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Discrete & Continuous Dynamical Systems - B

November 2019 , Volume 24 , Issue 11

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A note on the stochastic Ericksen-Leslie equations for nematic liquid crystals
Zdzisław Brzeźniak, Erika Hausenblas and Paul André Razafimandimby
2019, 24(11): 5785-5802 doi: 10.3934/dcdsb.2019106 +[Abstract](576) +[HTML](226) +[PDF](535.33KB)
Abstract:

In this note we prove the existence and uniqueness of local maximal smooth solution of the stochastic simplified Ericksen-Leslie systems modelling the dynamics of nematic liquid crystals under stochastic perturbations.

Almost periodic dynamical behaviors of the hematopoiesis model with mixed discontinuous harvesting terms
Fanchao Kong and Juan J. Nieto
2019, 24(11): 5803-5830 doi: 10.3934/dcdsb.2019107 +[Abstract](534) +[HTML](276) +[PDF](1530.12KB)
Abstract:

Discontinuous system is playing an increasingly important role in terms of both theory and applications. This paper presents a hematopoiesis model with mixed discontinuous harvesting terms. By using differential inclusions theory, the non-smooth analysis theory with Lyapunov-like approach, some new sufficient criteria are given to ascertain the existence, uniqueness and globally exponential stability of the bounded positive almost periodic solutions for the addressed model. Some previously known results are extended and complemented. Moreover, simulation results of two topical numerical examples are also delineated to demonstrate the effectiveness of the established theoretical results.

Strong convergence of neutral stochastic functional differential equations with two time-scales
Junhao Hu and Chenggui Yuan
2019, 24(11): 5831-5848 doi: 10.3934/dcdsb.2019108 +[Abstract](698) +[HTML](217) +[PDF](418.66KB)
Abstract:

The purpose of this paper is to discuss the strong convergence of neutral stochastic functional differential equations (NSFDEs) with two time-scales. The existence and uniqueness of invariant measure of the fast component is proved by using Wasserstein distance and the stability-in-distribution argument. The strong convergence between the slow component and the averaged component is also obtained by the the averaging principle in the spirit of Khasminskii's approach.

Modelling the effects of contaminated environments in mainland China on seasonal HFMD infections and the potential benefit of a pulse vaccination strategy
Jinyan Wang, Yanni Xiao and Robert A. Cheke
2019, 24(11): 5849-5870 doi: 10.3934/dcdsb.2019109 +[Abstract](604) +[HTML](306) +[PDF](802.81KB)
Abstract:

Substantial and increasing outbreaks of EV71-related hand, foot and mouth disease (HFMD) have occurred recently in mainland China with serious consequences for child health. The HFMD pathogens can survive for long periods outside the host in suitable conditions, and hence indirect transmission via free-living pathogens in the environment cannot be ignored. We propose a novel mathematical model of both periodic direct transmission and indirect transmission followed by incorporation of an impulsive vaccination strategy. By applying Floquet theory and the comparison theorem of impulsive differential equations, we obtained a threshold parameter which governs the extinction or the uniform persistence of the disease. The rate, frequency and timing of pulse vaccination were found to affect the basic reproduction number and the number of infected individuals significantly. In particular, frequent vaccination with a high coverage rate leads to declines in the basic reproduction number. Moreover, for a given rate of vaccination or frequency, numerical studies suggested that there was an optimal time (September, just before the start of new school terms) when the basic reproduction number and hence new HFMD infections could be minimised. Frequent high intensity vaccinations at a suitable time (e.g. September) and regular cleaning of the environment are effective measures for controlling HFMD infections.

Quasi sure exponential stabilization of nonlinear systems via intermittent $ G $-Brownian motion
Yong Ren and Wensheng Yin
2019, 24(11): 5871-5883 doi: 10.3934/dcdsb.2019110 +[Abstract](493) +[HTML](205) +[PDF](420.4KB)
Abstract:

This paper focuses on the quasi sure exponential stabilization of nonlinear systems. By virtue of exponential martingale inequality under \begin{document}$ G $\end{document}-framework and intermittent \begin{document}$ G $\end{document}-Brownian motion (in short, \begin{document}$ G $\end{document}-ISSs), we establish the sufficient conditions to guarantee quasi surely exponential stability. The efficiency of the proposed results is illustrated by the memristor-based Chua's oscillator.

