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Discrete & Continuous Dynamical Systems - B

June 2021 , Volume 26 , Issue 6

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Variational solutions of stochastic partial differential equations with cylindrical Lévy noise
Tomasz Kosmala and Markus Riedle
2021, 26(6): 2879-2898 doi: 10.3934/dcdsb.2020209 +[Abstract](716) +[HTML](314) +[PDF](398.97KB)
Abstract:

In this article, the existence of a unique solution in the variational approach of the stochastic evolution equation

driven by a cylindrical Lévy process \begin{document}$ L $\end{document} is established. The coefficients \begin{document}$ F $\end{document} and \begin{document}$ G $\end{document} are assumed to satisfy the usual monotonicity and coercivity conditions. The noise is modelled by a cylindrical Lévy processes which is assumed to belong to a certain subclass of cylindrical Lévy processes and may not have finite moments.

Convergence rate of solutions toward stationary solutions to the isentropic micropolar fluid model in a half line
Haibo Cui and Haiyan Yin
2021, 26(6): 2899-2920 doi: 10.3934/dcdsb.2020210 +[Abstract](638) +[HTML](294) +[PDF](436.74KB)
Abstract:

In this paper, we study the asymptotic behavior of solutions to the initial boundary value problem for the one-dimensional compressible isentropic micropolar fluid model in a half line \begin{document}$ \mathbb{R}_{+}: = (0, \infty). $\end{document} We mainly investigate the unique existence, the asymptotic stability and convergence rates of stationary solutions to the outflow problem for this model. We obtain the convergence rates of global solutions towards corresponding stationary solutions if the initial perturbation belongs to the weighted Sobolev space. The proof is based on the weighted energy method by taking into account the effect of the microrotational velocity on the viscous compressible fluid.

Periodic solutions to non-autonomous evolution equations with multi-delays
Pengyu Chen
2021, 26(6): 2921-2939 doi: 10.3934/dcdsb.2020211 +[Abstract](640) +[HTML](290) +[PDF](379.78KB)
Abstract:

In this paper, we provide some sufficient conditions for the existence, uniqueness and asymptotic stability of time \begin{document}$ \omega $\end{document}-periodic mild solutions for a class of non-autonomous evolution equation with multi-delays. This work not only extend the autonomous evolution equation with multi-delays studied in [37] to non-autonomous cases, but also greatly weaken the condition presented in [37] even for the case \begin{document}$ a(t)\equiv a $\end{document} by establishing a general abstract framework to find time \begin{document}$ \omega $\end{document}-periodic mild solutions for non-autonomous evolution equation with multi-delays. Finally, one illustrating example is supplied.

Global phase portraits and bifurcation diagrams for reversible equivariant Hamiltonian systems of linear plus quartic homogeneous polynomials
Yuzhou Tian and Yulin Zhao
2021, 26(6): 2941-2956 doi: 10.3934/dcdsb.2020214 +[Abstract](853) +[HTML](361) +[PDF](2049.56KB)
Abstract:

This paper is devoted to the complete classification of global phase portraits for reversible equivariant Hamiltonian systems of linear plus quartic homogeneous polynomials. Such system is affinely equivalent to one of five normal forms by an algebraic classification of its infinite singular points. Then, we classify the global phase portraits of these normal forms on the Poincaré disc. There are exactly \begin{document}$ 13 $\end{document} different global topological structures on the Poincaré disc. Finally we provide the bifurcation diagrams for the corresponding global phase portraits.

Optimal control of leachate recirculation for anaerobic processes in landfills
Marzia Bisi, Maria Groppi, Giorgio Martalò and Romina Travaglini
2021, 26(6): 2957-2976 doi: 10.3934/dcdsb.2020215 +[Abstract](744) +[HTML](335) +[PDF](733.68KB)
Abstract:

A mathematical model for the degradation of the organic fraction of solid waste in landfills, by means of an anaerobic bacterial population, is proposed. Additional phenomena, like hydrolysis of insoluble substrate and biomass decay, are taken into account. The evolution of the system is monitored by controlling the effects of leachate recirculation on the hydrolytic process. We investigate the optimal strategies to minimize substrate concentration and recirculation operation costs. Analytical and numerical results are presented and discussed for linear and quadratic cost functionals.

