilibrium when the movement (migration) rate of the susceptible or infected population tends to zero. Compared to the case that there is no spontaneous infection, our study suggests that spontaneous infection can enhance persistence of infectious disease, and hence the disease becomes more threatening.

Detailed analytic study of the compact pairwise model for SIS epidemic propagation on networks
Noémi Nagy and Péter L. Simon
2020, 25(1): 99-115 doi: 10.3934/dcdsb.2019174 +[Abstract](264) +[HTML](144) +[PDF](489.47KB)

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 \begin{document}$ \tau = \tau _c = \frac{\gamma n}{\langle n^{2}\rangle-n} $\end{document}, where \begin{document}$ \tau $\end{document} and \begin{document}$ \gamma $\end{document} are infection and recovery rates, respectively, \begin{document}$ n $\end{document} is the average degree of the network and \begin{document}$ \langle n^{2}\rangle $\end{document} is the second moment of the degree distribution. For subcritical values of \begin{document}$ \tau $\end{document} the disease-free steady state is stable, while for supercritical values a unique stable endemic equilibrium appears. We also prove that for subcritical values of \begin{document}$ \tau $\end{document} the disease-free steady state is globally stable under certain assumptions on the graph that cover a wide class of networks.

Stochastic partial differential equation models for spatially dependent predator-prey equations
Nhu N. Nguyen and George Yin
2020, 25(1): 117-139 doi: 10.3934/dcdsb.2019175 +[Abstract](374) +[HTML](213) +[PDF](495.74KB)

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.

Krylov implicit integration factor method for a class of stiff reaction-diffusion systems with moving boundaries
Shuang Liu and Xinfeng Liu
2020, 25(1): 141-159 doi: 10.3934/dcdsb.2019176 +[Abstract](349) +[HTML](186) +[PDF](526.55KB)

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.

Some remarks on the Robust Stackelberg controllability for the heat equation with controls on the boundary
Víctor Hernández-Santamaría and Liliana Peralta
2020, 25(1): 161-190 doi: 10.3934/dcdsb.2019177 +[Abstract](291) +[HTML](167) +[PDF](616.26KB)

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.

Boundedness and stabilization in a two-species chemotaxis system with two chemicals
Liangchen Wang, Jing Zhang, Chunlai Mu and Xuegang Hu
2020, 25(1): 191-221 doi: 10.3934/dcdsb.2019178 +[Abstract](367) +[HTML](191) +[PDF](627.66KB)

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

under homogeneous Neumann boundary conditions in a bounded domain \begin{document}$ \Omega\subset \mathbb{R}^n $\end{document} (\begin{document}$ n\geq1 $\end{document}), where the parameters \begin{document}$ d_1,d_2,d_3,d_4>0 $\end{document}, \begin{document}$ \mu_1,\mu_2>0 $\end{document}, \begin{document}$ a_1,a_2>0 $\end{document} and \begin{document}$ \alpha, \beta>0 $\end{document}. The chemotactic function \begin{document}$ \chi_i $\end{document} (\begin{document}$ i = 1,2 $\end{document}) and the signal production function \begin{document}$ f_i $\end{document} (\begin{document}$ i = 1,2 $\end{document}) are smooth. If \begin{document}$ n = 2 $\end{document}, it is shown that this system possesses a unique global bounded classical solution provided that \begin{document}$ |\chi'_i| $\end{document} (\begin{document}$ i = 1,2 $\end{document}) are bounded. If \begin{document}$ n\leq3 $\end{document}, this system possesses a unique global bounded classical solution provided that \begin{document}$ \mu_i $\end{document} (\begin{document}$ i = 1,2 $\end{document}) are sufficiently large. Specifically, we first obtain an explicit formula \begin{document}$ \mu_{i0}>0 $\end{document} such that this system has no blow-up whenever \begin{document}$ \mu_i>\mu_{i0} $\end{document}.