On the limit cycles of planar discontinuous piecewise linear differential systems with a unique equilibrium
Shimin Li and Jaume Llibre
2019, 24(11): 5885-5901 doi: 10.3934/dcdsb.2019111 +[Abstract](622) +[HTML](336) +[PDF](626.92KB)
Abstract:

This paper deals with planar discontinuous piecewise linear differential systems with two zones separated by a vertical straight line \begin{document}$ x = k $\end{document}. We assume that the left linear differential system (\begin{document}$ x<k $\end{document}) and the right linear differential system (\begin{document}$ x>k $\end{document}) share the same equilibrium, which is located at the origin \begin{document}$ O(0, 0) $\end{document} without loss of generality.

Our results show that if \begin{document}$ k = 0 $\end{document}, that is when the unique equilibrium \begin{document}$ O(0, 0) $\end{document} is located on the line of discontinuity, then the discontinuous piecewise linear differential systems have no crossing limit cycles. While for the case \begin{document}$ k\neq 0 $\end{document} we provide lower and upper bounds for the number of limit cycles of these planar discontinuous piecewise linear differential systems depending on the type of their linear differential systems, i.e. if those systems have foci, centers, saddles or nodes, see Table 2.

Analysis of minimizers of the Lawrence-Doniach energy for superconductors in applied fields
Patricia Bauman and Guanying Peng
2019, 24(11): 5903-5926 doi: 10.3934/dcdsb.2019112 +[Abstract](501) +[HTML](203) +[PDF](490.9KB)
Abstract:

We analyze minimizers of the Lawrence-Doniach energy for layered superconductors with Josephson constant \begin{document}$ \lambda $\end{document} and Ginzburg-Landau parameter \begin{document}$ 1/\epsilon $\end{document} in a bounded generalized cylinder \begin{document}$ D = \Omega\times[0, L] $\end{document} in \begin{document}$ \mathbb{R}^3 $\end{document}, where \begin{document}$ \Omega $\end{document} is a bounded simply connected Lipschitz domain in \begin{document}$ \mathbb{R}^2 $\end{document}. Our main result is that in an applied magnetic field \begin{document}$ \vec{H}_{ex} = h_{ex}\vec{e}_{3} $\end{document} which is perpendicular to the layers with \begin{document}$ \left|\ln\epsilon\right|\ll h_{ex}\ll\epsilon^{-2} $\end{document}, the minimum Lawrence-Doniach energy is given by \begin{document}$ \frac{|D|}{2}h_{ex}\ln\frac{1}{\epsilon\sqrt{h_{ex}}}(1+o_{\epsilon, s}(1)) $\end{document} as \begin{document}$ \epsilon $\end{document} and the interlayer distance \begin{document}$ s $\end{document} tend to zero. We also prove estimates on the behavior of the order parameters, induced magnetic field, and vorticity in this regime. Finally, we observe that as a consequence of our results, the same asymptotic formula holds for the minimum anisotropic three-dimensional Ginzburg-Landau energy in \begin{document}$ D $\end{document} with anisotropic parameter \begin{document}$ \lambda $\end{document} and \begin{document}$ o_{\epsilon, s}(1) $\end{document} replaced by \begin{document}$ o_{\epsilon}(1) $\end{document}.

Almost periodic solutions and stable solutions for stochastic differential equations
Yong Li, Zhenxin Liu and Wenhe Wang
2019, 24(11): 5927-5944 doi: 10.3934/dcdsb.2019113 +[Abstract](585) +[HTML](213) +[PDF](420.42KB)
Abstract:

In this paper, we discuss the relationships between stability and almost periodicity for solutions of stochastic differential equations. Our essential idea is to get stability of solutions or systems by some inherited properties of Lyapunov functions. Under suitable conditions besides Lyapunov functions, we obtain the existence of almost periodic solutions in distribution.

