# American Institute of Mathematical Sciences

ISSN:
1531-3492

eISSN:
1553-524X

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## Discrete & Continuous Dynamical Systems - B

January 2022 , Volume 27 , Issue 1

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2022, 27(1): 1-44 doi: 10.3934/dcdsb.2021031 +[Abstract](873) +[HTML](434) +[PDF](952.88KB)
Abstract:

The paper deals with the dynamics of conservative \begin{document}$N$\end{document}-degree-of-freedom vibro-impact systems involving one unilateral contact condition and a linear free flow. Among all possible trajectories, grazing orbits exhibit a contact occurrence with vanishing incoming velocity which generates mathematical difficulties. Such problems are commonly tackled through the definition of a Poincaré section and the attendant First Return Map. It is known that the First Return Time to the Poincaré section features a square-root singularity near grazing. In this work, a non-orthodox yet natural and intrinsic Poincaré section is chosen to revisit the square-root singularity. It is based on the unilateral condition but is not transverse to the grazing orbits. A detailed investigation of the proposed Poincaré section is provided. Higher-order singularities in the First Return Time are exhibited. Also, activation coefficients of the square-root singularity for the First Return Map are defined. For the linear and periodic grazing orbits from which bifurcate nonlinear modes, one of these coefficients is necessarily non-vanishing. The present work is a step towards the stability analysis of grazing orbits, which still stands as an open problem.

2022, 27(1): 45-72 doi: 10.3934/dcdsb.2021032 +[Abstract](852) +[HTML](395) +[PDF](473.41KB)
Abstract:

We study the long-time behaviour of the solutions to a nonlinear damped anisotropic sixth-order Schrödinger type equation in \begin{document}$\mathbb{R}^2$\end{document} that reads

We prove that this behaviour is described by the existence of regular global attractor in an anisotropic Sobolev space with finite fractal dimension.

2022, 27(1): 73-101 doi: 10.3934/dcdsb.2021033 +[Abstract](933) +[HTML](397) +[PDF](429.56KB)
Abstract:

This work is concerned with a stochastic wave equation driven by a white noise. Borrowing from the invariant random cone and employing the backward solvability argument, this wave system is approximated by a finite dimensional wave equation with a white noise. Especially, the finite dimension is explicit, accurate and determined by the coefficient of this wave system; and further originating from an Ornstein-Uhlenbek process and applying Banach space norm estimation, this wave system is approximated by a finite dimensional wave equation with a smooth colored noise.

2022, 27(1): 103-140 doi: 10.3934/dcdsb.2021034 +[Abstract](813) +[HTML](406) +[PDF](1033.42KB)
Abstract:

The process of binder removal from green ceramic components-a reaction-gas transport problem in porous media-has been analyzed with a number of mathematical techniques: 1) non-dimensionalization of the governing decomposition-reaction ordinary differential equation (ODE) and of the reaction gas-permeability partial differential equation (PDE); 2) development of a pseudo steady state approximation (PSSA) for the PDE, including error analysis via \begin{document}$L^2$\end{document} norm and singular perturbation methods; 3) derivation and analysis of a discrete model approximation; and 4) development of a time optimal control strategy to minimize processing time with temperature and pressure constraints. Theoretical analyses indicate the conditions under which the PSSA and discrete models are viable approximations. Numerical results indicate that under a range of conditions corresponding to practical binder burnout conditions, utilization of the optimal temperature protocol leads to shorter cycle times as compared to typical industrial practice.

2022, 27(1): 141-165 doi: 10.3934/dcdsb.2021035 +[Abstract](979) +[HTML](406) +[PDF](2737.52KB)
Abstract:

In this paper, we study two-dimensional, three-dimensional monotonic and nonmonotonic immune responses in viral infection systems. Our results show that the viral infection systems with monotonic immune response has no bistability appear. However, the systems with nonmonotonic immune response has bistability appear under some conditions. For immune intensity, we got two important thresholds, post-treatment control threshold and elite control threshold. When immune intensity is less than post-treatment control threshold, the virus will be rebound. The virus will be under control when immune intensity is larger than elite control threshold. While between the two thresholds is a bistable interval. When immune intensity is in the bistable interval, the system can have bistability appear. Select the rate of immune cells stimulated by the viruses as a bifurcation parameter for nonmonotonic immune responses, we prove that the system exhibits saddle-node bifurcation and transcritical bifurcation.