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

\begin{document}$ \bullet $\end{document} If \begin{document}$ a_1,a_2\in(0,1) $\end{document} and \begin{document}$ \mu_1 $\end{document} and \begin{document}$ \mu_2 $\end{document} are sufficiently large, then any global bounded solution exponentially converges to \begin{document}$\bigg(\frac{1-a_1}{1-a_1a_2},f_1(\frac{1-a_2}{1-a_1a_2})/\alpha,\frac{1-a_2}{1-a_1a_2},$\end{document}\begin{document}$ f_2(\frac{1-a_1}{1-a_1a_2})/\beta\bigg)$\end{document} as \begin{document}$ t\rightarrow\infty $\end{document};

\begin{document}$ \bullet $\end{document} If \begin{document}$ a_1>1>a_2>0 $\end{document} and \begin{document}$ \mu_2 $\end{document} is sufficiently large, then any global bounded solution exponentially converges to \begin{document}$ (0,f_1(1)/\alpha,1,0) $\end{document} as \begin{document}$ t\rightarrow\infty $\end{document};

\begin{document}$ \bullet $\end{document} If \begin{document}$ a_1 = 1>a_2>0 $\end{document} and \begin{document}$ \mu_2 $\end{document} is sufficiently large, then any global bounded solution algebraically converges to \begin{document}$ (0,f_1(1)/\alpha,1,0) $\end{document} as \begin{document}$ t\rightarrow\infty $\end{document}.

Global attractor of multi-valued operators with applications to a strongly damped nonlinear wave equation without uniqueness
Zhiming Liu and Zhijian Yang
2020, 25(1): 223-240 doi: 10.3934/dcdsb.2019179 +[Abstract](383) +[HTML](188) +[PDF](433.95KB)

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.

Analytical formula and dynamic profile of mRNA distribution
Feng Jiao, Jian Ren and Jianshe Yu
2020, 25(1): 241-257 doi: 10.3934/dcdsb.2019180 +[Abstract](266) +[HTML](184) +[PDF](630.88KB)

The stochasticity of transcription can be quantified by mRNA distribution \begin{document}$ P_m(t) $\end{document}, the probability that there are \begin{document}$ m $\end{document} mRNA molecules for the gene at time \begin{document}$ t $\end{document} in one cell. However, it still lacks of a standard method to calculate \begin{document}$ P_m(t) $\end{document} in a transparent formula. Here, we employ an infinite series method to express \begin{document}$ P_m(t) $\end{document} based on the classical two-state model. Intriguingly, we observe that a unimodal distribution of mRNA numbers at steady-state could be transformed from a dynamical bimodal distribution. This indicates that "bet hedging" strategy can be still achieved for the gene that generates phenotypic homogeneity of the cell population. Moreover, the formation and duration of such bimodality are tightly correlated with mRNA synthesis rate, reinforcing the modulation scenario of some inducible genes that manipulates mRNA synthesis rate in response to environmental change. More generally, the method presented here may be implemented to the other stochastic transcription models with constant rates.

Analysis of time-domain Maxwell's equations in biperiodic structures
Gang Bao, Bin Hu, Peijun Li and Jue Wang
2020, 25(1): 259-286 doi: 10.3934/dcdsb.2019181 +[Abstract](355) +[HTML](225) +[PDF](521.27KB)

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.

Advances in the LaSalle-type theorems for stochastic functional differential equations with infinite delay
Ya Wang, Fuke Wu, Xuerong Mao and Enwen Zhu
2020, 25(1): 287-300 doi: 10.3934/dcdsb.2019182 +[Abstract](313) +[HTML](217) +[PDF](376.51KB)

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.

Pullback exponential attractors for differential equations with variable delays
Mohamed Ali Hammami, Lassaad Mchiri, Sana Netchaoui and Stefanie Sonner
2020, 25(1): 301-319 doi: 10.3934/dcdsb.2019183 +[Abstract](392) +[HTML](193) +[PDF](428.51KB)

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.

