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Discrete & Continuous Dynamical Systems - B

February 2021 , Volume 26 , Issue 2

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Stabilization of stochastic differential equations driven by G-Lévy process with discrete-time feedback control
Guangjun Shen, Xueying Wu and Xiuwei Yin
2021, 26(2): 755-774 doi: 10.3934/dcdsb.2020133 +[Abstract](799) +[HTML](315) +[PDF](477.48KB)

The stabilization of stochastic differential equations driven by Brownian motion (G-Brownian motion) with discrete-time feedback controls under Lipschitz conditions has been discussed by several authors. In this paper, we first give the sufficient condition for the mean square exponential instability of stochastic differential equations driven by G-Lévy process with non-Lipschitz coefficients. Second, we design a discrete-time feedback control in the drift part and obtain the mean square exponential stability and quasi-sure exponential stability for the controlled systems. At last, we give an example to verify the obtained theory.

Stabilities and dynamic transitions of the Fitzhugh-Nagumo system
Chao Xing, Zhigang Pan and Quan Wang
2021, 26(2): 775-794 doi: 10.3934/dcdsb.2020134 +[Abstract](645) +[HTML](307) +[PDF](470.73KB)

The article aims to examine the dynamic transition of the reaction-diffusion Fitzhugh-Nagumo system defined on a thin spherical shell and a 2D-rectangular domain. The mathematical tool employed is the theory of phase transition dynamics established for dissipative dynamical systems. The main results in this paper include two parts. First, for the system on a thin spherical shell, we only focus on the transition from a real simple eigenvalue. More precisely, if the first eigenspace is three–dimensional, the system undergoes either a continuous transition or a jump transition. Besides, a mix transition is also allowed if the first eigenspace is one–dimensional. Second, for the system on a rectangular domain, both the transitions from a simple real eigenvalue and a pair of simple complex eigenvalues are considered. Our results imply that two steady-state solutions bifurcate, which are either attractors or saddle points, and a Hopf bifurcation is also possible in the system on the rectangular domain.

Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations
Xiaopeng Zhao and Yong Zhou
2021, 26(2): 795-813 doi: 10.3934/dcdsb.2020142 +[Abstract](640) +[HTML](274) +[PDF](525.85KB)

The global well-posedness and large time behavior of solutions for the Cauchy problem of the three-dimensional generalized Navier-Stokes equations are studied. We first construct a local continuous solution, then by combining some a priori estimates and the continuity argument, the local continuous solution is extended to all \begin{document}$ t>0 $\end{document} step by step provided that the initial data is sufficiently small. In addition, by using Strauss's inequality, generalized interpolation type lemma and a bootstrap argument, we establish the \begin{document}$ L^p $\end{document} decay estimate for the solution \begin{document}$ u(\cdot,t) $\end{document} and all its derivatives for generalized Navier-Stokes equations with \begin{document}$ \max\{1,\frac{3+q}6\}<\alpha\leq\frac12+\min\{\frac3q-\frac3p,\frac3{2p}\} $\end{document}.

Fully discrete finite element approximation of the 2D/3D unsteady incompressible magnetohydrodynamic-Voigt regularization flows
Xiaoli Lu, Pengzhan Huang and Yinnian He
2021, 26(2): 815-845 doi: 10.3934/dcdsb.2020143 +[Abstract](622) +[HTML](299) +[PDF](382.98KB)

We devote the present paper to a fully discrete finite element scheme for the 2D/3D nonstationary incompressible magnetohydrodynamic-Voigt regularization model. This scheme is based on a finite element approximation for space discretization and the Crank-Nicolson-type scheme for time discretization, which is a two-step method. Moreover, we study stability and convergence of the fully discrete finite element scheme and obtain unconditional stability and error estimates of velocity and magnetic fields, respectively. Finally, several numerical experiments are investigated to confirm our theoretical findings.

Chaos control in a special pendulum system for ultra-subharmonic resonance
Xianwei Chen, Xiangling Fu and Zhujun Jing
2021, 26(2): 847-860 doi: 10.3934/dcdsb.2020144 +[Abstract](532) +[HTML](265) +[PDF](285.41KB)

In this paper, we study the chaos control of pendulum system with vibration of suspension axis for ultra-subharmonic resonance by using Melnikov methods, and give a necessary condition for controlling heteroclinic chaos and homoclinic chaos, respectively. We give some bifurcation diagrams by numerical simulations, which indicate that the chaos behaviors for ultra-subharmonic resonance may be inhibited to periodic orbits by adjusting phase-difference of parametric excitation, and prove that results obtained are very effective in inhibiting chaos for ultra-subharmonic resonance.