2022, 27(1): 167-198 doi: 10.3934/dcdsb.2021036 +[Abstract](775) +[HTML](400) +[PDF](414.82KB)
Abstract:

At the very beginning of the theory of finite dynamical systems, it was discovered that some relatively simple systems, even of ordinary differential equations, can generate very complicated (chaotic) behaviors. Furthermore these systems are extremely sensitive to perturbations, in the sense that trajectories with close but different initial data may diverge exponentially. Very often, the trajectories of such chaotic systems are localized, up to some transient process, in some subset of the phase space, the so-called strange attractors. Such subset have a very complicated geometric structure. They accumulate the nontrivial dynamics of the system.

For a distributed system, whose time evolution is usually governed by partial differential equations (PDEs), the phase space X is (a subset of) an infinite dimensional function space. We will thus speak of infinite dimensional dynamical systems. Since the global existence and uniqueness of solutions has been proven for a large class of PDEs arising from different domains as Mechanics and Physics, it is therefore natural to investigate whether the features, in particular the notion of attractor, obtained for dynamical systems generated by systems of ODEs generalizes to systems of PDEs.

In this paper we give a positive aftermath by proving the existence of pullback \begin{document}$\mathcal{D}$\end{document}-attractor. The key point is to find a bounded family of pullback \begin{document}$\mathcal{D}$\end{document}-absorbing sets then we apply the decomposition techniques and a method used in previous works to verify the pullback \begin{document}$w$\end{document}-\begin{document}$\mathcal{D}$\end{document}-limit compactness. It is based on the concept of the Kuratowski measure of noncompactness of a bounded set as well as some new estimates of the equicontinuity of the solutions.

2022, 27(1): 199-228 doi: 10.3934/dcdsb.2021037 +[Abstract](961) +[HTML](424) +[PDF](624.36KB)
Abstract:

According to the report of the WHO, there is a strong relationship between AIDS and tuberculosis (TB). Therefore, it is very important to study how to control TB in the context of the global AIDS epidemic. In this paper, we establish an age structured mathematical model of HIV-TB co-infection to study the transmission dynamics of this co-infection, and consider awareness in the modeling. We give the basic reproduction numbers for each of the two diseases and find four equilibria, namely, disease-free equilibrium, TB-free equilibrium, HIV-free equilibrium and endemic disease equilibrium. Then we discuss the local stability of the equilibria according to the range of values of the two basic reproduction numbers, and find the endemic equilibrium is unstable. We also discuss the global stability of the disease-free equilibrium and the TB-free equilibrium. Based on the new HIV-positive cases and TB cases data in China, the best-fit parameter values and initial values of the model are identified by the MCMC algorithm. Then we perform uncertainty and sensitivity analysis to identify the parameters that have significant impact on the basic reproduction number \begin{document}$\mathcal{R}_{T}$\end{document}. Finally, combined with the established model, we give some measures that may help China achieve the goal of WHO of reducing the incidence of TB by 80% by 2030 compared to 2015.

2022, 27(1): 229-243 doi: 10.3934/dcdsb.2021038 +[Abstract](865) +[HTML](429) +[PDF](334.93KB)
Abstract:

In this paper, we consider a generalized predator-prey system described by a reaction-diffusion system with spatio-temporal delays. We study the local stability for the constant equilibria of predator-prey system with the generalized delay kernels. Moreover, using the specific delay kernels, we perform a qualitative analysis of the solutions, including existence, uniqueness, and boundedness of the solutions; global stability, and Hopf bifurcation of the nontrivial equilibria.

