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Discrete & Continuous Dynamical Systems - B

April 2020 , Volume 25 , Issue 4

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Existence of a global attractor for fractional differential hemivariational inequalities
Yirong Jiang, Nanjing Huang and Zhouchao Wei
2020, 25(4): 1193-1212 doi: 10.3934/dcdsb.2019216 +[Abstract](525) +[HTML](272) +[PDF](415.14KB)

In this paper, we introduce and study a new class of fractional differential hemivariational inequalities ((FDHVIs), for short) formulated by an initial-value fractional evolution inclusion and a hemivariational inequality in infinite Banach spaces. First, by applying measure of noncompactness, a fixed point theorem of a condensing multivalued map, we obtain the nonemptiness and compactness of the mild solution set for (FDHVIs). Further, we apply the obtained results to establish an existence theorem of the mild solution of a global attractor for the semiflow governed by a fractional differential hemivariational inequality ((FDHVI), for short). Finally, we provide an example to demonstrate the main results.

Instability of the standing waves for a Benney-Roskes/Zakharov-Rubenchik system and blow-up for the Zakharov equations
José R. Quintero and Juan C. Cordero
2020, 25(4): 1213-1240 doi: 10.3934/dcdsb.2019217 +[Abstract](415) +[HTML](187) +[PDF](436.7KB)

In this paper we establish the nonlinear orbital instability of ground state standing waves for a Benney-Roskes/Zakharov-Rubenchik system that models the interaction of low amplitude high frequency waves, acustic type waves in \begin{document}$ N = 2 $\end{document} and \begin{document}$ N = 3 $\end{document} spatial directions. For \begin{document}$ N = 2 $\end{document}, we follow M. Weinstein's approach used in the case of the Schrödinger equation, by establishing a virial identity that relates the second variation of a momentum type functional with the energy (Hamiltonian) on a class of solutions for the Benney-Roskes/Zakharov-Rubenchik system. From this identity, it is possible to show that solutions for the Benney-Roskes/Zakharov-Rubenchik system blow up in finite time, in the case that the energy (Hamiltonian) of the initial data is negative, indicating a possible blow-up result for non radial solutions to the Zakharov equations. For \begin{document}$ N = 3 $\end{document}, we establish the instability by using a scaling argument and the existence of invariant regions under the flow due to a concavity argument.

Attractors for first order lattice systems with almost periodic nonlinear part
Ahmed Y. Abdallah
2020, 25(4): 1241-1255 doi: 10.3934/dcdsb.2019218 +[Abstract](417) +[HTML](166) +[PDF](405.64KB)

We study the existence of the uniform global attractor for a family of infinite dimensional first order non-autonomous lattice dynamical systems of the following form:

with initial data

The nonlinear part of the system \begin{document}$ f\left( u,t\right) $\end{document} presents the main difficultly of this work. To overcome this difficulty we introduce a suitable Banach space \begin{document}$ W $\end{document} of functions satisfying (3)-(7) with norm (8) such that \begin{document}$ f_{0}\left( \cdot ,t\right) $\end{document} is an almost periodic function of \begin{document}$ t $\end{document} with values in \begin{document}$ W $\end{document} and \begin{document}$ \left( g,f\right) \in \mathcal{H}\left( \left( g_{0},f_{0}\right) \right) $\end{document}.

A criterion for the existence of relaxation oscillations with applications to predator-prey systems and an epidemic model
Ting-Hao Hsu and Gail S. K. Wolkowicz
2020, 25(4): 1257-1277 doi: 10.3934/dcdsb.2019219 +[Abstract](594) +[HTML](229) +[PDF](952.88KB)

We derive characteristic functions to determine the number and stability of relaxation oscillations for a class of planar systems. Applying our criterion, we give conditions under which the chemostat predator-prey system has a globally orbitally asymptotically stable limit cycle. Also we demonstrate that a prescribed number of relaxation oscillations can be constructed by varying the perturbation for an epidemic model studied by Li et al. [SIAM J. Appl. Math, 2016].

On regularity of stochastic convolutions of functional linear differential equations with memory
Kai Liu
2020, 25(4): 1279-1298 doi: 10.3934/dcdsb.2019220 +[Abstract](354) +[HTML](157) +[PDF](382.96KB)

In this work, we consider the regularity property of stochastic convolutions for a class of abstract linear stochastic retarded functional differential equations with unbounded operator coefficients. We first establish some useful estimates on fundamental solutions which are time delay versions of those on \begin{document}$ C_0 $\end{document}-semigroups. To this end, we develop a scheme of constructing the resolvent operators for the integrodifferential equations of Volterra type since the equation under investigation is of this type in each subinterval describing the segment of its solution. Then we apply these estimates to stochastic convolutions of our equations to obtain the desired regularity property.