Error analysis of discontinuous Galerkin method for the time fractional KdV equation with weak singularity solution
Na An, Chaobao Huang and Xijun Yu
2020, 25(1): 321-334 doi: 10.3934/dcdsb.2019185 +[Abstract](132) +[HTML](82) +[PDF](499.89KB)

In this work, the time fractional KdV equation with Caputo time derivative of order \begin{document}$ \alpha \in (0,1) $\end{document} is considered. The solution of this problem has a weak singularity near the initial time \begin{document}$ t = 0 $\end{document}. A fully discrete discontinuous Galerkin (DG) method combining the well-known L1 discretisation in time and DG method in space is proposed to approximate the time fractional KdV equation. The unconditional stability result and O\begin{document}$ (N^{-\min \{r\alpha,2-\alpha\}}+h^{k+1}) $\end{document} convergence result for \begin{document}$ P^k \; (k\geq 2) $\end{document} polynomials are obtained. Finally, numerical experiments are presented to illustrate the efficiency and the high order accuracy of the proposed scheme.

Multi-scale analysis for highly anisotropic parabolic problems
Thomas Blanc and Mihaï Bostan
2020, 25(1): 335-399 doi: 10.3934/dcdsb.2019186 +[Abstract](107) +[HTML](60) +[PDF](752.03KB)

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.

Bi-center problem and Hopf cyclicity of a Cubic Liénard system
Min Hu, Tao Li and Xingwu Chen
2020, 25(1): 401-414 doi: 10.3934/dcdsb.2019187 +[Abstract](121) +[HTML](77) +[PDF](1778.6KB)

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 \begin{document}$ 9 $\end{document} possible styles of small amplitude limit cycles surrounding these two center-foci and \begin{document}$ 6 $\end{document} styles of them can occur, from which the total Hopf cyclicity is no more than \begin{document}$ 4 $\end{document} and no less than \begin{document}$ 2 $\end{document}.

A nonlinear fourth-order PDE for multi-frame image super-resolution enhancement
Amine Laghrib, Abdelkrim Chakib, Aissam Hadri and Abdelilah Hakim
2020, 25(1): 415-442 doi: 10.3934/dcdsb.2019188 +[Abstract](149) +[HTML](133) +[PDF](6444.21KB)

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.

A hybrid model of collective motion of discrete particles under alignment and continuum chemotaxis
Ezio Di Costanzo, Marta Menci, Eleonora Messina, Roberto Natalini and Antonia Vecchio
2020, 25(1): 443-472 doi: 10.3934/dcdsb.2019189 +[Abstract](126) +[HTML](135) +[PDF](1021.95KB)

In this paper we propose and study a hybrid discrete–continuous mathematical model of collective motion under alignment and chemotaxis effect. Starting from paper [23], in which the Cucker-Smale model [22] was coupled with other cell mechanisms, to describe the cell migration and self-organization in the zebrafish lateral line primordium, we introduce a simplified model in which the coupling between an alignment and chemotaxis mechanism acts on a system of interacting particles. In particular we rely on a hybrid description in which the agents are discrete entities, while the chemoattractant is considered as a continuous signal. The proposed model is then studied both from an analytical and a numerical point of view. From the analytic point of view we prove, globally in time, existence and uniqueness of the solution. Then, the asymptotic behaviour of a linearised version of the system is investigated. Through a suitable Lyapunov functional we show that for 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.

On the forward dynamical behavior of nonautonomous systems
Chunqiu Li, Desheng Li and Xuewei Ju
2020, 25(1): 473-487 doi: 10.3934/dcdsb.2019190 +[Abstract](151) +[HTML](68) +[PDF](403.87KB)

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 \begin{document}$ \{\mathcal{A}_\varepsilon(p)\}_{p\in P} $\end{document} of phase space \begin{document}$ X $\end{document}, which is forward invariant such that \begin{document}$ \{\mathcal {A}_\varepsilon(p)\}_{p\in P} $\end{document} uniformly forward attracts each bounded subset of \begin{document}$ X $\end{document}. Furthermore, we can also prove that \begin{document}$ \{\mathcal{A}_\varepsilon(p)\}_{p\in P} $\end{document} forward attracts each bounded set at an exponential rate.

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