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Discrete & Continuous Dynamical Systems - B

August 2021 , Volume 26 , Issue 8

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Complex dynamics in a quasi-periodic plasma perturbations model
Xin Zhang and Shuangling Yang
2021, 26(8): 4013-4043 doi: 10.3934/dcdsb.2020272 +[Abstract](709) +[HTML](285) +[PDF](981.3KB)

In this paper, the complex dynamics of a quasi-periodic plasma perturbations (QPP) model, which governs the interplay between a driver associated with pressure gradient and relaxation of instability due to magnetic field perturbations in Tokamaks, are studied. The model consists of three coupled ordinary differential equations (ODEs) and contains three parameters. This paper consists of three parts: (1) We study the stability and bifurcations of the QPP model, which gives the theoretical interpretation of various types of oscillations observed in [Phys. Plasmas, 18(2011):1-7]. In particular, assuming that there exists a finite time lag \begin{document}$ \tau $\end{document} between the plasma pressure gradient and the speed of the magnetic field, we also study the delay effect in the QPP model from the point of view of Hopf bifurcation. (2) We provide some numerical indices for identifying chaotic properties of the QPP system, which shows that the QPP model has chaotic behaviors for a wide range of parameters. Then we prove that the QPP model is not rationally integrable in an extended Liouville sense for almost all parameter values, which may help us distinguish values of parameters for which the QPP model is integrable. (3) To understand the asymptotic behavior of the orbits for the QPP model, we also provide a complete description of its dynamical behavior at infinity by the Poincaré compactification method. Our results show that the input power \begin{document}$ h $\end{document} and the relaxation of the instability \begin{document}$ \delta $\end{document} do not affect the global dynamics at infinity of the QPP model and the heat diffusion coefficient \begin{document}$ \eta $\end{document} just yield quantitative, but not qualitative changes for the global dynamics at infinity of the QPP model.

Effect of diffusion in a spatial SIS epidemic model with spontaneous infection
Yachun Tong, Inkyung Ahn and Zhigui Lin
2021, 26(8): 4045-4057 doi: 10.3934/dcdsb.2020273 +[Abstract](495) +[HTML](246) +[PDF](361.7KB)

This paper is concerned with an SIS epidemic reaction-diffusion model with mass-action incidence incorporating spontaneous infection in a spatially heterogeneous environment. The main goal of this article is to study the influence of spontaneous infection on the endemic equilibrium (EE) of the model. To achieve this, first the existence of EE is investigated. Furthermore, we discuss the asymptotic behavior of endemic equilibrium if the migration rate of the susceptible or infected population is sufficiently small. Compared to the case without spontaneous infection, our theoretical results show that spontaneous infection can enhance persistence of infectious disease.

A theoretical approach to understanding rumor propagation dynamics in a spatially heterogeneous environment
Linhe Zhu, Wenshan Liu and Zhengdi Zhang
2021, 26(8): 4059-4092 doi: 10.3934/dcdsb.2020274 +[Abstract](520) +[HTML](245) +[PDF](1440.35KB)

Most of the previous work on rumor propagation either focus on ordinary differential equations with temporal dimension or partial differential equations (PDE) with only consideration of spatially independent parameters. Little attention has been given to rumor propagation models in a spatiotemporally heterogeneous environment. This paper is dedicated to investigating a SCIR reaction-diffusion rumor propagation model with a general nonlinear incidence rate in both heterogeneous and homogeneous environments. In spatially heterogeneous case, the well-posedness of global solutions is established first. The basic reproduction number \begin{document}$ R_0 $\end{document} is introduced, which can be used to reveal the threshold-type dynamics of rumor propagation: if \begin{document}$ R_0 < 1 $\end{document}, the rumor-free steady state is globally asymptotically stable, while \begin{document}$ R_0 > 1 $\end{document}, the rumor is uniformly persistent. In spatially homogeneous case, after introducing the time delay, the stability properties have been extensively studied. Finally, numerical simulations are presented to illustrate the validity of the theoretical analysis and the influence of spatial heterogeneity on rumor propagation is further demonstrated.