2022, 27(1): 245-256 doi: 10.3934/dcdsb.2021039 +[Abstract](977) +[HTML](487) +[PDF](439.43KB)
Abstract:

In this paper we study the dynamics of the Higgins–Selkov system

where \begin{document}$\alpha$\end{document} is a real parameter and \begin{document}$\gamma>1$\end{document} is an integer. We classify the phase portraits of this system for \begin{document}$\gamma = 3, 4, 5, 6,$\end{document} in the Poincaré disc for all the values of the parameter \begin{document}$\alpha$\end{document}. Moreover, we determine in function of the parameter \begin{document}$\alpha$\end{document} the regions of the phase space with biological meaning.

2022, 27(1): 257-275 doi: 10.3934/dcdsb.2021040 +[Abstract](792) +[HTML](400) +[PDF](365.46KB)
Abstract:

We consider random perturbations of a general eco-epidemiological model. We prove the existence of a global random attractor, the persistence of susceptibles preys and provide conditions for the simultaneous extinction of infectives and predators. We also discuss the dynamics of the corresponding random epidemiological \begin{document}$SI$\end{document} and predator-prey models. We obtain for this cases a global random attractor, prove the prevalence of susceptibles/preys and provide conditions for the extinctions of infectives/predators.

2022, 27(1): 277-300 doi: 10.3934/dcdsb.2021042 +[Abstract](848) +[HTML](416) +[PDF](534.56KB)
Abstract:

A fluid-particle interaction model with magnetic field is studied in this paper. When the initial vacuum and the far field vacuum of the fluid and the particles are contained, the constant shear viscosity \begin{document}$\mu$\end{document} and the bulk viscosity \begin{document}$\lambda$\end{document} are \begin{document}$\mu>0$\end{document} \begin{document}$\lambda = \rho^\beta$\end{document} for any \begin{document}$\beta\geq 0$\end{document}, the strong solutions of the 2D Cauchy problem for the coupled system are established applying the method of weighted estimates in Li-Liang's paper on Navier-Stokes equations.

2022, 27(1): 301-310 doi: 10.3934/dcdsb.2021043 +[Abstract](830) +[HTML](405) +[PDF](127.8KB)
Abstract:

Effect of additive fading noise on a behavior of the solution of a stochastic difference equation with continuous time is investigated. It is shown that if the zero solution of the initial stochastic difference equation is asymptotically mean square quasistable and the level of additive stochastic perturbations is given by square summable sequence, then the solution of a perturbed difference equation remains to be an asymptotically mean square quasitrivial. The obtained results are formulated in terms of Lyapunov functionals and linear matrix inequalities (LMIs). It is noted that the study of the situation, when an additive stochastic noise fades on the infinity not so quickly, remains an open problem.

Feng Dai and
2022, 27(1): 311-341 doi: 10.3934/dcdsb.2021044 +[Abstract](798) +[HTML](468) +[PDF](504.13KB)
Abstract:

This paper is concerned with the two species cancer invasion haptotaxis model with tissue remodeling