Asymptotic behavior of an SIR reaction-diffusion model with a linear source
Keng Deng
2019, 24(11): 5945-5957 doi: 10.3934/dcdsb.2019114 +[Abstract](629) +[HTML](225) +[PDF](375.49KB)
Abstract:

In this paper, we consider an SIR reaction-diffusion model with a linear external source in spatially heterogeneous environment. We first study the global stability of the disease-free equilibrium in spatially heterogeneous environment and the global stability of the endemic equilibrium in spatially homogeneous environment. We then investigate the asymptotic profiles of the endemic equilibrium in spatially heterogeneous environment for small and large diffusion rates.

Upper semicontinuity of random attractors for the stochastic non-autonomous suspension bridge equation with memory
Ling Xu, Jianhua Huang and Qiaozhen Ma
2019, 24(11): 5959-5979 doi: 10.3934/dcdsb.2019115 +[Abstract](599) +[HTML](205) +[PDF](524.07KB)
Abstract:

This paper is devoted to the well-posedness and long-time behavior of a stochastic suspension bridge equation with memory effect. The existence of the random attractor for the stochastic suspension bridge equation with memory is established. Moreover, the upper semicontinuity of random attractors is also provided when the coefficient of random term approaches zero.

Stable solution induced by domain geometry in the heat equation with nonlinear boundary conditions on surfaces of revolution
Maicon Sônego
2019, 24(11): 5981-5988 doi: 10.3934/dcdsb.2019116 +[Abstract](607) +[HTML](212) +[PDF](305.4KB)
Abstract:

In this paper we consider the heat equation on surfaces of revolution subject to nonlinear Neumann boundary conditions. We provide a sufficient condition on the geometry of the surface in order to ensure the existence of an asymptotically stable nonconstant solution.

Dynamic transitions and stability for the acetabularia whorl formation
Yiqiu Mao, Dongming Yan and ChunHsien Lu
2019, 24(11): 5989-6004 doi: 10.3934/dcdsb.2019117 +[Abstract](458) +[HTML](207) +[PDF](2244.68KB)
Abstract:

Dynamical transitions of the Acetabularia whorl formation caused by outside calcium concentration is carefully analyzed using a chemical reaction diffusion model on a thin annulus. Restricting ourselves with Turing instabilities, we found all three types of transition, continuous, catastrophic and random can occur under different parameter regimes. Detailed linear analysis and numerical investigations are also provided. The main tool used in the transition analysis is Ma & Wang's dynamical transition theory including the center manifold reduction.

Bautin bifurcation in delayed reaction-diffusion systems with application to the segel-jackson model
Yuxiao Guo and Ben Niu
2019, 24(11): 6005-6024 doi: 10.3934/dcdsb.2019118 +[Abstract](542) +[HTML](239) +[PDF](810.75KB)
Abstract:

In this paper, we present an algorithm for deriving the normal forms of Bautin bifurcations in reaction-diffusion systems with time delays and Neumann boundary conditions. On the center manifold near a Bautin bifurcation, the first and second Lyapunov coefficients are calculated explicitly, which completely determine the dynamical behavior near the bifurcation point. As an example, the Segel-Jackson predator-prey model is studied. Near the Bautin bifurcation we find the existence of fold bifurcation of periodic orbits, as well as subcritical and supercritical Hopf bifurcations. Both theoretical and numerical results indicate that solutions with small (large) initial conditions converge to stable periodic orbits (diverge to infinity).

Applying battelli-fečkan's method to transversal heteroclinic bifurcation in piecewise smooth systems
Yurong Li and Zhengdong Du
2019, 24(11): 6025-6052 doi: 10.3934/dcdsb.2019119 +[Abstract](592) +[HTML](217) +[PDF](617.63KB)
Abstract:

In the last few years, Battelli and Fečkan have developed a functional analytic method to rigorously prove the existence of chaotic behaviors in time-perturbed piecewise smooth systems whose unperturbed part has a piecewise continuous homoclinic solution. In this paper, by applying their method, we study the appearance of chaos in time-perturbed piecewise smooth systems with discontinuities on finitely many switching manifolds whose unperturbed part has a hyperbolic saddle in each subregion and a heteroclinic orbit connecting those saddles that crosses every switching manifold transversally exactly once. We obtain a set of Melnikov type functions whose zeros correspond to the occurrence of chaos of the system. Furthermore, the Melnikov functions for planar piecewise smooth systems are explicitly given. As an application, we present an example of quasiperiodically excited three-dimensional piecewise linear system with four zones.

