# American Institute of Mathematical Sciences

ISSN:
1531-3492

eISSN:
1553-524X

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## Discrete and Continuous Dynamical Systems - B

April 2019 , Volume 24 , Issue 4

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2019, 24(4): 1411-1448 doi: 10.3934/dcdsb.2018214 +[Abstract](4366) +[HTML](508) +[PDF](655.54KB)
Abstract:

The aim of the paper is to systematically introduce thermodynamic potentials for thermodynamic systems and Hamiltonian energies for quantum systems of condensates. The study is based on the rich previous work done by pioneers in the related fields. The main ingredients of the study consist of 1) SO(3) symmetry of thermodynamical potentials, 2) theory of fundamental interaction of particles, 3) the statistical theory of heat developed recently [23], 4) quantum rules for condensates that we postulate in Quantum Rule 4.1, and 5) the dynamical transition theory developed by Ma and Wang [20]. The statistical and quantum systems we study in this paper include conventional thermodynamic systems, thermodynamic systems of condensates, as well as quantum condensate systems. The potentials and Hamiltonian energies that we derive are based on first principles, and no mean-field theoretic expansions are used.

2019, 24(4): 1449-1467 doi: 10.3934/dcdsb.2018215 +[Abstract](4221) +[HTML](501) +[PDF](441.98KB)
Abstract:

Based on a recent result in [13], in this paper, we extend it to stochastic evolution equations with jumps in Hilbert spaces. This is done via Galerkin type finite-dimensional approximations of the infinite-dimensional stochastic evolution equations with jumps in a manner that one could then link the characterisation of the path-independence for finite-dimensional jump type SDEs to that for the infinite-dimensional settings. Our result provides an intrinsic link of infinite-dimensional stochastic evolution equations with jumps to infinite-dimensional (nonlinear) integro-differential equations.

2019, 24(4): 1469-1483 doi: 10.3934/dcdsb.2018216 +[Abstract](4921) +[HTML](702) +[PDF](671.5KB)
Abstract:

This paper presents an SEIRVS epidemic model with different vaccination strategies to investigate the elimination of the chronic disease. The mixed vaccination strategy, a combination of constant vaccination and pulse vaccination, is a future development tendency of disease control. Theoretical analysis and threshold conditions for eradicating the disease are given. Then we propose an optimal control problem and solve the optimal scheduling of the mixed vaccination strategy through the combined multiple shooting and collocation (CMSC) method. Theoretical results and numerical simulations can help to design the final mixed vaccination strategy for the optimal control of the chronic disease once the new vaccine comes into use.

2019, 24(4): 1485-1510 doi: 10.3934/dcdsb.2018217 +[Abstract](3907) +[HTML](558) +[PDF](3171.01KB)
Abstract:

In this work we prove continuity of solutions with respect to initial conditions and a couple of parameters and we prove upper semicontinuity of a family of pullback attractors for the problem

under homogeneous Neumann boundary conditions, \begin{document}$u_{τ s}∈ H: = L^2(Ω),$\end{document} \begin{document}$Ω\subset\mathbb{R}^n$\end{document}(\begin{document}$n≥ 1$\end{document}) is a smooth bounded domain, \begin{document}$B:H \to H$\end{document} is a globally Lipschitz map with Lipschitz constant \begin{document}$L≥ 0$\end{document}, \begin{document}$D_s∈[1,∞)$\end{document}, \begin{document}$C(·)∈ L^{∞}([τ, T];\mathbb{R}^+)$\end{document} is bounded from above and below and is monotonically nonincreasing in time, \begin{document}$p_s(·)∈ C(\bar{Ω})$\end{document}, \begin{document}$p_s^-: = \textrm{min}_{x∈\bar{Ω}}\;p_s(x)≥ p,$\end{document} \begin{document}$p_s^+: = \textrm{max}_{x∈\bar{Ω}}\;p_s(x)≤ a,$\end{document} for all \begin{document}$s∈ \mathbb{N},$\end{document} when \begin{document}$p_s(·) \to p$\end{document} in \begin{document}$L^∞(Ω)$\end{document} and \begin{document}$D_s \to ∞$\end{document} as \begin{document}$s \to∞,$\end{document} with \begin{document}$a,p>2$\end{document} positive constants.