in a bounded and smooth domain \begin{document}$\Omega\subset\mathbb{R}^N\;(N\geq1)$\end{document} with zero-flux boundary conditions for \begin{document}$c_1,c_2$\end{document} and \begin{document}$m$\end{document}, where \begin{document}$\chi_i,\mu_i,r_i>0\;(i = 1,2)$\end{document}, \begin{document}$\eta>0$\end{document}, \begin{document}$\kappa\geq1$\end{document}, \begin{document}$\tau\in\{0,1\}$\end{document}, and \begin{document}$\mu_{\rm EMT} = \mu_{ \rm EMT}\left(c_1,c_2,m,v\right):[0,\infty)^4\rightarrow [0,\infty)$\end{document} is the epithelial-mesenchymal transition rate function such that \begin{document}$\mu_{\rm EMT}\leq\mu_M$\end{document} with some constant \begin{document}$\mu_M>0$\end{document}. When \begin{document}$\kappa = 1$\end{document} and \begin{document}$N = 3$\end{document}, by rasing the coupled a priori estimates of \begin{document}$c_1$\end{document} and \begin{document}$c_2$\end{document} in the following way \begin{document}$L^1(\Omega)\rightarrow L^2(\Omega)\rightarrow L^p(\Omega)\rightarrow L^\infty(\Omega)$\end{document} with any \begin{document}$p>2$\end{document}, it is shown that for some appropriately regular and small initial data, the associated initial-boundary value problem possesses a unique globally bounded classical solution for suitably small \begin{document}$r_i$\end{document} and \begin{document}$\mu_M$\end{document}. When \begin{document}$\kappa>1$\end{document} and \begin{document}$N\geq1$\end{document}, by rasing the coupled a priori estimates of \begin{document}$c_1$\end{document} and \begin{document}$c_2$\end{document} from \begin{document}$L^1(\Omega)$\end{document} to \begin{document}$L^p(\Omega)$\end{document} with any \begin{document}$p>1$\end{document}, then to \begin{document}$L^\infty(\Omega)$\end{document}, it is proved that for any reasonably regular initial data, the corresponding initial-boundary value problem admits a unique globally bounded classical solution for arbitrary \begin{document}$r_i$\end{document} and \begin{document}$\mu_M$\end{document}. The result for \begin{document}$\kappa = 1$\end{document} complements previously known one, and the result for \begin{document}$\kappa>1$\end{document} is new.

2022, 27(1): 343-360 doi: 10.3934/dcdsb.2021045 +[Abstract](914) +[HTML](425) +[PDF](398.65KB)
Abstract:

In this work we consider the chemotaxis system with singular sensitivity and signal production in a two dimensional bounded domain. We present the global existence of weak solutions under appropriate regularity assumptions on the initial data. Our results generalize some well-known results in the literature.

2022, 27(1): 361-391 doi: 10.3934/dcdsb.2021047 +[Abstract](944) +[HTML](601) +[PDF](469.84KB)
Abstract:

This paper deals with the following two-species chemotaxis system with nonlinear diffusion, sensitivity, signal secretion and (without or with) logistic source

under homogeneous Neumann boundary conditions in a bounded domain \begin{document}$\Omega\subset \mathbb{R}^n$\end{document} with \begin{document}$n\geq2$\end{document}. The diffusion functions \begin{document}$D_{i}(s) \in C^{2}([0,\infty))$\end{document} and the chemotactic sensitivity functions \begin{document}$S_{i}(s) \in C^{2}([0,\infty))$\end{document} are given by

where \begin{document}$C_{d_{i}},C_{s_{i}}>0$\end{document} and \begin{document}$\alpha_i,\beta_{i} \in \mathbb{R}$\end{document} \begin{document}$(i = 1,2)$\end{document}. The logistic source functions \begin{document}$f_{i}(s) \in C^{0}([0,\infty))$\end{document} and the nonlinear signal secretion functions \begin{document}$g_{i}(s) \in C^{1}([0,\infty))$\end{document} are given by

where \begin{document}$r_{i} \in \mathbb{R}$\end{document}, \begin{document}$\mu_{i},\gamma_{i} > 0$\end{document} and \begin{document}$k_{i} > 1$\end{document} \begin{document}$(i = 1,2)$\end{document}. With the assumption of proper initial data regularity, the global boundedness of solution is established under the some specific conditions with or without the logistic functions \begin{document}$f_{i}(s)$\end{document}.

Moreover, in case \begin{document}$r_{i}>0$\end{document}, for the large time behavior of the smooth bounded solution, by constructing the appropriate energy functions, under the conditions \begin{document}$\mu_{i}$\end{document} are sufficiently large, it is shown that the global bounded solution exponentially converges to \begin{document}$\left((\frac{r_{1}}{\mu_{1}})^{\frac{1}{k_{1}-1}}, (\frac{r_{2}}{\mu_{2}})^{\frac{\gamma_{1}}{k_{2}-1}}, (\frac{r_{2}}{\mu_{2}})^{\frac{1}{k_{2}-1}}, (\frac{r_{1}}{\mu_{1}})^{\frac{\gamma_{2}}{k_{1}-1}}\right)$\end{document} as \begin{document}$t\rightarrow\infty$\end{document}.