Coexisting hidden attractors in a 5D segmented disc dynamo with three types of equilibria
Jianghong Bao, Dandan Chen, Yongjian Liu and Hongbo Deng
2019, 24(11): 6053-6069 doi: 10.3934/dcdsb.2019130 +[Abstract](328) +[HTML](159) +[PDF](5087.54KB)
Abstract:

Little seems to be known about coexisting hidden attractors in hyperchaotic systems with three types of equilibria. Based on the segmented disc dynamo, this paper proposes a new 5D hyperchaotic system which possesses the properties. This new system can generate hidden hyperchaos and chaos when initial conditions vary, as well as self-excited chaotic and hyperchaotic attractors when parameters vary. Furthermore, the paper proves that the Hopf bifurcation and pitchfork bifurcation occur in the system. Numerical simulations demonstrate the emergence of the two bifurcations. The MATLAB simulation results are further confirmed and validated by circuit implementation using NI Multisim.

Ground state solutions of fractional Schrödinger equations with potentials and weak monotonicity condition on the nonlinear term
Chao Ji
2019, 24(11): 6071-6089 doi: 10.3934/dcdsb.2019131 +[Abstract](553) +[HTML](215) +[PDF](453.84KB)
Abstract:

In this paper we are concerned with the fractional Schrödinger equation \begin{document}$ (-\Delta)^{\alpha} u+V(x)u = f(x, u) $\end{document}, \begin{document}$ x\in {{\mathbb{R}}^{N}} $\end{document}, where \begin{document}$ f $\end{document} is superlinear, subcritical growth and \begin{document}$ u\mapsto\frac{f(x, u)}{\vert u\vert} $\end{document} is nondecreasing. When \begin{document}$ V $\end{document} and \begin{document}$ f $\end{document} are periodic in \begin{document}$ x_{1},\ldots, x_{N} $\end{document}, we show the existence of ground states and the infinitely many solutions if \begin{document}$ f $\end{document} is odd in \begin{document}$ u $\end{document}. When \begin{document}$ V $\end{document} is coercive or \begin{document}$ V $\end{document} has a bounded potential well and \begin{document}$ f(x, u) = f(u) $\end{document}, the ground states are obtained. When \begin{document}$ V $\end{document} and \begin{document}$ f $\end{document} are asymptotically periodic in \begin{document}$ x $\end{document}, we also obtain the ground states solutions. In the previous research, \begin{document}$ u\mapsto\frac{f(x, u)}{\vert u\vert} $\end{document} was assumed to be strictly increasing, due to this small change, we are forced to go beyond methods of smooth analysis.

A Max-Cut approximation using a graph based MBO scheme
Blaine Keetch and Yves Van Gennip
2019, 24(11): 6091-6139 doi: 10.3934/dcdsb.2019132 +[Abstract](293) +[HTML](183) +[PDF](1758.9KB)
Abstract:

The Max-Cut problem is a well known combinatorial optimization problem. In this paper we describe a fast approximation method. Given a graph \begin{document}$ G $\end{document}, we want to find a cut whose size is maximal among all possible cuts. A cut is a partition of the vertex set of \begin{document}$ G $\end{document} into two disjoint subsets. For an unweighted graph, the size of the cut is the number of edges that have one vertex on either side of the partition; we also consider a weighted version of the problem where each edge contributes a nonnegative weight to the cut.

We introduce the signless Ginzburg–Landau functional and prove that this functional \begin{document}$ \Gamma $\end{document}-converges to a Max-Cut objective functional. We approximately minimize this functional using a graph based signless Merriman–Bence–Osher (MBO) scheme, which uses a signless Laplacian. We derive a Lyapunov functional for the iterations of our signless MBO scheme. We show experimentally that on some classes of graphs the resulting algorithm produces more accurate maximum cut approximations than the current state-of-the-art approximation algorithm. One of our methods of minimizing the functional results in an algorithm with a time complexity of \begin{document}$ \mathcal{O}(|E|) $\end{document}, where \begin{document}$ |E| $\end{document} is the total number of edges on \begin{document}$ G $\end{document}.