2019, 24(4): 1511-1541 doi: 10.3934/dcdsb.2018218 +[Abstract](4520) +[HTML](551) +[PDF](497.41KB)
Abstract:

This paper is concerned with the traveling waves for a three-species competitive system with nonlocal dispersal. It has been shown by Dong, Li and Wang (DCDS 37 (2017) 6291-6318) that there exists a minimal wave speed such that a traveling wave exists if and only if the wave speed is above this minimal wave speed. In this paper, we first investigate the asymptotic behavior of traveling waves at negative infinity by a modified version of Ikehara's Theorem. Then we prove the uniqueness of traveling waves by applying the stronger comparison principle and the sliding method. Finally, we establish the exponential stability of traveling waves with large speeds by the weighted-energy method and the comparison principle, when the initial perturbation around the traveling wavefront decays exponentially as x → -∞, but can be arbitrarily large in other locations.

2019, 24(4): 1543-1568 doi: 10.3934/dcdsb.2018219 +[Abstract](4340) +[HTML](744) +[PDF](828.35KB)
Abstract:

Randomness in gene transcription can result in fluctuations (noise) of messenger RNA (mRNA) levels, leading to phenotypic plasticity in the isogenic populations of cells. Recent experimental studies indicate that multiple pathway activation mechanism plays an important role in the regulation of transcription noise and cell-to-cell variability. Previous theoretical studies on transcription noise have been emphasized on exact solutions and analysis for models with a single pathway or two cross-talking pathways. For stochastic models with more than two pathways, however, exact analytical results for fluctuations of mRNA levels have not been obtained yet. We develop a gene transcription model to examine the impact of multiple pathways on transcription noise for which the exact fluctuations of mRNA distributions are obtained. It is nontrivial to determine the analytical results for transcription fluctuations due to the high dimension of system parameter space. At the heart of our method lies the usage of the model's intrinsic symmetry to simplify the complicated calculations. We show the symmetric relation among system parameters, which allows us to derive the analytical expressions of the dynamical and steady-state fluctuations and to characterize the nature of transcription noise. Our results not only can be reduced to previous ones in limiting cases but also indicate some differences between the three or more pathway model and the single or two pathway one. Our analytical approaches provide new insights into the role of multiple pathways in noise regulation and optimization. A further study for stochastic gene transcription involving multiple pathways may shed light on the relation between transcription fluctuation and genetic network architecture.

2019, 24(4): 1569-1587 doi: 10.3934/dcdsb.2018220 +[Abstract](5052) +[HTML](858) +[PDF](1079.83KB)
Abstract:

This paper deals with the following two-species chemotaxis system

under homogeneous Neumann boundary conditions in a bounded domain \begin{document}$Ω\subset\mathbb{R}^{n}$\end{document} with smooth boundary. The parameters in the system are positive and the signal production function h is a prescribed C1-regular function. The main objectives of this paper are two-fold: One is the existence and boundedness of global solutions, the other is the large time behavior of the global bounded solutions in three competition cases (i.e., a weak competition case, a partially strong competition case and a fully strong competition case). It is shown that the unique positive spatially homogeneous equilibrium \begin{document}$(u_{*}, v_{*}, w_{*}, z_{*})$\end{document} may be globally attractive in the weak competition case (i.e., \begin{document}$0 < a_{1}, a_{2} < 1$\end{document}), while the constant stationary solution (0, h(1), 1, 0) may be globally attractive and globally stable in the partially strong competition case (i.e., \begin{document}$a_{1}>1>a_{2}>0$\end{document}). In the fully strong competition case (i.e. \begin{document}$a_{1}, a_{2}>1$\end{document}), however, we can only obtain the local stability of the two semi-trivial stationary solutions (0, h(1), 1, 0) and (1, 0, 0, h(1)) and the instability of the positive spatially homogeneous \begin{document}$(u_{*}, v_{*}, w_{*}, z_{*})$\end{document}. The matter which species ultimately wins out depends crucially on the starting advantage each species has.