Existence of pullback attractors for the non-autonomous suspension bridge equation with time delay
Suping Wang and Qiaozhen Ma
2020, 25(4): 1299-1316 doi: 10.3934/dcdsb.2019221 +[Abstract](434) +[HTML](178) +[PDF](352.5KB)

We investigate the long-time behavior of solutions for the suspension bridge equation when the forcing term containing some hereditary characteristic. Existence of pullback attractor is shown by using the contractive function methods.

A global well-posedness and asymptotic dynamics of the kinetic Winfree equation
Seung-Yeal Ha, Jinyeong Park and Xiongtao Zhang
2020, 25(4): 1317-1344 doi: 10.3934/dcdsb.2019229 +[Abstract](175) +[HTML](71) +[PDF](423.36KB)

We study a global well-posedness and asymptotic dynamics of measure-valued solutions to the kinetic Winfree equation. For this, we introduce a second-order extension of the first-order Winfree model on an extended phase-frequency space. We present the uniform(-in-time) \begin{document}$ \ell_p $\end{document}-stability estimate with respect to initial data and the equivalence relation between the original Winfree model and its second-order extension. For this extended model, we present uniform-in-time mean-field limit and large-time behavior of measure-valued solution for the second-order Winfree model. Using stability and asymptotic estimates for the extended model and the equivalence relation, we recover the uniform mean-field limit and large-time asymptotics for the Winfree model. 200 words.

Global weak solution and smooth solution of the periodic initial value problem for the generalized Landau-Lifshitz-Bloch equation in high dimensions
Jing Li, Boling Guo, Lan Zeng and Yitong Pei
2020, 25(4): 1345-1360 doi: 10.3934/dcdsb.2019230 +[Abstract](194) +[HTML](72) +[PDF](330.84KB)

In this paper, by using the Gal\begin{document}$ {\rm\ddot{e}} $\end{document}rkin method and energy estimates, the global weak solution and the smooth solution to the generalized Landau-Lifshitz-Bloch (GLLB) equation in high dimensions are obtained.

Ergodicity of non-autonomous discrete systems with non-uniform expansion
Pablo G. Barrientos and Abbas Fakhari
2020, 25(4): 1361-1382 doi: 10.3934/dcdsb.2019231 +[Abstract](213) +[HTML](92) +[PDF](583.68KB)

We study the ergodicity of non-autonomous discrete dynamical systems with non-uniform expansion. As an application we get that any uniformly expanding finitely generated semigroup action of \begin{document}$ C^{1+\alpha} $\end{document} local diffeomorphisms of a compact manifold is ergodic with respect to the Lebesgue measure. Moreover, we will also prove that every exact non-uniform expandable finitely generated semigroup action of conformal \begin{document}$ C^{1+\alpha} $\end{document} local diffeomorphisms of a compact manifold is Lebesgue ergodic.

Periodic solutions of differential-algebraic equations
Yingjie Bi, Siyu Liu and Yong Li
2020, 25(4): 1383-1395 doi: 10.3934/dcdsb.2019232 +[Abstract](208) +[HTML](68) +[PDF](2241.96KB)

In this paper, we study the existence of periodic solutions for a class of differential-algebraic equation

where \begin{document}$ h(t, x) = A(t)x(t) $\end{document}, \begin{document}$ h(t, x) $\end{document} and \begin{document}$ f(t, x) $\end{document} are \begin{document}$ T $\end{document}-periodic in first variable. Via the topological degree theory, and the method of guiding functions, some existence theorems are presented. To our knowledge, this is the first approach to periodic solutions of differential-algebraic equations. Some examples and numerical simulations are given to illustrate our results.

Bautin bifurcation in a minimal model of immunoediting
Joaquín Delgado, Eymard Hernández–López and Lucía Ivonne Hernández–Martínez
2020, 25(4): 1397-1414 doi: 10.3934/dcdsb.2019233 +[Abstract](162) +[HTML](69) +[PDF](1176.89KB)

One of the simplest model of immune surveillance and neoplasia was proposed by Delisi and Resigno [6]. Later Liu et al [10] proved the existence of non-degenerate Takens-Bogdanov (BT) bifurcations defining a surface in the whole set of five positive parameters. In this paper we prove the existence of Bautin bifurcations completing the scenario of possible codimension two bifurcations that occur in this model. We give an interpretation of our results in terms of the Immuno Edition Theory (IET) of three phases: elimination, equilibrium and escape.