Global weak solution to the quantum Navier-Stokes-Landau-Lifshitz equations with density-dependent viscosity
Guangwu Wang and Boling Guo
2019, 24(11): 6141-6166 doi: 10.3934/dcdsb.2019133 +[Abstract](343) +[HTML](207) +[PDF](502.3KB)
Abstract:

In this paper we investigate the global existence of the weak solutions to the quantum Navier-Stokes-Landau-Lifshitz equations with density dependent viscosity in two dimensional case. We research the model with singular pressure and the dispersive term. The main technique is using the uniform energy estimates and B-D entropy estimates to prove the convergence of the solutions to the approximate system. We also use some convergent theorems in Sobolev space.

Linearized stability for abstract functional differential equations subject to state-dependent delays with applications
Jitai Liang, Ben Niu and Junjie Wei
2019, 24(11): 6167-6188 doi: 10.3934/dcdsb.2019134 +[Abstract](380) +[HTML](144) +[PDF](516.55KB)
Abstract:

In this paper, the linearized stability for a class of abstract functional differential equations (FDE) with state-dependent delays (SD) is investigated. In particular, such equations contain more general delay terms which not only cover the discrete delay and distributed delay as special cases, but also extend the SD to abstract integro-differential equation that the states belong to some infinite-dimensional space. The principle of linearized stability for such equations is established, which is nontrivial compared with ordinary differential equations with SD. Moreover, it should be stressed that such topic is untreated in the literatures up to date. Finally, we present an example to show the effectiveness of the proposed results.

An interface-free multi-scale multi-order model for traffic flow
Emiliano Cristiani and Elisa Iacomini
2019, 24(11): 6189-6207 doi: 10.3934/dcdsb.2019135 +[Abstract](329) +[HTML](146) +[PDF](4090.54KB)
Abstract:

In this paper we present a new multi-scale method for reproducing traffic flow which couples a first-order macroscopic model with a second-order microscopic model, avoiding any interface or boundary conditions between them. The multi-scale model is characterized by the fact that microscopic and macroscopic descriptions are not spatially separated. On the contrary, the macro-scale is always active while the micro-scale is activated only if needed by the traffic conditions. The Euler-Godunov scheme associated to the model is conservative and it is able to reproduce typical traffic phenomena like stop & go waves.

Portfolio optimization and model predictive control: A kinetic approach
Torsten Trimborn, Lorenzo Pareschi and Martin Frank
2019, 24(11): 6209-6238 doi: 10.3934/dcdsb.2019136 +[Abstract](368) +[HTML](186) +[PDF](1462.38KB)
Abstract:

In this paper, we introduce a large system of interacting financial agents in which all agents are faced with the decision of how to allocate their capital between a risky stock or a risk-less bond. The investment decision of investors, derived through an optimization, drives the stock price. The model has been inspired by the econophysical Levy-Levy-Solomon model [30]. The goal of this work is to gain insights into the stock price and wealth distribution. We especially want to discover the causes for the appearance of power-laws in financial data. We follow a kinetic approach similar to [33] and derive the mean field limit of the microscopic agent dynamics. The novelty in our approach is that the financial agents apply model predictive control (MPC) to approximate and solve the optimization of their utility function. Interestingly, the MPC approach gives a mathematical connection between the two opposing economic concepts of modeling financial agents to be rational or boundedly rational. Furthermore, this is to our knowledge the first kinetic portfolio model which considers a wealth and stock price distribution simultaneously. Due to the kinetic approach, we can study the wealth and price distribution on a mesoscopic level. The wealth distribution is characterized by a log-normal law. For the stock price distribution, we can either observe a log-normal behavior in the case of long-term investors or a power-law in the case of high-frequency trader. Furthermore, the stock return data exhibit a fat-tail, which is a well known characteristic of real financial data.