2019, 24(4): 1589-1615 doi: 10.3934/dcdsb.2018221 +[Abstract](4570) +[HTML](629) +[PDF](7372.96KB)
Abstract:

In this article, we propose and analyze some novel spectral methods for the Schödinger equation (including the associated eigenvalue problem) with an inverse square potential on an arbitrary whole space \begin{document}$\mathbb{R}^d$\end{document} for any dimension \begin{document}$d$\end{document}. We start from the investigation that the radial component of the eigenfunctions, corresponding to spherical harmonics of degree \begin{document}$n$\end{document}, of the Schrödinger operator \begin{document}$\displaystyle -Δ u + \frac{c^2}{r^2}u$\end{document} can be expressed by Bessel functions of fractional orders \begin{document}$α_n = \sqrt{c^2+(n+d/2-1)^2}$\end{document} together with the multiplier \begin{document}$r^{1-\frac{d}{2}}$\end{document}. This knowledge helps us to construct the Müntz-Hermite functions as the basis functions to fit the singularities of the eigenfunctions. In return, a novel spectral method is then proposed for solving the Schrödinger eigenvalue problem efficiently. Further, a Galerkin spectral approximation using genuine Hermite functions with a distinct Müntz sequence \begin{document}$\{α_n = α+n+d/2-1\}$\end{document} is also proposed for the Schrödinger source problem with a singular solution of type \begin{document}$r^{α}$\end{document}. Optimal error estimates are then established rigorously for both the source and eigenvalue problems. In contrast to classic Hermite spectral methods using tensorial basis functions, our new methods possess an exponential order of convergence for such singular problems while offer a banded structure of the stiffness and mass matrices. Finally, numerical experiments illustrate the efficiency and spectral accuracy of our new methods.

2019, 24(4): 1617-1625 doi: 10.3934/dcdsb.2018222 +[Abstract](3650) +[HTML](537) +[PDF](351.97KB)
Abstract:

In the past years, there were very few works on the existence of nonconstant periodic solutions with fixed energy of singular second-order Hamiltonian systems, and now we attempt to ingeniously use Ekeland's variational principle to prove the existence of nonconstant periodic solutions with any fixed energy for singular second-order Hamiltonian systems, and our results greatly generalize some well known results such as [1,Theorem 3.6]. Moreover, we exhibit two simple and instructive singular potential examples to make our result more clear, which have not been solved by known results.

2019, 24(4): 1627-1652 doi: 10.3934/dcdsb.2018223 +[Abstract](4710) +[HTML](688) +[PDF](508.88KB)
Abstract:

This paper is devoted to a spatial heterogeneous SIS model with the infected group equipped with a free boundary. Our main aim is to determine whether the disease is spreading forever or extinct eventually, and to illustrate, under the nonhomogeneous spatial environment, free boundaries can have a large influence on the infected behavior at the large time. For this purpose, we first introduce a basic reproduction number and then establish a spreading-vanishing dichotomy. Then by investigating the effect of the diffusion rate, initial domain and spreading speed on the asymptotic behavior of the infected group, we establish some sufficient conditions and even necessary and sufficient conditions for disease spreading or vanishing.

2019, 24(4): 1653-1675 doi: 10.3934/dcdsb.2018286 +[Abstract](3561) +[HTML](483) +[PDF](510.08KB)
Abstract:

In this article, the problem of finite-time stabilization of two strings connected by point mass is discussed. We use the so-called Riemann coordinates to convert the study system into four transport equations coupled with the dynamic of the charge. We act by Bhat-Bernstein feedbacks in various positions (two extremities, the point mass and one of boundaries, only on the point mass, ...) and we show that in some cases the nature of the stability depends sensitively on the physical parameters of the system.