Lyapunov exponents of discrete quasi-periodic gevrey Schrödinger equations
Wenmeng Geng and Kai Tao
2021, 26(6): 2977-2996 doi: 10.3934/dcdsb.2020216 +[Abstract](706) +[HTML](344) +[PDF](378.51KB)
Abstract:

In the study of the continuity of the Lyapunov exponent for the discrete quasi-periodic Schrödinger operators, there is a pioneering result by Wang-You [21] that the authors constructed examples whose Lyapunov exponent is discontinuous in the potential with the \begin{document}$ C^0 $\end{document} norm for non-analytic potentials. In this paper, we consider this operators for some Gevrey potential, which is an analytic function having a Gevrey small perturbation, with Diophantine frequency. We prove that in the large coupling regions, the Lyapunov exponent is positive and jointly continuous in all parameters, such as the energy, the frequency and the potential. Note that all analytic functions are also Gevrey ones. Therefore, we also obtain that all of the large analytic potentials are the non-perturbative weak Hölder continuous points of the Lyapunov exponent in the Gevrey topology with \begin{document}$ C^0 $\end{document} norm. It is the first result about the continuity in non-analytic potential with this norm and is complementary to Wang-You's result.

Asymptotic profiles of the endemic equilibrium of a reaction-diffusion-advection SIS epidemic model with saturated incidence rate
Renhao Cui
2021, 26(6): 2997-3022 doi: 10.3934/dcdsb.2020217 +[Abstract](794) +[HTML](327) +[PDF](476.0KB)
Abstract:

In this paper, we consider a reaction-diffusion SIS epidemic model with saturated incidence rate in advective heterogeneous environments. The existence of the endemic equilibrium (EE) is established when the basic reproduction number is greater than one. We further investigate the effects of diffusion, advection and saturation on asymptotic profiles of the endemic equilibrium. The individuals concentrate at the downstream end when the advection rate tends to infinity. As the the diffusion rate of the susceptible individuals tends to zero, a certain portion of the susceptible population concentrates at the downstream end, and the remaining portion of the susceptible population distributes in the habitat in a non-homogeneous way; on the other hand, the density of infected population is positive on the entire habitat. The density of the infected vanishes on the habitat for small diffusion rate of infected individuals or the large saturation. The results may provide some implications on disease control and prediction.

The Keller-Segel system with logistic growth and signal-dependent motility
Hai-Yang Jin and Zhi-An Wang
2021, 26(6): 3023-3041 doi: 10.3934/dcdsb.2020218 +[Abstract](716) +[HTML](294) +[PDF](386.14KB)
Abstract:

The paper is concerned with the following chemotaxis system with nonlinear motility functions

subject to homogeneous Neumann boundary conditions in a bounded domain \begin{document}$ \Omega\subset \mathbb{R}^2 $\end{document} with smooth boundary, where the motility functions \begin{document}$ \gamma(v) $\end{document} and \begin{document}$ \chi(v) $\end{document} satisfy the following conditions

\begin{document}$ (\gamma, \chi)\in [C^2[0, \infty)]^2 $\end{document} with \begin{document}$ \gamma(v)>0 $\end{document} and \begin{document}$ \frac{|\chi(v)|^2}{\gamma(v)} $\end{document} is bounded for all \begin{document}$ v\geq 0 $\end{document}.

By employing the method of energy estimates, we establish the existence of globally bounded solutions of ($\ast$) with \begin{document}$ \mu>0 $\end{document} for any \begin{document}$ u_0 \in W^{1, \infty}(\Omega) $\end{document} with \begin{document}$ u_0 \geq (\not\equiv) 0 $\end{document}. Then based on a Lyapunov function, we show that all solutions \begin{document}$ (u, v) $\end{document} of ($\ast$) will exponentially converge to the unique constant steady state \begin{document}$ (1, 1) $\end{document} provided \begin{document}$ \mu>\frac{K_0}{16} $\end{document} with \begin{document}$ K_0 = \max\limits_{0\leq v \leq \infty}\frac{|\chi(v)|^2}{\gamma(v)} $\end{document}.