When do Keller–Segel systems with heterogeneous logistic sources admit generalized solutions?
Jianlu Yan and Mario Fuest
2021, 26(8): 4093-4109 doi: 10.3934/dcdsb.2020275 +[Abstract](517) +[HTML](261) +[PDF](415.68KB)

We construct global generalized solutions to the chemotaxis system

in smooth, bounded domains \begin{document}$ \Omega \subset \mathbb R^n $\end{document}, \begin{document}$ n \geq 2 $\end{document}, for certain choices of \begin{document}$ \lambda, \mu $\end{document} and \begin{document}$ \kappa $\end{document}.

Here, inter alia, the selections \begin{document}$ \mu(x) = |x|^\alpha $\end{document} with \begin{document}$ \alpha < 2 $\end{document} and \begin{document}$ \kappa = 2 $\end{document} as well as \begin{document}$ \mu \equiv \mu_1 > 0 $\end{document} and \begin{document}$ \kappa > \min\{\frac{2n-2}{n}, \frac{2n+4}{n+4}\} $\end{document} are admissible (in both cases for any sufficiently smooth \begin{document}$ \lambda $\end{document}).

While the former case appears to be novel in general, in the two- and three-dimensional setting, the latter improves on a recent result by Winkler (Adv. Nonlinear Anal. 9 (2019), no. 1,526–566), where the condition \begin{document}$ \kappa > \frac{2n+4}{n+4} $\end{document} has been imposed. In particular, for \begin{document}$ n = 2 $\end{document}, our result shows that taking any \begin{document}$ \kappa > 1 $\end{document} suffices to exclude the possibility of collapse into a persistent Dirac distribution.

Time-domain analysis of forward obstacle scattering for elastic wave
Lu Zhao, Heping Dong and Fuming Ma
2021, 26(8): 4111-4130 doi: 10.3934/dcdsb.2020276 +[Abstract](629) +[HTML](252) +[PDF](205.08KB)

This paper concerns a time-domain scattering problem of elastic plane wave by a rigid obstacle, which is immersed in an open space filled with homogeneous and isotropic elastic medium in two dimensions. A new compressed coordinate transformation is developed to reduce the scattering problem into an initial boundary value problem in a bounded domain over a finite time interval. The well-posednesss is established for the reduced problem. This paper adopts Galerkin method to prove the uniqueness results and employs energy method to derive stability results for the scattering problem. Furthermore, we achieve a priori estimate with explicit time dependence.

A new weak gradient for the stabilizer free weak Galerkin method with polynomial reduction
Xiu Ye and Shangyou Zhang
2021, 26(8): 4131-4145 doi: 10.3934/dcdsb.2020277 +[Abstract](446) +[HTML](258) +[PDF](311.28KB)

The weak Galerkin (WG) finite element method is an effective and flexible general numerical technique for solving partial differential equations. It is a natural extension of the classic conforming finite element method for discontinuous approximations, which maintains simple finite element formulation. Stabilizer free weak Galerkin methods further simplify the WG methods and reduce computational complexity. This paper explores the possibility of optimal combination of polynomial spaces that minimize the number of unknowns in the stabilizer free WG schemes without compromising the accuracy of the numerical approximation. A new stabilizer free weak Galerkin finite element method is proposed and analyzed with polynomial degree reduction. To achieve such a goal, a new definition of weak gradient is introduced. Error estimates of optimal order are established for the corresponding WG approximations in both a discrete \begin{document}$ H^1 $\end{document} norm and the standard \begin{document}$ L^2 $\end{document} norm. The numerical examples are tested on various meshes and confirm the theory.

Optimal control of an avian influenza model with multiple time delays in state and control variables
Ting Kang, Qimin Zhang and Haiyan Wang
2021, 26(8): 4147-4171 doi: 10.3934/dcdsb.2020278 +[Abstract](553) +[HTML](288) +[PDF](2534.34KB)

In this paper, we consider an optimal control model governed by a class of delay differential equation, which describe the spread of avian influenza virus from the poultry to human. We take three control variables into the optimal control model, namely: slaughtering to the susceptible and infected poultry (\begin{document}$ u_{1}(t) $\end{document}), educational campaign to the susceptible human population (\begin{document}$ u_{2}(t) $\end{document}) and treatment to infected population (\begin{document}$ u_{3}(t) $\end{document}). The model involves two time delays that stand for the incubation periods of avian influenza virus in the infective poultry and human populations. We derive first order necessary conditions for existence of the optimal control and perform several numerical simulations. Numerical results show that different control strategies have different effects on controlling the outbreak of avian influenza. At the same time, we discuss the influence of time delays on objective function and conclude that the spread of avian influenza will slow down as the time delays increase.