2019, 24(4): 1677-1695 doi: 10.3934/dcdsb.2018287 +[Abstract](3549) +[HTML](460) +[PDF](409.92KB)
Abstract:

This paper explores the class of equations of the Non-linear Schrödinger (NLS) type by employing both geometrical and spectral analysis methods. The work is developed in three stages. First, the geometrical method (AKS theorem) is used to derive different equations of the Non-linear Schrödinger (NLS) and Derivative Non-linear Schrödinger (DNLS) families. Second, the spectral technique (Tu method) is applied to obtain the hierarchies of equations belonging to these types. Third, the trace identity along with other techniques is used to obtain the corresponding Hamiltonian structures. It is found that the spectral method provides a simple algorithmic procedure to obtain the hierarchy as well as the Hamiltonian structure. Finally, the connection between the two formalisms is discussed and it is pointed out how application of these two techniques in unison can facilitate the understanding of integrable systems. In concurrence with Tu's method, Gesztesy and Holden also formulated a method of derivation of the trace formulas for integrable nonlinear evolution equations, this method is based on a contour-integration technique.

2019, 24(4): 1697-1723 doi: 10.3934/dcdsb.2018288 +[Abstract](3453) +[HTML](525) +[PDF](937.23KB)
Abstract:

We study the convex geometry of certain invariant manifolds, known as carrying simplices, for 3-species competitive Kolmogorov-type maps. We show that if all planes whose normal bundles are contained in a fixed closed and solid convex cone are rendered convex (concave) surfaces by the map, then, if there is a carrying simplex, it is a convex (concave) surface. We apply our results to the May-Leonard map.

2019, 24(4): 1725-1741 doi: 10.3934/dcdsb.2018233 +[Abstract](3583) +[HTML](494) +[PDF](2715.98KB)
Abstract:

The present research work recalls a control-theoretic approach to the synchronization of a first-order master/slave oscillators pair on \begin{document}$\mathbb{R}^3$\end{document} and extends such technique to the case of curved Riemannian manifolds. As theoretical results, this paper proves the asymptotic convergence of the feedback controller and studies the entity of the 'control effort'. As a case study, the complete equations for the controller of a slave oscillator on the unit hypersphere \begin{document}$\mathbb{S}^{n-1}$\end{document} are laid out and are illustrated by numerical examples for \begin{document}$n = 3$\end{document} and \begin{document}$n = 10$\end{document}, even in the hypothesis of noisy master-system state measurement.

2019, 24(4): 1743-1767 doi: 10.3934/dcdsb.2018235 +[Abstract](3883) +[HTML](460) +[PDF](538.58KB)
Abstract:

It has been recently established that a deterministic infinite horizon discounted optimal control problem in discrete time is closely related to a certain infinite dimensional linear programming problem and its dual, the latter taking the form of a certain max-min problem. In the present paper, we use these results to establish necessary and sufficient optimality conditions for this optimal control problem and to investigate a way how the latter can be used for the construction of a near optimal control.

2019, 24(4): 1769-1784 doi: 10.3934/dcdsb.2018236 +[Abstract](4386) +[HTML](580) +[PDF](438.09KB)
Abstract:

We apply the averaging theory of high order for computing the limit cycles of discontinuous piecewise quadratic and cubic polynomial perturbations of a linear center. These discontinuous piecewise differential systems are formed by two either quadratic, or cubic polynomial differential systems separated by a straight line.

We compute the maximum number of limit cycles of these discontinuous piecewise polynomial perturbations of the linear center, which can be obtained by using the averaging theory of order \begin{document}$n$\end{document} for \begin{document}$n = 1, 2, 3, 4, 5$\end{document}. Of course these limit cycles bifurcate from the periodic orbits of the linear center. As it was expected, using the averaging theory of the same order, the results show that the discontinuous quadratic and cubic polynomial perturbations of the linear center have more limit cycles than the ones found for continuous and discontinuous linear perturbations.

Moreover we provide sufficient and necessary conditions for the existence of a center or a focus at infinity if the discontinuous piecewise perturbations of the linear center are general quadratic polynomials or cubic quasi-homogenous polynomials.

2019, 24(4): 1785-1814 doi: 10.3934/dcdsb.2018237 +[Abstract](3332) +[HTML](550) +[PDF](556.24KB)
Abstract:

A model of \begin{document}$m$\end{document} species competing for a single growth-limiting resource is considered. We aim to use the dynamics of such a problem to describe the invasion and spread of \begin{document}$m$\end{document} species which are introduced localized in space \begin{document}$\mathbb{R}^N$\end{document}. The existence, uniqueness and uniform boundedness of the Cauchy problem are investigated by semigroup theory and local \begin{document}$L^p$\end{document}-estimates. The asymptotic speed of spread is achieved by uniform persistence ideas. The existence of traveling wave is obtained by upper-lower solutions and sliding techniques. Our result shows that the asymptotic speed of spread for \begin{document}$m$\end{document} species is characterized by the minimum wave speed of the positive traveling wave solutions associated with this system.