The impact of toxins on competition dynamics of three species in a polluted aquatic environment
Yuyue Zhang, Jicai Huang and Qihua Huang
2021, 26(6): 3043-3068 doi: 10.3934/dcdsb.2020219 +[Abstract](641) +[HTML](345) +[PDF](1233.99KB)
Abstract:

Accurately assessing the risks of toxins in polluted ecosystems and finding factors that determine population persistence and extirpation are important from both environmental and conservation perspectives. In this paper, we develop and study a toxin-mediated competition model for three species that live in the same polluted aquatic environment and compete for the same resources. Analytical analysis of positive invariance, existence and stability of equilibria, sensitivity of equilibria to toxin are presented. Bifurcation analysis is used to understand how the environmental toxins, plus distinct vulnerabilities of three species to toxins, affect the competition outcomes. Our results reveal that while high concentrations lead to extirpation of all species, sublethal levels of toxins affect competition outcomes in many counterintuitive ways, which include boosting coexistence of species by reducing the abundance of the predominant species, inducing many different types of bistability and even tristability, generating and reducing population oscillations, and exchanging roles of winner and loser in competition. The findings in this work provide a sound theoretical foundation for understanding and assessing population or community effects of toxicity.

An almost periodic Dengue transmission model with age structure and time-delayed input of vector in a patchy environment
Jing Feng and Bin-Guo Wang
2021, 26(6): 3069-3096 doi: 10.3934/dcdsb.2020220 +[Abstract](624) +[HTML](349) +[PDF](803.51KB)
Abstract:

In this paper, we propose an almost periodic multi-patch SIR-SEI model with age structure and time-delayed input of vector. The existence of the almost periodic disease-free solution and the definition of the basic reproduction ratio \begin{document}$ R_{0} $\end{document} are given. It is shown that the disease is uniformly persistent if \begin{document}$ R_0>1 $\end{document}, and it dies out if \begin{document}$ R_0<1 $\end{document} under the assumptions that there exists a small invasion and the same travel rate of susceptible, infective and recovered host population in different patches. Finally, we illustrate the above results by numerical simulations. In addition, a simple example shows that the basic reproduction ratio may be underestimated or overestimated if an almost periodic coefficient is approximated by a periodic one.

Mean-square delay-distribution-dependent exponential synchronization of chaotic neural networks with mixed random time-varying delays and restricted disturbances
Quan Hai and Shutang Liu
2021, 26(6): 3097-3118 doi: 10.3934/dcdsb.2020221 +[Abstract](671) +[HTML](293) +[PDF](469.18KB)
Abstract:

This paper investigates the delay-distribution-dependent exponential synchronization problem for a class of chaotic neural networks with mixed random time-varying delays as well as restricted disturbances. Given the probability distribution of the time-varying delay, stochastic variable that satisfying Bernoulli distribution is formulated to produce a new system which includes the information of the probability distribution. Based on the Lyapunov-Krasovskii functional method, the Jensen's integral inequality theory and linear matrix inequality (LMI) technique, several delay-distribution-dependent sufficient conditions are developed to guarantee that the chaotic neural networks with mixed random time-varying delays are exponentially synchronized in mean square. Furthermore, the derived results are given in terms of simplified LMI, which can be straightforwardly solved by Matlab. Finally, two numerical examples are proposed to demonstrate the feasibility and the effectiveness of the presented synchronization scheme.

A subgrid stabilizing postprocessed mixed finite element method for the time-dependent Navier-Stokes equations
Yueqiang Shang and Qihui Zhang
2021, 26(6): 3119-3142 doi: 10.3934/dcdsb.2020222 +[Abstract](621) +[HTML](339) +[PDF](886.55KB)
Abstract:

A postprocessed mixed finite element method based on a subgrid model is presented for the simulation of time-dependent incompressible Navier-Stokes equations. This method consists of two steps: the first step is to solve a subgrid stabilized nonlinear Navier-Stokes system on a coarse grid to obtain an approximate solution \begin{document}$ u_{H}(x,T) $\end{document} at the final time \begin{document}$ T $\end{document}, and the second step is to postprocess \begin{document}$ u_{H}(x,T) $\end{document} by solving a stabilized Stokes problem on a finer grid or by higher-order finite element elements defined on the same coarse grid. Stability of the method and error estimates of the processing solution are analyzed. Numerical results on an example with known analytic solution and the flow around a circular cylinder are given to verify the theoretical predictions and demonstrate the effectiveness of the proposed method.

