ISSN:

1531-3492

eISSN:

1553-524X

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## Discrete & Continuous Dynamical Systems - B

February 2021 , Volume 26 , Issue 2

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*+*[Abstract](919)

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**Abstract:**

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.

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**Abstract:**

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.

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**Abstract:**

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

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**Abstract:**

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.

*+*[Abstract](631)

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**Abstract:**

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.

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**Abstract:**

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

where

*+*[Abstract](594)

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**Abstract:**

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.

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**Abstract:**

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

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**Abstract:**

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.

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**Abstract:**

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.

*+*[Abstract](550)

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**Abstract:**

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

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**Abstract:**

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.

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**Abstract:**

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

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**Abstract:**

We study the replicator equations, also known as mean-field equations, for a simple model of cyclic dominance with any number

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**Abstract:**

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 [

*+*[Abstract](814)

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**Abstract:**

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

*+*[Abstract](798)

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**Abstract:**

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.

*+*[Abstract](654)

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**Abstract:**

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 **(ⅰ)** When no tumorous equilibrium exists, the tumour-free equilibrium is globally asymptotically stable. **(ⅱ)** When **(ⅲ)** When **(ⅳ)** When

*+*[Abstract](582)

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**Abstract:**

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.

*+*[Abstract](734)

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**Abstract:**

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

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**Abstract:**

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

*+*[Abstract](1026)

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**Abstract:**

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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