2019, 24(4): 1815-1842 doi: 10.3934/dcdsb.2018238 +[Abstract](3486) +[HTML](586) +[PDF](1360.54KB)
Abstract:

We consider an aggregation model with nonlinear diffusion in domains with boundaries and investigate the zero diffusion limit of its solutions. We establish the convergence of weak solutions for fixed times, as well as the convergence of energy minimizers in this limit. Numerical simulations that support the analytical results are presented. A second key scope of the numerical studies is to demonstrate that adding small nonlinear diffusion rectifies a flaw of the plain aggregation model in domains with boundaries, which is to evolve into unstable equilibria (non-minimizers of the energy).

2019, 24(4): 1843-1865 doi: 10.3934/dcdsb.2018240 +[Abstract](4273) +[HTML](444) +[PDF](3808.79KB)
Abstract:

In this paper, multiple information diffusion in online social networks with free boundary condition is investigated. We prove a spreading-vanishing dichotomy for the problem: i.e., either at least one piece of information lasts forever or all information suspend in finite time. The criterion for spreading and vanishing is established, it is related to the initial spreading area and the expansion capacity. We also obtain several cases of the asymptotic behavior of the information under different conditions. When the information spreads, we provide some upper and lower bounds of the spreading speed corresponding to different cases of asymptotic behavior of the information. In addition, numerical examples are given to illustrate the impacts of the initial spreading area and the expansion capacity on the free boundary, and all cases of the asymptotic behavior of the information.

2019, 24(4): 1867-1874 doi: 10.3934/dcdsb.2018243 +[Abstract](3420) +[HTML](451) +[PDF](334.45KB)
Abstract:

In this work we will consider a family of nonautonomous dynamical systems \begin{document}$x_{k+1} = f_k(x_k,\lambda)$\end{document}, \begin{document}$\lambda \in [-1,1]^{\mathbb{N}_0}$\end{document}, generated by a one-parameter family of flat-topped tent maps \begin{document}$g_{\alpha}(x)$\end{document}, i.e., \begin{document}$f_k(x,\lambda) = g_{\lambda_k}(x)$\end{document} for all \begin{document}$k\in \mathbb{N}_0$\end{document}. We will reinterpret the concept of attractive periodic orbit in this context, through the existence of some periodic, invariant and attractive nonautonomous sets and establish sufficient conditions over the parameter sequences for the existence of such periodic attractors.

2019, 24(4): 1875-1887 doi: 10.3934/dcdsb.2018245 +[Abstract](3162) +[HTML](505) +[PDF](462.6KB)
Abstract:

Two-patch models are used to mimic the unidirectional movement of organisms in continuous, advective environments. We assume that species can move between two patches, with patch 1 as the upper stream patch and patch 2 as the downstream patch. Species disperse between two patches with the same rate, and species in patch 1 is transported to patch 2 by drift, but not vice versa. We also mimic no-flux boundary conditions at the upstream and zero Dirichlet boundary conditions at the downstream. The criteria for the persistence of a single species is established. For two competing species model, we show that there is an intermediate dispersal rate which is evolutionarily stable. These results support the conjecture in [6], initially proposed for reaction-diffusion models with continuous advective environments.

2019, 24(4): 1889-1917 doi: 10.3934/dcdsb.2018247 +[Abstract](4052) +[HTML](483) +[PDF](520.5KB)
Abstract:

In this paper we study asymptotic behavior of a class of stochastic plate equations with rotational inertia and Kelvin-Voigt dissipative term. First we introduce a continuous random dynamical system 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.