A mathematical model to restore water quality in urban lakes using Phoslock
Pankaj Kumar Tiwari, Rajesh Kumar Singh, Subhas Khajanchi, Yun Kang and Arvind Kumar Misra
2021, 26(6): 3143-3175 doi: 10.3934/dcdsb.2020223 +[Abstract](844) +[HTML](342) +[PDF](4962.44KB)
Abstract:

Urban lakes are the life lines for the population residing in the city. Excessive amounts of phosphate entering water courses through household discharges is one of the main causes of deterioration of water quality in these lakes because of the way it drives algal productivity and undesirable changes in the balance of aquatic life. The ability to remove biologically available phosphorus in a lake is therefore a major step towards improving water quality. By removing phosphate from the water column using Phoslock essentially deprives algae and its proliferation. In view of this, we develop a mathematical model to investigate whether the application of Phoslock would significantly reduce the bio-availability of phosphate in the water column. We consider phosphorus, algae, detritus and Phoslock as dynamical variables. In the modeling process, the introduction rate of Phoslock is assumed to be proportional to the concentration of phosphorus in the lake. Further, we consider a discrete time delay which accounts for the time lag involved in the application of Phoslock. Moreover, we investigate behavior of the system by assuming the application rate of Phoslock as a periodic function of time. Our results evoke that Phoslock essentially reduces the concentration of phosphorus and density of algae, and plays crucial role in restoring the quality of water in urban lakes. We observe that for the gradual increase in the magnitude of the delay involved in application of Phoslock, the autonomous system develops limit cycle oscillations through a Hopf-bifurcation while the corresponding nonautonomous system shows chaotic dynamics through quasi-periodic oscillations.

Revisit of the Peierls-Nabarro model for edge dislocations in Hilbert space
Yuan Gao, Jian-Guo Liu, Tao Luo and Yang Xiang
2021, 26(6): 3177-3207 doi: 10.3934/dcdsb.2020224 +[Abstract](896) +[HTML](298) +[PDF](547.94KB)
Abstract:

In this paper, we revisit the mathematical validation of the Peierls–Nabarro (PN) models, which are multiscale models of dislocations that incorporate the detailed dislocation core structure. We focus on the static and dynamic PN models of an edge dislocation in Hilbert space. In a PN model, the total energy includes the elastic energy in the two half-space continua and a nonlinear potential energy, which is always infinite, across the slip plane. We revisit the relationship between the PN model in the full space and the reduced problem on the slip plane in terms of both governing equations and energy variations. The shear displacement jump is determined only by the reduced problem on the slip plane while the displacement fields in the two half spaces are determined by linear elasticity. We establish the existence and sharp regularities of classical solutions in Hilbert space. For both the reduced problem and the full PN model, we prove that a static solution is a global minimizer in a perturbed sense. We also show that there is a unique classical, global in time solution of the dynamic PN model.

Reversible polynomial Hamiltonian systems of degree 3 with nilpotent saddles
Montserrat Corbera and Claudia Valls
2021, 26(6): 3209-3233 doi: 10.3934/dcdsb.2020225 +[Abstract](604) +[HTML](279) +[PDF](4216.56KB)
Abstract:

We provide normal forms and the global phase portraits in the Poincaré disk for all Hamiltonian planar polynomial vector fields of degree 3 symmetric with respect to the \begin{document}$ x- $\end{document}axis having a nilpotent saddle at the origin.

Invariant measures of stochastic delay lattice systems
Zhang Chen, Xiliang Li and Bixiang Wang
2021, 26(6): 3235-3269 doi: 10.3934/dcdsb.2020226 +[Abstract](604) +[HTML](319) +[PDF](468.24KB)
Abstract:

This paper is concerned with the existence and uniqueness of invariant measures for infinite-dimensional stochastic delay lattice systems defined on the entire integer set. For Lipschitz drift and diffusion terms, we prove the existence of invariant measures of the systems by showing the tightness of a family of probability distributions of solutions in the space of continuous functions from a finite interval to an infinite-dimensional space, based on the idea of uniform tail-estimates, the technique of diadic division and the Arzela-Ascoli theorem. We also show the uniqueness of invariant measures when the Lipschitz coefficients of the nonlinear drift and diffusion terms are sufficiently small.