Dynamics of charged elastic bodies under diffusion at large strains
Tomáš Roubíček and Giuseppe Tomassetti
2020, 25(4): 1415-1437 doi: 10.3934/dcdsb.2019234 +[Abstract](151) +[HTML](77) +[PDF](474.81KB)

We present a model for the dynamics of elastic or poroelastic bodies with monopolar repulsive long-range (electrostatic) interactions at large strains. Our model respects (only) locally the non-self-interpenetration condition but can cope with possible global self-interpenetration, yielding thus a certain justification of most of engineering calculations which ignore these effects in the analysis of elastic structures. These models necessarily combines Lagrangian (material) description with Eulerian (actual) evolving configuration evolving in time. Dynamical problems are studied by adopting the concept of nonlocal nonsimple materials, applying the change of variables formula for Lipschitz-continuous mappings, and relying on a positivity of determinant of deformation gradient thanks to a result by Healey and Krömer.

Periodicity and stabilization control of the delayed Filippov system with perturbation
Zuowei Cai, Jianhua Huang, Liu Yang and Lihong Huang
2020, 25(4): 1439-1467 doi: 10.3934/dcdsb.2019235 +[Abstract](322) +[HTML](66) +[PDF](694.29KB)

By employing Leray-Schauder alternative theorem of set-valued maps and non-Lyapunov method (non-smooth analysis, inequality analysis, matrix theory), this paper investigates the problems of periodicity and stabilization for time-delayed Filippov system with perturbation. Several criteria are obtained to ensure the existence of periodic solution of time-delayed Filippov system by using differential inclusion. By designing appropriate switching state-feedback controller, the asymptotic stabilization and exponential stabilization control of Filippov system are realized. Applying these criteria and control design method to a class of time-delayed neural networks with perturbation and discontinuous activation functions under a periodic environment. The developed theorems improve the existing results and their effectiveness are demonstrated by numerical example.

Traveling waves for a nonlocal dispersal vaccination model with general incidence
Jinling Zhou, Yu Yang and Cheng-Hsiung Hsu
2020, 25(4): 1469-1495 doi: 10.3934/dcdsb.2019236 +[Abstract](183) +[HTML](67) +[PDF](432.69KB)

This paper is concerned with the existence and asymptotic behavior of traveling wave solutions for a nonlocal dispersal vaccination model with general incidence. We first apply the Schauder's fixed point theorem to prove the existence of traveling wave solutions when the wave speed is greater than a critical speed \begin{document}$ c^* $\end{document}. Then we investigate the boundary asymptotic behaviour of traveling wave solutions at \begin{document}$ +\infty $\end{document} by using an appropriate Lyapunov function. Applying the method of two-sided Laplace transform, we further prove the non-existence of traveling wave solutions when the wave speed is smaller than \begin{document}$ c^* $\end{document}. From our work, one can see that the diffusion rate and nonlocal dispersal distance of the infected individuals can increase the critical speed \begin{document}$ c^* $\end{document}, while vaccination reduces the critical speed \begin{document}$ c^* $\end{document}. In addition, two specific examples are provided to verify the validity of our theoretical results, which cover and improve some known results.

An SICR rumor spreading model in heterogeneous networks
Jinxian Li, Ning Ren and Zhen Jin
2020, 25(4): 1497-1515 doi: 10.3934/dcdsb.2019237 +[Abstract](183) +[HTML](71) +[PDF](4277.85KB)

This article discusses the spread of rumors in heterogeneous networks. Using the probability generating function method and the approximation theory, we establish an SICR rumor model and calculate the threshold conditions for the outbreak of the rumor. We also compare the speed of the rumors spreading with different initial conditions. The numerical simulations of the SICR model in this paper fit well with the stochastic simulations, which means that the model is reliable. Moreover the effects of the parameters in the model on the transmission of rumors are studied numerically.