Analysis of a reaction diffusion model for a reservoir supported spread of infectious disease
W. E. Fitzgibbon and J. J. Morgan
2019, 24(11): 6239-6259 doi: 10.3934/dcdsb.2019137 +[Abstract](278) +[HTML](132) +[PDF](427.85KB)
Abstract:

Motivated by recent outbreaks of the Ebola Virus, we are concerned with the role that a vector reservoir plays in supporting the spatio-temporal spread of a highly lethal disease through a host population. In our context, the reservoir is a species capable of harboring and sustaining the pathogen. We develop models that describe the horizontal spread of the disease among the host population when the host population is in contact with the reservoir and when it is not in contact with the host population. These models are of reaction diffusion type, and they are analyzed, and their long term asymptotic behavior is determined.

Dynamics of a chemostat system with two patches
Shikun Wang
2019, 24(11): 6261-6278 doi: 10.3934/dcdsb.2019138 +[Abstract](255) +[HTML](166) +[PDF](255.99KB)
Abstract:

This paper studies a diffusion model with two patches, which is derived from experiments and includes exploitable resources. Our aim is to provide theoretical proof for experimental observations and extend previous theory to consumer-resource systems with external resource inputs. First, we exhibit nonnegativeness and boundedness of solutions of the model. For one-patch subsystems, we demonstrate the global dynamics by excluding periodic solutions. For the two-patch system, we exhibit uniform persistence of the system and asymptotic stability of the positive equilibria, while the equilibria converge to a unique positive point as the diffusion tends to infinity. Then we demonstrate that homogeneously distributed resources support higher total population abundance than heterogeneously distributed resources with diffusion, which coincides with empirical observation but refutes previous theory. Meanwhile, we exhibit new conditions under which populations diffusing in heterogeneous environments can reach higher total size than if non-diffusing. A new finding of our study is that these results hold even with source-sink populations, and varying the diffusion rate can result in survival/extinction of the species. Our results are consistent with experimental observations and provide new insights.

Bifurcation analysis of a mosquito population model for proportional releasing sterile mosquitoes
Liming Cai, Jicai Huang, Xinyu Song and Yuyue Zhang
2019, 24(11): 6279-6295 doi: 10.3934/dcdsb.2019139 +[Abstract](402) +[HTML](212) +[PDF](1692.51KB)
Abstract:

To reduce or eradicate mosquito-borne diseases, one of effective methods is to control the wild mosquito populations by using the sterile insect technique. Dynamical models with different releasing strategies of sterile mosquitoes have been proposed and investigated in the recent work by Cai et al. [SIAM. J. Appl. Math. 75(2014)], where some basic analysis on the dynamics are given and some complicated dynamical behaviors are found by numerical simulations. While their findings seem exciting and promising, yet the models could exhibit much more complex dynamics than it has been observed. In this paper, to further study the impact of the sterile insect technique on controlling the wild mosquito populations, we systematically study bifurcations and dynamics of the model with a proportional release rate of sterile mosquitoes by bifurcation method. We show that the model undergoes saddle-node bifurcation, subcritical and supercritical Hopf bifurcations, and Bogdanov-Takens bifurcation as the values of parameters vary. Some numerical simulations, including the bifurcation diagram and phase portraits, are also presented to illustrate the theoretical conclusions. These rich and complicated bifurcation phenomena can be regarded as a complement to the work by Cai et al. [SIAM. J. Appl. Math. 75(2014)].

Effects of migration on vector-borne diseases with forward and backward stage progression
Derdei Mahamat Bichara
2019, 24(11): 6297-6323 doi: 10.3934/dcdsb.2019140 +[Abstract](361) +[HTML](191) +[PDF](957.72KB)
Abstract:

Is it possible to break the host-vector chain of transmission when there is an influx of infectious hosts into a naïve population and competent vector? To address this question, a class of vector-borne disease models with an arbitrary number of infectious stages that account for immigration of infective individuals is formulated. The proposed model accounts for forward and backward progression, capturing the mitigation and aggravation to and from any stages of the infection, respectively. The model has a rich dynamic, which depends on the patterns of infected immigrant influx into the host population and connectivity of the transfer between infectious classes. We provide conditions under which the answer of the initial question is positive.

2018  Impact Factor: 1.008

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