Global well-posedness of a 3D Stokes-Magneto equations with fractional magnetic diffusion
Yingdan Ji and Wen Tan
2021, 26(6): 3271-3278 doi: 10.3934/dcdsb.2020227 +[Abstract](593) +[HTML](272) +[PDF](346.6KB)
Abstract:

This paper is devoted to the global well-posedness of a three-dimensional Stokes-Magneto equations with fractional magnetic diffusion. It is proved that the equations admit a unique global-in-time strong solution for arbitrary initial data when the fractional index \begin{document}$ \alpha\ge\frac32 $\end{document}. This result might have a potential application in the theory of magnetic relaxtion.

Dynamic observers for unknown populations
Chris Guiver, Nathan Poppelreiter, Richard Rebarber, Brigitte Tenhumberg and Stuart Townley
2021, 26(6): 3279-3302 doi: 10.3934/dcdsb.2020232 +[Abstract](555) +[HTML](315) +[PDF](421.89KB)
Abstract:

Dynamic observers are considered in the context of structured-population modeling and management. Roughly, observers combine a known measured variable of some process with a model of that process to asymptotically reconstruct the unknown state variable of the model. We investigate the potential use of observers for reconstructing population distributions described by density-independent (linear) models and a class of density-dependent (nonlinear) models. In both the density-dependent and -independent cases, we show, in several ecologically reasonable circumstances, that there is a natural, optimal construction of these observers. Further, we describe the robustness these observers exhibit with respect to disturbances and uncertainty in measurement.

Asymptotic behavior of non-autonomous random Ginzburg-Landau equation driven by colored noise
Lingyu Li and Zhang Chen
2021, 26(6): 3303-3333 doi: 10.3934/dcdsb.2020233 +[Abstract](589) +[HTML](304) +[PDF](440.33KB)
Abstract:

This paper investigates mainly the long term behavior of the non-autonomous random Ginzburg-Landau equation driven by nonlinear colored noise on unbounded domains. Due to the noncompactness of Sobolev embeddings on unbounded domains, pullback asymptotic compactness of random dynamical system associated with such random Ginzburg-Landau equation is proved by the tail-estimates method. Moreover, it is proved that the pullback random attractor of the non-autonomous random Ginzburg-Landau equation driven by a linear multiplicative colored noise converges to that of the corresponding stochastic system driven by a linear multiplicative white noise.

Entropy-dissipating finite-difference schemes for nonlinear fourth-order parabolic equations
Marcel Braukhoff and Ansgar Jüngel
2021, 26(6): 3335-3355 doi: 10.3934/dcdsb.2020234 +[Abstract](739) +[HTML](264) +[PDF](438.46KB)
Abstract:

Structure-preserving finite-difference schemes for general nonlinear fourth-order parabolic equations on the one-dimensional torus are derived. Examples include the thin-film and the Derrida–Lebowitz–Speer–Spohn equations. The schemes conserve the mass and dissipate the entropy. The scheme associated to the logarithmic entropy also preserves the positivity. The idea of the derivation is to reformulate the equations in such a way that the chain rule is avoided. A central finite-difference discretization is then applied to the reformulation. In this way, the same dissipation rates as in the continuous case are recovered. The strategy can be extended to a multi-dimensional thin-film equation. Numerical examples in one and two space dimensions illustrate the dissipation properties.

Dynamics at infinity and Jacobi stability of trajectories for the Yang-Chen system
Yongjian Liu, Qiujian Huang and Zhouchao Wei
2021, 26(6): 3357-3380 doi: 10.3934/dcdsb.2020235 +[Abstract](728) +[HTML](363) +[PDF](958.44KB)
Abstract:

The present work is devoted to giving new insights into a chaotic system with two stable node-foci, which is named Yang-Chen system. Firstly, based on the global view of the influence of equilibrium point on the complexity of the system, the dynamic behavior of the system at infinity is analyzed. Secondly, the Jacobi stability of the trajectories for the system is discussed from the viewpoint of Kosambi-Cartan-Chern theory (KCC-theory). The dynamical behavior of the deviation vector near the whole trajectories (including all equilibrium points) is analyzed in detail. The obtained results show that in the sense of Jacobi stability, all equilibrium points of the system, including those of the two linear stable node-foci, are Jacobi unstable. These studies show that one might witness chaotic behavior of the system trajectories before they enter in a neighborhood of equilibrium point or periodic orbit. There exists a sort of stability artifact that cannot be found without using the powerful method of Jacobi stability analysis.