Critical and super-critical abstract parabolic equations
Tomasz Dlotko, Tongtong Liang and Yejuan Wang
2020, 25(4): 1517-1541 doi: 10.3934/dcdsb.2019238 +[Abstract](199) +[HTML](79) +[PDF](412.26KB)

Our purpose is to formulate an abstract result, motivated by the recent paper [8], allowing to treat the solutions of critical and super-critical equations as limits of solutions to their regularizations. In both cases we are improving the viscosity, making it stronger, solving the obtained regularizations with the use of Dan Henry's technique, then passing to the limit in the improved viscosity term to get a solution of the limit problem. While in case of the critical problems we will just consider a 'bit higher' fractional power of the viscosity term, for super-critical problems we need to use a version of the 'vanishing viscosity technique' that comes back to the considerations of E. Hopf, O.A. Oleinik, P.D. Lax and J.-L. Lions from 1950th. In both cases, the key to that method are the uniform with respect to the parameter estimates of the approximating solutions. The abstract result is illustrated with the Navier-Stokes equation in space dimensions 2 to 4, and with the 2-D quasi-geostrophic equation. Various technical estimates related to that problems and their fractional generalizations are also presented in the paper.

Efficient Legendre dual-Petrov-Galerkin methods for odd-order differential equations
Shan Li, Shi-Mi Yan and Zhong-Qing Wang
2020, 25(4): 1543-1563 doi: 10.3934/dcdsb.2019239 +[Abstract](171) +[HTML](69) +[PDF](429.1KB)

Efficient Legendre dual-Petrov-Galerkin methods for solving odd-order differential equations are proposed. Some Sobolev bi-orthogonal basis functions are constructed which lead to the diagonalization of discrete systems. Accordingly, both the exact solutions and the approximate solutions can be represented as infinite and truncated Fourier-like series. Numerical results indicate that the suggested methods are extremely accurate and efficient, and suitable for the odd-order equations.

Finite dimensional reducing and smooth approximating for a class of stochastic partial differential equations
Min Yang and Guanggan Chen
2020, 25(4): 1565-1581 doi: 10.3934/dcdsb.2019240 +[Abstract](166) +[HTML](64) +[PDF](349.13KB)

This work provides a finite dimensional reducing and a smooth approximating for a class of stochastic partial differential equations with an additive white noise. Using the invariant random cone to show the asymptotical completion, this stochastic partial differential equation is reduced to a stochastic ordinary differential equation on a random invariant manifold. Furthermore, after deriving the finite dimensional reducing for another stochastic partial differential equation driven by a Wong-Zakai scheme via a smooth colored noise, it is proved that when the smooth colored noise tends to the white noise, the solution and the finite dimensional reducing of the approximate system converge pathwisely to those of the original system.

G-neutral stochastic differential equations with variable delay and non-Lipschitz coefficients
Mahmoud Abouagwa and Ji Li
2020, 25(4): 1583-1606 doi: 10.3934/dcdsb.2019241 +[Abstract](215) +[HTML](102) +[PDF](401.21KB)

This paper has two parts. In part Ⅰ, existence and uniqueness theorem is established for solutions of neutral stochastic differential equations with variable delays driven by \begin{document}$ G $\end{document}-Brownian motion (VNSDDEGs in short) under global Carathéodory conditions. In part Ⅱ, a simplified VNSDDEGs for the original one is proposed. And the convergence both in \begin{document}$ L^p $\end{document}-sense and capacity between the solutions of the simplified and original VNSDDEGs are established in view of the approximation theorems. Two examples are conducted to justify the theoretical results of the approximation theorems.

Investigating the effects of intervention strategies in a spatio-temporal anthrax model
Buddhi Pantha, Judy Day and Suzanne Lenhart
2020, 25(4): 1607-1622 doi: 10.3934/dcdsb.2019242 +[Abstract](267) +[HTML](92) +[PDF](1484.08KB)

In this paper, we extend our previous work on optimal control applied in an anthrax outbreak in wild animals. We use a system of ordinary differential equation (ODE) and partial differential equations (PDEs) to track the change in susceptible, infected and vaccinated animals as well as the infected carcasses. In addition to the assumption that the infected animals and the infected carcasses are the main source of infection, we consider the animal movement by diffusion and see its effects in disease transmission. Two controls: vaccinating susceptible animals and disposing infected carcasses properly are applied in the model and these controls depend on both space and time. We formulate an optimal control problem to investigate the effect of intervention strategies in our spatio-temporal model in controlling the outbreak at minimum cost. Finally some numerical results for the optimal control problem are presented.

2018  Impact Factor: 1.008




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