Limit cycles of planar system defined by the sum of two quasi-homogeneous vector fields
Jianfeng Huang and Haihua Liang
2021, 26(2): 861-873 doi: 10.3934/dcdsb.2020145 +[Abstract](507) +[HTML](268) +[PDF](432.83KB)

In this paper we consider the limit cycles of the planar system

where \begin{document}$ \boldsymbol X_n $\end{document} and \begin{document}$ \boldsymbol X_m $\end{document} are quasi-homogeneous vector fields of degree \begin{document}$ n $\end{document} and \begin{document}$ m $\end{document} respectively. We prove that under a new hypothesis, the maximal number of limit cycles of the system is \begin{document}$ 1 $\end{document}. We also show that our result can be applied to some systems when the previous results are invalid. The proof is based on the investigations for the Abel equation and the generalized-polar equation associated with the system, respectively. Usually these two kinds of equations need to be dealt with separately, and for both equations, an efficient approach to estimate the number of periodic solutions is constructing suitable auxiliary functions. In the present paper we introduce a formula on the divergence, which allows us to construct an auxiliary function of one equation with the auxiliary function of the other equation, and vice versa.

On the rigid-lid approximation of shallow water Bingham
Bilal Al Taki, Khawla Msheik and Jacques Sainte-Marie
2021, 26(2): 875-905 doi: 10.3934/dcdsb.2020146 +[Abstract](526) +[HTML](279) +[PDF](644.21KB)

This paper discusses the well posedness of an initial value problem describing the motion of a Bingham fluid in a basin with a degenerate bottom topography. A physical interpretation of such motion is discussed. The system governing such motion is obtained from the Shallow Water-Bingham models in the regime where the Froude number degenerates, i.e taking the limit of such equations as the Froude number tends to zero. Since we are considering equations with degenerate coefficients, then we shall work with weighted Sobolev spaces in order to establish the existence of a weak solution. In order to overcome the difficulty of the discontinuity in Bingham's constitutive law, we follow a similar approach to that introduced in [G. DUVAUT and J.-L. LIONS, Springer-Verlag, 1976]. We study also the behavior of this solution when the yield limit vanishes. Finally, a numerical scheme for the system in 1D is furnished.

Global attractor for a suspension bridge problem with a nonlinear delay term in the internal feedback
Wenjun Liu and Hefeng Zhuang
2021, 26(2): 907-942 doi: 10.3934/dcdsb.2020147 +[Abstract](565) +[HTML](261) +[PDF](610.86KB)

In the present paper, we consider a suspension bridge problem with a nonlinear delay term in the internal feedback. Namely, we investigate the following equation:

together with some suitable initial data and boundary conditions. We prove the global existence of solutions by means of the energy method combined with the Faedo-Galerkin procedure under a certain relation between the weight of the delay term in the feedback and the weight of the nonlinear frictional damping term without delay. Moreover, we establish the existence of a global attractor for the above-mentioned system by proving the existence of an absorbing set and the asymptotic smoothness of the semigroup \begin{document}$ S(t) $\end{document}.

A modified May–Holling–Tanner predator-prey model with multiple Allee effects on the prey and an alternative food source for the predator
Claudio Arancibia-Ibarra, José Flores, Michael Bode, Graeme Pettet and Peter van Heijster
2021, 26(2): 943-962 doi: 10.3934/dcdsb.2020148 +[Abstract](684) +[HTML](322) +[PDF](1814.31KB)

We study a predator-prey model with Holling type Ⅰ functional response, an alternative food source for the predator, and multiple Allee effects on the prey. We show that the model has at most two equilibrium points in the first quadrant, one is always a saddle point while the other can be a repeller or an attractor. Moreover, there is always a stable equilibrium point that corresponds to the persistence of the predator population and the extinction of the prey population. Additionally, we show that when the parameters are varied the model displays a wide range of different bifurcations, such as saddle-node bifurcations, Hopf bifurcations, Bogadonov-Takens bifurcations and homoclinic bifurcations. We use numerical simulations to illustrate the impact changing the predation rate, or the non-fertile prey population, and the proportion of alternative food source have on the basins of attraction of the stable equilibrium point in the first quadrant (when it exists). In particular, we also show that the basin of attraction of the stable positive equilibrium point in the first quadrant is bigger when we reduce the depensation in the model.