Dynamics of the food-chain population in a polluted environment with impulsive input of toxicant
Yu Mu and Wing-Cheong Lo
2021, 26(8): 4173-4190 doi: 10.3934/dcdsb.2020279 +[Abstract](474) +[HTML](283) +[PDF](3649.31KB)

Some industrial behaviors, such as wasting outputs and inadequately treated and stored hazardous materials, may pollute our environment, so some populations in the polluted habitats are at the edge of extinction. In this work, we develop a mathematical model that validates the dynamics of the food-chain population in a polluted environment with impulsive toxicant input. Based on the model, we obtain a sufficient condition for the extinction of populations. When the concentration of toxicants surpasses the threshold, it will contribute to the extinction of populations in the related environment. Also, sufficient conditions for the permanence of populations are obtained in our analysis. Several numerical simulations validate the theoretical conclusions and further reflect the influence of toxicants.

Dynamic analysis of an $ SEIR $ epidemic model with a time lag in awareness allocated funds
Riane Hajjami, Mustapha El Jarroudi, Aadil Lahrouz, Adel Settati, Mohamed EL Fatini and Kai Wang
2021, 26(8): 4191-4225 doi: 10.3934/dcdsb.2020285 +[Abstract](482) +[HTML](248) +[PDF](3063.9KB)

The spread of infectious diseases is often accompanied by a rise in the awareness programs to educate the general public about the infection risk and suggest necessary preventive practices. In the present paper we propose to study the impact of awareness on the dynamics of the classical \begin{document}$ SEIR $\end{document} by considering the budget allocation to warn people as a new dynamic variable. In the model formulation, it is assumed that the susceptible individuals contract the infection via a direct contact with infected individuals, and that the transmission rate is presented by a general decreasing function of the availability of funds. We further introduced a time delay in the growth rate of the budget allocation related to the number of reported infected cases. The existence and the stability criteria of the equilibrium states are obtained in terms of the basic reproduction number \begin{document}$ \mathcal{R}_{a} $\end{document}. It is shown that \begin{document}$ \mathcal{R}_{a} \leq 1 $\end{document} is a necessary and sufficient condition for the global stability of the disease-free equilibrium, and by application of the geometric approach based on the third additive compound matrix we derived sufficient conditions for the global stability of the positive equilibrium state in the absence of delay. Our analysis reveals that awareness programs have the ability to reduce the infection prevalence. However, delay in providing funds destabilizes the system and give rise to periodic oscillations through Hopf-bifurcation. The direction and the stability of the bifurcating periodic solutions are investigated by using the normal form theory and central manifold theorem. Numerical simulations and sensitive analysis are provided to illustrate the theoretical findings..

Emergent behaviors of the generalized Lohe matrix model
Seung-Yeal Ha and Hansol Park
2021, 26(8): 4227-4261 doi: 10.3934/dcdsb.2020286 +[Abstract](480) +[HTML](227) +[PDF](460.82KB)

We present a first-order aggregation model on the space of complex matrices which can be derived from the Lohe tensor model on the space of tensors with the same rank and size. We call such matrix-valued aggregation model as "the generalized Lohe matrix model". For the proposed matrix model with two cubic coupling terms, we study several structural properties such as the conservation laws, solution splitting property. In particular, for the case of only one coupling, we reformulate the reduced Lohe matrix model into the Lohe matrix model with a diagonal frustration, and provide several sufficient frameworks leading to the complete and practical aggregations. For the estimates of collective dynamics, we use a nonlinear functional approach using an ensemble diameter which measures the degree of aggregation.

A flow on $ S^2 $ presenting the ball as its minimal set
Tiago Carvalho and Luiz Fernando Gonçalves
2021, 26(8): 4263-4280 doi: 10.3934/dcdsb.2020287 +[Abstract](649) +[HTML](274) +[PDF](591.63KB)

The main goal of this paper is to present the existence of a vector field tangent to the unit sphere \begin{document}$ S^2 $\end{document} such that \begin{document}$ S^2 $\end{document} itself is a minimal set. This is reached using a piecewise smooth (discontinuous) vector field and following the Filippov's convention on the switching manifold. As a consequence, none regularization process applied to the initial model can be topologically equivalent to it and we obtain a vector field tangent to \begin{document}$ S^2 $\end{document} without equilibria.