2022, 27(1): 393-420 doi: 10.3934/dcdsb.2021048 +[Abstract](963) +[HTML](458) +[PDF](703.92KB)
Abstract:

Dengue, a mosquito-borne disease, poses a tremendous burden to human health with about 390 million annual dengue infections worldwide. The environmental temperature plays a major role in the mosquito life-cycle as well as the mosquito-human-mosquito dengue transmission cycle. While previous studies have provided useful insights into the understanding of dengue diseases, there is little emphasis put on the role of environmental temperature variation, especially diurnal variation, in the mosquito vector and dengue dynamics. In this study, we develop a mathematical model to investigate the impact of seasonal and diurnal temperature variations on the persistence of mosquito vector and dengue. Importantly, using a threshold dynamical system approach to our model, we formulate the mosquito reproduction number and the infection invasion threshold, which completely determine the global threshold dynamics of mosquito population and dengue transmission, respectively. Our model predicts that both seasonal and diurnal variations of the environmental temperature can be determinant factors for the persistence of mosquito vector and dengue. In general, our numerical estimates of the mosquito reproduction number and the infection invasion threshold show that places with higher diurnal or seasonal temperature variations have a tendency to suffer less from the burden of mosquito population and dengue epidemics. Our results provide novel insights into the theoretical understanding of the role of diurnal temperature, which can be beneficial for the control of mosquito vector and dengue spread.

2022, 27(1): 421-441 doi: 10.3934/dcdsb.2021049 +[Abstract](761) +[HTML](417) +[PDF](1063.05KB)
Abstract:

Denote by CH, CSH, CQH, and CSQH the planar cubic homogeneous, cubic semi-homogeneous, cubic quasi-homogeneous and cubic semi-quasi-homogeneous differential systems, respectively. The problems on limit cycles and global dynamics of these systems have been solved for CH, and partially for CSH. This paper studies the same problems for CQH and CSQH. We prove that CQH have no limit cycles and CSQH can have at most one limit cycle with the limit cycle realizable. Moreover, we classify all the global phase portraits of CSQH.

2022, 27(1): 443-468 doi: 10.3934/dcdsb.2021050 +[Abstract](749) +[HTML](398) +[PDF](478.52KB)
Abstract:

In this paper we study asymptotic behavior of a class of stochastic plate equations with memory and additive noise. First we introduce a continuous cocycle for the equation and establish the pullback asymptotic compactness of solutions. Second we consider the existence and upper semicontinuity of random attractors for the equation.

2022, 27(1): 469-486 doi: 10.3934/dcdsb.2021051 +[Abstract](918) +[HTML](383) +[PDF](398.63KB)
Abstract:

This paper investigates the global existence of weak solutions for the incompressible \begin{document}$p$\end{document}-Navier-Stokes equations in \begin{document}$\mathbb{R}^d$\end{document} \begin{document}$(2\leq d\leq p)$\end{document}. The \begin{document}$p$\end{document}-Navier-Stokes equations are obtained by adding viscosity term to the \begin{document}$p$\end{document}-Euler equations. The diffusion added is represented by the \begin{document}$p$\end{document}-Laplacian of velocity and the \begin{document}$p$\end{document}-Euler equations are derived as the Euler-Lagrange equations for the action represented by the Benamou-Brenier characterization of Wasserstein-\begin{document}$p$\end{document} distances with constraint density to be characteristic functions.