2019, 24(4): 1919-1942 doi: 10.3934/dcdsb.2018249 +[Abstract](3831) +[HTML](527) +[PDF](544.22KB)
Abstract:

We study the convergence rates of solutions to the two-species chemotaxis-Navier-Stokes system with Lotka-Volterra competitive kinetics:

under homogeneous Neumann boundary conditions for \begin{document}$n_1,n_2,c$\end{document} and no-slip boundary condition for \begin{document}$u$\end{document} in a bounded domain \begin{document}$\Omega \subset \mathbb{R}^d(d\in\{2,3\})$\end{document} with smooth boundary. The global existence, boundedness and stabilization of solutions have been obtained in \begin{document}$2$\end{document}-D [8] and \begin{document}$3$\end{document}-D for \begin{document}$\kappa = 0$\end{document} and \begin{document}$\frac{\max\{\chi_1,\chi_2\}}{\min\{\mu_1,\mu_2\}}\|c_0\|_{L^\infty(\Omega)}$\end{document} being sufficiently small [4]. Here, we examine further convergence and derive the explicit rates of convergence for any supposedly given global bounded classical solution \begin{document}$(n_1, n_2, c, u)$\end{document}; more specifically, in \begin{document}$L^\infty$\end{document}-topology, we show that

In either cases, the \begin{document}$c$\end{document}-solution component converges exponentially to \begin{document}$0$\end{document}.

Moreover, it is shown that only the rate of convergence for \begin{document}$u$\end{document} is expressed in terms of the model parameters and the first eigenvalue of \begin{document}$-\Delta$\end{document} in \begin{document}$\Omega$\end{document} under homogeneous Dirichlet boundary conditions, and all other rates of convergence are explicitly expressed only in terms of the model parameters \begin{document}$a_i, \mu_i, \alpha$\end{document} and \begin{document}$\beta$\end{document} and the space dimension \begin{document}$d$\end{document}.

2019, 24(4): 1943-1960 doi: 10.3934/dcdsb.2018256 +[Abstract](3890) +[HTML](491) +[PDF](521.2KB)
Abstract:

Lyapunov type inequalities for (linear or nonlinear) Hammerstein integral equations are established and applied to second order differential equations (ODEs) with general separated boundary conditions. These new inequalities provide necessary conditions for the Hammerstein integral equations and these boundary value problems to have nonzero nonnegative solutions. As applications of these inequalities for nonlinear ODEs, we obtain extinction criteria and optimal locations of favorable habitats for populations inhabiting one dimensional heterogeneous environments governed by reaction-diffusion equations with spatially varying growth rates and external forcing.

2019, 24(4): 1961-1987 doi: 10.3934/dcdsb.2018257 +[Abstract](3459) +[HTML](419) +[PDF](509.43KB)
Abstract:

We first prove the existence of a compact positively invariant set which exponentially attracts any bounded set for abstract multi-valued semidynamical systems. Then, we apply the abstract theory to handle retarded ordinary differential equations and lattice dynamical systems, as well as reactiondiffusion equations with infinite delays. We do not assume any Lipschitz condition on the nonlinear term, just a continuity assumption together with growth and dissipative conditions, so that uniqueness of the Cauchy problem fails to be true.

2019, 24(4): 1989-2015 doi: 10.3934/dcdsb.2019026 +[Abstract](4716) +[HTML](218) +[PDF](556.31KB)
Abstract:

Trapped dynamics widely appears in nature, e.g., the motion of particles in viscous cytoplasm. The famous continuous time random walk (CTRW) model with power law waiting time distribution (having diverging first moment) describes this phenomenon. Because of the finite lifetime of biological particles, sometimes it is necessary to temper the power law measure such that the waiting time measure has convergent first moment. Then the time operator of the Fokker-Planck equation corresponding to the CTRW model with tempered waiting time measure is the so-called tempered fractional derivative. This paper focus on discussing the properties of the time tempered fractional derivative, and studying the well-posedness and the Jacobi-predictor-corrector algorithm for the tempered fractional ordinary differential equation. By adjusting the parameter of the proposed algorithm, high convergence order can be obtained and the computational cost linearly increases with time. The numerical results shows that our algorithm converges with order \begin{document}$N_I$\end{document}, where \begin{document}$N_I$\end{document} is the number of used interpolating points.

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