Asymptotics in a two-species chemotaxis system with logistic source
Wenji Zhang and Pengcheng Niu
2021, 26(8): 4281-4298 doi: 10.3934/dcdsb.2020288 +[Abstract](388) +[HTML](215) +[PDF](344.04KB)

This paper deals with nonnegative solutions of a fully parabolic two-species chemotaxis system with competitive kinetics under homogeneous Neumann boundary conditions in a N-dimensional bounded smooth domain with reasonably smooth nonnegative initial data. In a previous paper of Bai & Winkler (2016), the equilibrium of the global bounded classical solution was shown in both coexistence and extinction cases. We extend this result to weak solutions and prove these solutions globally exist and finally converge to the same semi-trivial steady state in a certain sense.

On initial value and terminal value problems for subdiffusive stochastic Rayleigh-Stokes equation
Tomás Caraballo, Tran Bao Ngoc, Tran Ngoc Thach and Nguyen Huy Tuan
2021, 26(8): 4299-4323 doi: 10.3934/dcdsb.2020289 +[Abstract](570) +[HTML](260) +[PDF](484.63KB)

In this paper, we study two stochastic problems for time-fractional Rayleigh-Stokes equation including the initial value problem and the terminal value problem. Here, two problems are perturbed by Wiener process, the fractional derivative are taken in the sense of Riemann-Liouville, the source function and the time-spatial noise are nonlinear and satisfy the globally Lipschitz conditions. We attempt to give some existence results and regularity properties for the mild solution of each problem.

Upper semi-continuity of attractors for non-autonomous fractional stochastic parabolic equations with delay
Pengyu Chen and Xuping Zhang
2021, 26(8): 4325-4357 doi: 10.3934/dcdsb.2020290 +[Abstract](422) +[HTML](208) +[PDF](505.29KB)

This paper is concerned with the asymptotic behavior of the solutions to a class of non-autonomous nonlocal fractional stochastic parabolic equations with delay defined on bounded domain. We first prove the existence of a continuous non-autonomous random dynamical system for the equations as well as the uniform estimates of solutions with respect to the delay time and noise intensity. We then show pullback asymptotical compactness of solutions as well as the existence and uniqueness of tempered random attractors by utilizing the Arzela-Ascoli theorem and the uniform estimates of solutions in fractional Sobolev space \begin{document}$ H^\alpha(\mathbb{R}^n) $\end{document} with \begin{document}$ \alpha\in (0,1) $\end{document} as well as their time derivatives in \begin{document}$ L^2(\mathbb{R}^n) $\end{document}. Finally, we establish the upper semi-continuity of the random attractors when noise intensity and time delay approaches zero, respectively.

Density function analysis for a stochastic SEIS epidemic model with non-degenerate diffusion
Qun Liu and Qingmei Chen
2021, 26(8): 4359-4373 doi: 10.3934/dcdsb.2020291 +[Abstract](394) +[HTML](623) +[PDF](333.83KB)

In this paper, we construct a stochastic SEIS epidemic model that incorporates constant recruitment, non-degenerate diffusion and infectious force in the latent period and infected period. By solving the corresponding Fokker-Planck equation, we obtain the exact expression of the density function around the endemic equilibrium of the deterministic system provided that the basic reproduction number is greater than one. Our work greatly improves the result of Chen [A new idea on density function and covariance matrix analysis of a stochastic SEIS epidemic model with degenerate diffusion, Appl. Math. Lett., 2020, 106200].

Modulation approximation for the quantum Euler-Poisson equation
Dongfen Bian, Huimin Liu and Xueke Pu
2021, 26(8): 4375-4405 doi: 10.3934/dcdsb.2020292 +[Abstract](416) +[HTML](229) +[PDF](474.0KB)

The nonlinear Schrödinger (NLS) equation is used to describe the envelopes of slowly modulated spatially and temporally oscillating wave packet-like solutions, which can be derived as a formal approximation equation of the quantum Euler-Poisson equation. In this paper, we rigorously justify such an approximation by taking a modified energy functional and a space-time resonance method to overcome the difficulties induced by the quadratic terms, resonance and quasilinearity.