Asymmetric diffusion in a two-patch mutualism system characterizing exchange of resource for resource
Yuanshi Wang
2021, 26(2): 963-985 doi: 10.3934/dcdsb.2020149 +[Abstract](637) +[HTML](278) +[PDF](464.68KB)

This paper considers a two-patch mutualism system derived from exchange of resource for resource, where the obligate mutualist can diffuse asymmetrically between patches. First, we give a complete analysis on dynamics of the system without diffusion, which exhibit how resource production of the obligate mutualist leads to its survival/extinction. Using monotone dynamics theory, we show global stability of a positive equilibrium in the three-dimensional system with diffusion. A novel finding of this work is that the obligate species' final abundance is explicitly expressed as a function of the diffusion rate and asymmetry, which demonstrates precise mechanisms by which the diffusion and asymmetry lead to the abundance higher than if non-diffusing, even though the facultative species declines. It is shown that for a fixed diffusion rate, intermediate asymmetry is favorable while extremely large asymmetry is unfavorable; For a fixed asymmetry, small diffusion is favorable while extremely large asymmetry is unfavorable. Initial densities of the species are also shown to be important in species' persistence and abundance. Numerical simulations confirm and extend our results.

Optimal convergence rates of the magnetohydrodynamic model for quantum plasmas with potential force
Xiuli Xu and Xueke Pu
2021, 26(2): 987-1010 doi: 10.3934/dcdsb.2020150 +[Abstract](478) +[HTML](258) +[PDF](468.13KB)

In this paper, we consider the quantum magnetohydrodynamic model for quantum plasmas with potential force. We prove the optimal decay rates for the solution to the stationary state in the whole space in the \begin{document}$ L^{q}-L^{2} $\end{document} norm with \begin{document}$ 1\leq q\leq2 $\end{document}. The proof is based on the optimal decay of the linearized equations, multi-frequency decompositions and nonlinear energy estimates.

Time scale-induced asynchronous discrete dynamical systems
Stefan Siegmund and Petr Stehlík
2021, 26(2): 1011-1029 doi: 10.3934/dcdsb.2020151 +[Abstract](579) +[HTML](264) +[PDF](561.42KB)

We study two coupled discrete-time equations with different (asynchronous) periodic time scales. The coupling is of the type sample and hold, i.e., the state of each equation is sampled at its update times and held until it is read as an input at the next update time for the other equation. We construct an interpolating two-dimensional complex-valued system on the union of the two time scales and an extrapolating four-dimensional system on the intersection of the two time scales. We discuss stability by several results, examples and counterexamples in various frameworks to show that the asynchronicity can have a significant impact on the dynamical properties.

Entire solutions originating from monotone fronts for nonlocal dispersal equations with bistable nonlinearity
Fang-Di Dong, Wan-Tong Li, Shi-Liang Wu and Li Zhang
2021, 26(2): 1031-1060 doi: 10.3934/dcdsb.2020152 +[Abstract](496) +[HTML](287) +[PDF](595.17KB)

This paper mainly focuses on the entire solutions of nonlocal dispersal equations with bistable nonlinearity. Under certain assumptions of wave speed, firstly constructing appropriate super- and sub-solutions and applying corresponding comparison principle, we established the existence and related properties of entire solutions formed by the collision of three and four traveling wave solutions. Then by introducing the definition of terminated sequence, it is proved that there has no entire solutions formed by \begin{document}$ k $\end{document} traveling wave solutions that collide with each other as long as \begin{document}$ k\geq5 $\end{document}. Finally, based on the classical weighted energy approach, we obtain the global exponentially stability of the entire solutions in some weighted space.

Boundary dynamics of the replicator equations for neutral models of cyclic dominance
Eric Foxall
2021, 26(2): 1061-1082 doi: 10.3934/dcdsb.2020153 +[Abstract](492) +[HTML](260) +[PDF](858.66KB)

We study the replicator equations, also known as mean-field equations, for a simple model of cyclic dominance with any number \begin{document}$ m $\end{document} of strategies, generalizing the rock-paper-scissors model which corresponds to the case \begin{document}$ m = 3 $\end{document}. Previously the dynamics were solved for \begin{document}$ m\in\{3,4\} $\end{document} by consideration of \begin{document}$ m-2 $\end{document} conserved quantities. Here we show that for any \begin{document}$ m $\end{document}, the boundary of the phase space is partitioned into heteroclinic networks for which we give a precise description. A set of \begin{document}$ {\lfloor} m/2{\rfloor} $\end{document} conserved quantities plays an important role in the analysis. We also discuss connections to the well-mixed stochastic version of the model.