Qualitative properties and bifurcations of a leaf-eating herbivores model
Jiyu Zhong
2021, 26(6): 3381-3407 doi: 10.3934/dcdsb.2020236 +[Abstract](539) +[HTML](282) +[PDF](1525.08KB)
Abstract:

In this paper, we discuss the dynamics of a discrete-time leaf-eating herbivores model. First of all, to investigate the bifurcations of the model, we study the qualitative properties of a fixed point, including hyperbolic and non-hyperbolic. Secondly, applying the center manifold theorem, we give the conditions that the model produces a supercritical flip bifurcation and a subcritical flip bifurcation respectively, from which we find a generalized flip bifurcation point. And then, we prove rigorously that the model undergoes a generalized flip bifurcation and give three parameter regions that the model possesses two period-two cycles, one period-two cycles and none respectively. Next, computing the normal form, we prove that the model undergoes a subcritical Neimark-Sacker bifurcation and produces a unique unstable invariant circle near the fixed point. Finally, by numerical simulations, we not only verify our results but also show a saddle period-five cycle and a saddle period-six cycle on the invariant circle.

Global existence and Gevrey regularity to the Navier-Stokes-Nernst-Planck-Poisson system in critical Besov-Morrey spaces
Jinyi Sun, Zunwei Fu, Yue Yin and Minghua Yang
2021, 26(6): 3409-3425 doi: 10.3934/dcdsb.2020237 +[Abstract](599) +[HTML](276) +[PDF](415.63KB)
Abstract:

The paper is concerned with the Navier-Stokes-Nernst-Planck-Poisson system arising from electrohydrodynamics in \begin{document}$ \mathbb{R}^d $\end{document}. By means of the implicit function theorem, we prove the global existence of mild solutions for Cauchy problem of this system with small initial data in critical Besov-Morrey spaces. In comparison to the previous works, our existence result provides a new class of initial data, for which the problem is global solvability. Meanwhile, based on the so-called Gevrey estimates, we verify that the obtained mild solutions are analytic in the spatial variables. As a byproduct, we show the asymptotic stability of solutions as the time goes to infinity. Furthermore, decay estimates of higher-order derivatives of solutions are deduced in Morrey spaces.

Dynamic analysis on an almost periodic predator-prey system with impulsive effects and time delays
Demou Luo and Qiru Wang
2021, 26(6): 3427-3453 doi: 10.3934/dcdsb.2020238 +[Abstract](555) +[HTML](284) +[PDF](440.05KB)
Abstract:

This article is concerned with a generalized almost periodic predator-prey model with impulsive effects and time delays. By utilizing comparison theorem and constructing a feasible Lyapunov functional, we obtain sufficient conditions to guarantee the permanence and global asymptotic stability of the system. By applying Arzelà-Ascoli theorem, we establish the existence and uniqueness of almost-periodic positive solutions. A feasible numerical simulation is provided to explain the suitability of our main criteria.

On 3d dipolar Bose-Einstein condensates involving quantum fluctuations and three-body interactions
Yongming Luo and Athanasios Stylianou
2021, 26(6): 3455-3477 doi: 10.3934/dcdsb.2020239 +[Abstract](742) +[HTML](276) +[PDF](430.32KB)
Abstract:

We study the following nonlocal mixed order Gross-Pitaevskii equation

where \begin{document}$ K $\end{document} is the classical dipole-dipole interaction kernel, \begin{document}$ \lambda_3>0 $\end{document} and \begin{document}$ p\in(4,6] $\end{document}; the case \begin{document}$ p = 6 $\end{document} being energy critical. For \begin{document}$ p = 5 $\end{document} the equation is considered currently as the state-of-the-art model for describing the dynamics of dipolar Bose-Einstein condensates (Lee-Huang-Yang corrected dipolar GPE). We prove existence and nonexistence of standing waves in different parameter regimes; for \begin{document}$ p\neq 6 $\end{document} we prove global well-posedness and small data scattering.

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