Bifurcations in an economic model with fractional degree
Shaowen Shi and Weinian Zhang
2021, 26(8): 4407-4431 doi: 10.3934/dcdsb.2020293 +[Abstract](496) +[HTML](225) +[PDF](2441.08KB)

A planar ODE system which models the industrialization of a small open economy is considered. Because fractional powers are involved, its interior equilibria are hardly found by solving a transcendental equation and the routine qualitative analysis is not applicable. We qualitatively discuss the transcendental equation, eliminating the transcendental term to polynomialize the expression of extreme value, so that we can compute polynomials to obtain the number of interior equilibria in all cases and complete their qualitative analysis. Orbits near the origin, at which the system cannot be extended differentiably, are investigated by using the GNS method. Then we display all bifurcations of equilibria such as saddle-node bifurcation, transcritical bifurcation and a codimension 2 bifurcation on a one-dimensional center manifold. Furthermore, we prove nonexistence of closed orbits, homoclinic loops and heteroclinic loops, exhibit global orbital structure of the system and analyze the tendency of the industrialization development.

Pullback attractors for a weakly damped wave equation with delays and sup-cubic nonlinearity
Kaixuan Zhu, Yongqin Xie and Xinyu Mei
2021, 26(8): 4433-4458 doi: 10.3934/dcdsb.2020294 +[Abstract](467) +[HTML](224) +[PDF](483.92KB)

In this paper, we consider the weakly damped wave equations with hereditary effects and the nonlinearity \begin{document}$ f $\end{document} satisfying sup-cubic growth. Based on the recent extension of the Strichartz estimates to the case of bounded domains, we establish the global well-posedness of the Shatah-Struwe solutions for the non-autonomous case. Then, we prove the existence of the pullback \begin{document}$ \mathcal{D} $\end{document}-attractors in \begin{document}$ C_{H_{0}^{1}(\Omega)}\times C_{L^{2}(\Omega)} $\end{document} for the solutions process \begin{document}$ \{U(t,\tau)\}_{t\geq\tau} $\end{document} by applying the idea of contractive functions.

Bifurcation analysis of a general activator-inhibitor model with nonlocal dispersal
Xiaoli Wang and Guohong Zhang
2021, 26(8): 4459-4477 doi: 10.3934/dcdsb.2020295 +[Abstract](477) +[HTML](215) +[PDF](591.06KB)

In this paper, we are mainly concerned with the effect of nonlocal diffusion and dispersal spread on bifurcations of a general activator-inhibitor system in which the activator has a nonlocal dispersal. We find that spatially inhomogeneous patterns always exist if the dispersal rate of the activator is sufficiently small, while a larger dispersal spread and an increase of the activator diffusion inhibit the formation of spatial patterns. Compared with the "spatial averaging" nonlocal dispersal model, our model admits a larger parameter region supporting pattern formations, which is also true if compared with the local reaction-diffusion one when the dispersal spread is small. We also study the existence of nonconstant positive steady states through bifurcation theory and find that there could exist finite or infinite steady state bifurcation points of the inhibitor diffusion constant. As an example of our results, we study a water-biomass model with nonlocal dispersal of plants and show that the water and plant distributions could be inphase and antiphase.

Lyapunov functions for disease models with immigration of infected hosts
Connell McCluskey
2021, 26(8): 4479-4491 doi: 10.3934/dcdsb.2020296 +[Abstract](432) +[HTML](221) +[PDF](380.96KB)

Recent work has produced examples where models of the spread of infectious disease with immigration of infected hosts are shown to be globally asymptotically stable through the use of Lyapunov functions. In each case, the Lyapunov function was similar to a Lyapunov function that worked for the corresponding model without immigration of infected hosts.

We distill the calculations from the individual examples into a general result, finding algebraic conditions under which the Lyapunov function for a model without immigration of infected hosts extends to be a valid Lyapunov function for the corresponding system with immigration of infected hosts.

Finally, the method is applied to a multi-group \begin{document}$ SIR $\end{document} model.