Numerical study of vanishing and spreading dynamics of chemotaxis systems with logistic source and a free boundary
Lei Yang and Lianzhang Bao
2021, 26(2): 1083-1109 doi: 10.3934/dcdsb.2020154 +[Abstract](525) +[HTML](298) +[PDF](1674.57KB)

The current paper is to investigate the numerical approximation of logistic type chemotaxis models in one space dimension with a free boundary. Such a model with a free boundary describes the spreading of a new or invasive species subject to the influence of some chemical substances in an environment with a free boundary representing the spreading front (see Bao and Shen [1], [2]). The main challenges in the numerical studies lie in tracking the moving free boundary and the nonlinear terms from the chemical. To overcome them, a front-fixing framework coupled with the finite difference method is introduced. The accuracy of the proposed method, the positivity of the solution, and the stability of the scheme are discussed. The numerical simulations agree well with theoretical results such as the vanishing spreading dichotomy, local persistence, and stability. These simulations also validate some conjectures in our future theoretical studies such as the dependence of the vanishing-spreading dichotomy on the initial value \begin{document}$ u_0 $\end{document}, initial habitat \begin{document}$ h_0 $\end{document}, the moving speed \begin{document}$ \nu $\end{document} and the chemotactic sensitivity coefficients \begin{document}$ \chi_1, \chi_2 $\end{document}.

Flocking of non-identical Cucker-Smale models on general coupling network
Yu-Jhe Huang, Zhong-Fu Huang, Jonq Juang and Yu-Hao Liang
2021, 26(2): 1111-1127 doi: 10.3934/dcdsb.2020155 +[Abstract](721) +[HTML](290) +[PDF](687.02KB)

The purpose of the paper is to investigate the flocking behavior of the discrete-time Cucker-Smale(C-S) model under general interaction network topologies with agents having their free-will accelerations. We prove theoretically that if the free-will accelerations of agents are summable, then, for any given initial conditions, the solution achieves flocking with a finite moving speed by suitably choosing the time step as well as the communication rate of the system or the strength of the interaction between agents. In particular, if the communication rate \begin{document}$ \beta $\end{document} of the system is subcritical, i.e., \begin{document}$ \beta $\end{document} is less than a critical value \begin{document}$ \beta_c $\end{document}, then flocking holds for any initial conditions regardless of the strength of the interaction between agents. While, if the communication rate is critical (\begin{document}$ \beta = \beta_c $\end{document}) or supercritical (\begin{document}$ \beta > \beta_c $\end{document}), then flocking can only be achieved by making the strength of the interaction large enough. We also present some numerical simulations to support our obtained theoretical results.

Global analysis of a model of competition in the chemostat with internal inhibitor
Mohamed Dellal and Bachir Bar
2021, 26(2): 1129-1148 doi: 10.3934/dcdsb.2020156 +[Abstract](715) +[HTML](297) +[PDF](989.13KB)

A model of two microbial species in a chemostat competing for a single resource in the presence of an internal inhibitor is considered. The model is a four-dimensional system of ordinary differential equations. Using general growth rate functions of the species, we give a complete analysis for the existence and local stability of all steady states. We describe the behavior of the system with respect to the operating parameters represented by the dilution rate and the input concentrations of the substrate. The operating diagram has the operating parameters as its coordinates and the various regions defined in it correspond to qualitatively different asymptotic behavior: washout, competitive exclusion of one species, coexistence of the species, bistability, multiplicity of positive steady states. This bifurcation diagram which determines the effect of the operating parameters, is very useful to understand the model from both the mathematical and biological points of view, and is often constructed in the mathematical and biological literature.

Global dynamics in a tumor-immune model with an immune checkpoint inhibitor
Shujing Shi, Jicai Huang and Yang Kuang
2021, 26(2): 1149-1170 doi: 10.3934/dcdsb.2020157 +[Abstract](569) +[HTML](254) +[PDF](1048.82KB)

In this paper, we fill several key gaps in the study of the global dynamics of a highly nonlinear tumor-immune model with an immune checkpoint inhibitor proposed by Nikolopoulou et al. (Letters in Biomathematics, 5 (2018), S137-S159). For this tumour-immune interaction model, it is known that the model has a unique tumour-free equilibrium and at most two tumorous equilibria. We present sufficient and necessary conditions for the global stability of the tumour-free equilibrium or the unique tumorous equilibrium. The global dynamics is obtained by employing a new Dulac function to establish the nonexistence of nontrivial positive periodic orbits. Our analysis shows that we can almost completely classify the global dynamics of the model with two critical values \begin{document}$ C_{K0}, C_{K1} (C_{K0}>C_{K1}) $\end{document} for the carrying capacity \begin{document}$ C_K $\end{document} of tumour cells and one critical value \begin{document}$ d_{T0} $\end{document} for the death rate \begin{document}$ d_{T} $\end{document} of T cells. Specifically, the following are true. (ⅰ) When no tumorous equilibrium exists, the tumour-free equilibrium is globally asymptotically stable. (ⅱ) When \begin{document}$ C_K \leq C_{K1} $\end{document} and \begin{document}$ d_T>d_{T0} $\end{document}, the unique tumorous equilibrium is globally asymptotically stable. (ⅲ) When \begin{document}$ C_K >C_{K1} $\end{document}, the model exhibits saddle-node bifurcation of tumorous equilibria. In this case, we show that when a unique tumorous equilibrium exists, tumor cells can persist for all positive initial densities, or can be eliminated for some initial densities and persist for other initial densities. When two distinct tumorous equilibria exist, we show that the model exhibits bistable phenomenon, and tumor cells have alternative fates depending on the positive initial densities. (ⅳ) When \begin{document}$ C_K > C_{K0} $\end{document} and \begin{document}$ d_T = d_{T0} $\end{document}, or \begin{document}$ d_T>d_{T0} $\end{document}, tumor cells will persist for all positive initial densities.