2022, 27(1): 487-522 doi: 10.3934/dcdsb.2021052 +[Abstract](879) +[HTML](398) +[PDF](534.21KB)
Abstract:

In this paper, we prove the global existence of the strong solutions to the vacuum free boundary problem of isentropic compressible magnetohydrodynamic equations with small initial data and axial symmetry, where the solutions are independent of the axial variable and the angular variable. The solutions capture the precise physical behavior that the sound speed is \begin{document}$C^{1/2}$\end{document}-Hölder continuous across the vacuum boundary provided that the adiabatic exponent \begin{document}$\gamma\in(1, 2)$\end{document}. The main difficulties of this problem lie in the singularity at the symmetry axis, the degeneracy of the system near the free boundary and the strong coupling of the magnetic field and the velocity. We overcome the obstacles by constructing some new weighted nonlinear functionals (involving both lower-order and higher-order derivatives) and establishing the uniform-in-time weighted energy estimates of solutions by delicate analysis, in which the balance of pressure and self-gravitation, and the dissipation of velocity are crucial.

2022, 27(1): 523-553 doi: 10.3934/dcdsb.2021053 +[Abstract](835) +[HTML](408) +[PDF](406.3KB)
Abstract:

This paper is concerned with the well-posedness of global solution and exponential stability to the Timoshenko system subject with time-varying weights and time-varying delay. We consider two problems: full and partially damped systems. We prove existence of global solution for both problems combining semigroup theory with the Kato's variable norm technique. To prove exponential stability, we apply the Energy Method. For partially damped system the exponential stability is proved under assumption of equal-speed wave propagation in the transversal and angular directions. For full damped system the exponential stability is obtained without the hypothesis of equal-speed wave propagation.

2022, 27(1): 555-568 doi: 10.3934/dcdsb.2021054 +[Abstract](675) +[HTML](381) +[PDF](1463.85KB)
Abstract:

We propose a generalization of the method of cyclic projections, which uses the lengths of projection steps carried out in the past to learn about the geometry of the problem and decides on this basis which projections to carry out in the future. We prove the convergence of this algorithm and illustrate its behavior in a first numerical study.

2022, 27(1): 569-581 doi: 10.3934/dcdsb.2021055 +[Abstract](910) +[HTML](448) +[PDF](389.41KB)
Abstract:

This paper is concerned with stablization of hybrid differential equations by intermittent control based on delay observations. By M-matrix theory and intermittent control strategy, we establish a sufficient stability criterion on intermittent hybrid stochastic differential equations. Meantime, we show that hybrid differential equations can be stabilized by intermittent control based on delay observations if the delay time \begin{document}$\tau$\end{document} is bounded by \begin{document}$\tau^*$\end{document}. Finally, an example is presented to illustrate our theory.

2022, 27(1): 583-600 doi: 10.3934/dcdsb.2021056 +[Abstract](752) +[HTML](381) +[PDF](366.35KB)
Abstract:

This work is devoted to studying the relations between the asymptotic behavior for a class of hyperbolic equations with super-cubic nonlinearity and a class of heat equations, where the problem is considered in a smooth bounded three dimensional domain. Based on the extension of the Strichartz estimates to the case of bounded domain, we show the regularity of the pullback, uniform, and cocycle attractors for the non-autonomous dynamical system given by hyperbolic equation. Then we prove that all types of non-autonomous attractors converge, upper semicontiously, to the natural extension global attractor of the limit parabolic equations.

2022, 27(1): 601-617 doi: 10.3934/dcdsb.2021057 +[Abstract](790) +[HTML](409) +[PDF](2352.47KB)
Abstract:

Turing-type reaction-diffusion systems on evolving domains arising in biology, chemistry and physics are considered in this paper. The evolving domain is transformed into a reference domain, on which we use a second order semi-implicit backward difference formula (SBDF2) for time integration and a meshless collocation method for space discretization. A global refinement strategy is proposed to reduce the computational cost. Numerical experiments are carried out for different evolving domains. The convergence behavior of the proposed algorithm and the effectiveness of the refinement strategy are verified numerically.

2020 Impact Factor: 1.327
5 Year Impact Factor: 1.492
2020 CiteScore: 2.2