Scattering and strong instability of the standing waves for dipolar quantum gases
Juan Huang
2021, 26(8): 4493-4513 doi: 10.3934/dcdsb.2020297 +[Abstract](461) +[HTML](216) +[PDF](412.19KB)

This paper concerns the nonlinear Schrödinger equation which describes the dipolar quantum gases. When the energy plus mass is lower than the mass of the ground state, we find we can use the kinetic energy and mass of the initial data to divide the subspace into two parts. If the initial data are in one of the parts, the solutions exist globally. Moreover, by using the Kening-Merle roadmap method, we find that these solutions will scatter. If initial data are in the other part, the solutions will collapse. And hence, the standing waves are strong unstable.

Guaranteed cost control of discrete-time switched saturated systems
Haijun Sun and Xinquan Zhang
2021, 26(8): 4515-4522 doi: 10.3934/dcdsb.2020300 +[Abstract](502) +[HTML](207) +[PDF](328.77KB)

The problem of guaranteed cost control is investigated for a class of discrete-time saturated switched systems. The purpose is to design the switched law and state feedback control law such that the closed-loop system is asymptotically stable and the upper-bound of the cost function is minimized. Based on the multiple Lyapunov functions approach, some sufficient conditions for the existence of guaranteed cost controllers are obtained. Furthermore, a convex optimization problem with linear matrix inequalities (LMI) constraints is formulated to determine the minimum upper-bound of the cost function. Finally, a numerical example is given to demonstrate the effectiveness of the proposed method.

Periodic solutions of a tumor-immune system interaction under a periodic immunotherapy
Gladis Torres-Espino and Claudio Vidal
2021, 26(8): 4523-4547 doi: 10.3934/dcdsb.2020301 +[Abstract](444) +[HTML](228) +[PDF](3883.95KB)

In this paper, we consider a mathematical model of a tumor-immune system interaction when a periodic immunotherapy treatment is applied. We give sufficient conditions, using averaging theory, for the existence and stability of periodic solutions in such system as a function of the six parameters associated to this problem. Finally, we provide examples where our results are applied.

The effect of caputo fractional difference operator on a novel game theory model
Amina-Aicha Khennaoui, A. Othman Almatroud, Adel Ouannas, M. Mossa Al-sawalha, Giuseppe Grassi and Viet-Thanh Pham
2021, 26(8): 4549-4565 doi: 10.3934/dcdsb.2020302 +[Abstract](564) +[HTML](255) +[PDF](10563.79KB)

It is well-known that fractional-order discrete-time systems have a major advantage over their integer-order counterparts, because they can better describe the memory characteristics and the historical dependence of the underlying physical phenomenon. This paper presents a novel fractional-order triopoly game with bounded rationality, where three firms producing differentiated products compete over a common market. The proposed game theory model consists of three fractional-order difference equations and is characterized by eight equilibria, including the Nash fixed point. When suitable values for the fractional order are considered, the stability of the Nash equilibrium is lost via a Neimark-Sacker bifurcation or via a flip bifurcation. As a consequence, a number of chaotic attractors appear in the system dynamics, indicating that the behaviour of the economic model becomes unpredictable, independently of the actions of the considered firm. The presence of chaos is confirmed via both the computation of the maximum Lyapunov exponent and the 0-1 test. Finally, an entropy algorithm is used to measure the complexity of the proposed game theory model.

Time periodic solutions for a two-species chemotaxis-Navier-Stokes system
Changchun Liu and Pingping Li
2021, 26(8): 4567-4585 doi: 10.3934/dcdsb.2020303 +[Abstract](758) +[HTML](283) +[PDF](328.4KB)

We consider a chemotaxis-Navier-Stokes system in two dimensional bounded domains. It is asserted that the chemotaxis system admits a time periodic solution under some conditions.

A dynamical theory for singular stochastic delay differential equations Ⅱ: nonlinear equations and invariant manifolds
Mazyar Ghani Varzaneh and Sebastian Riedel
2021, 26(8): 4587-4612 doi: 10.3934/dcdsb.2020304 +[Abstract](442) +[HTML](201) +[PDF](429.28KB)

Building on results obtained in [21], we prove Local Stable and Unstable Manifold Theorems for nonlinear, singular stochastic delay differential equations. The main tools are rough paths theory and a semi-invertible Multiplicative Ergodic Theorem for cocycles acting on measurable fields of Banach spaces obtained in [20].

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