Enhanced Backscattering of a partially coherent field from an anisotropic random lossy medium
Josselin Garnier and Knut Sølna
2021, 26(2): 1171-1195 doi: 10.3934/dcdsb.2020158 +[Abstract](511) +[HTML](274) +[PDF](642.25KB)

The weak localization or enhanced backscattering phenomenon has received a lot of attention in the literature. The enhanced backscattering cone refers to the situation that the wave backscattered by a random medium exhibits an enhanced intensity in a narrow cone around the incoming wave direction. This phenomenon can be analyzed by a formal path integral approach. Here a mathematical derivation of this result is given based on a system of equations that describes the second-order moments of the reflected wave. This system derives from a multiscale stochastic analysis of the wave field in the situation with high-frequency waves and propagation through a lossy medium with fine scale random microstructure. The theory identifies a duality relation between the spreading of the wave and the enhanced backscattering cone. It shows how the cone, its regularity and width relate to the statistical structure of the random medium. We discuss how this information in particular can be used to estimate the internal structure of the random medium based on observations of the reflected wave.

On the asymptotic behaviour of traveling wave solution for a discrete diffusive epidemic model
Ran Zhang and Shengqiang Liu
2021, 26(2): 1197-1204 doi: 10.3934/dcdsb.2020159 +[Abstract](644) +[HTML](261) +[PDF](325.05KB)

A recent paper [Y.-Y. Chen, J.-S. Guo, F. Hamel, Traveling waves for a lattice dynamical system arising in a diffusive endemic model, Nonlinearity, 30 (2017), 2334-2359] presented a discrete diffusive Kermack-McKendrick epidemic model, and the authors proved the existence of traveling wave solutions connecting the disease-free equilibrium to the endemic equilibrium. However, the boundary asymptotic behavior of the traveling waves converge to the endemic equilibrium at \begin{document}$ +\infty $\end{document} is still an open problem. In this paper, we investigate the above open problem and completely solve it by constructing suitable Lyapunov functional and employing Lebesgue dominated convergence theorem.

Forced oscillation of viscous Burgers' equation with a time-periodic force
Taige Wang and Bing-Yu Zhang
2021, 26(2): 1205-1221 doi: 10.3934/dcdsb.2020160 +[Abstract](516) +[HTML](270) +[PDF](493.11KB)

This paper is concerned about the existence of periodic solutions of the viscous Burgers' equation when a forced oscillation is prescribed. We establish the existence theory by contraction mapping in \begin{document}$ H^s[0,1] $\end{document} with \begin{document}$ s\ge 0 $\end{document}. Asymptotical periodicity is obtained as well, and the periodic solution is achieved by selecting a suitable function as initial data to generate a solution and passing time limit to infinity. Moreover, uniqueness and global stability is achieved for this periodic solution.

Collision-free flocking for a time-delay system
Maoli Chen, Xiao Wang and Yicheng Liu
2021, 26(2): 1223-1241 doi: 10.3934/dcdsb.2020251 +[Abstract](930) +[HTML](161) +[PDF](476.08KB)

The co-existence of collision avoidance and time-asymptotic flocking of multi-particle systems with measurement delay is considered. Based on Lyapunov stability theory and some auxiliary differential inequalities, a delay-related sufficient condition is established for this system to admit a time-asymptotic flocking and collision avoidance. The estimated range of the delay is given, which may affect the flocking performance of the system. An analytical expression was proposed to quantitatively analyze the upper bound of this delay. Under the flocking conditions, the exponential decay of the relative velocity of any two particles in the system is characterized. Particularly, the collision-free flocking conditions are also given for the case without delay. This work verifies that both collision avoidance and flocking behaviors can be achieved simultaneously in a delay system.

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