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Discrete & Continuous Dynamical Systems - B

2016 , Volume 21 , Issue 10

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Complex dynamics in the segmented disc dynamo
Jianghong Bao
2016, 21(10): 3301-3314 doi: 10.3934/dcdsb.2016098 +[Abstract](62) +[PDF](671.6KB)
The present work is devoted to giving new insights into the segmented disc dynamo. The integrability of the system is studied. The paper provides its first integrals for the parameter $r=0$. For $r>0$, the system has neither polynomial first integrals nor exponential factors, and it is also further proved not to be Darboux integrable. In addition, by choosing an appropriate bifurcation parameter, the paper proves that Hopf bifurcations occur in the system and presents the formulae for determining the direction of the Hopf bifurcations and the stability of bifurcating periodic solutions.
Minimizing $\mathcal R_0$ for in-host virus model with periodic combination antiviral therapy
Cameron J. Browne and  Sergei S. Pilyugin
2016, 21(10): 3315-3330 doi: 10.3934/dcdsb.2016099 +[Abstract](37) +[PDF](420.7KB)
The aim of this article is to provide formulas and properties of the basic reproduction number, $\mathcal R_0$, for a within-host virus model with periodic combination drug treatment. In particular, we extend and further results about how the phase difference between drug treatment can critically affect the asymptotic dynamics of model and corresponding $\mathcal R_0$. Our main theorem establishes that $\mathcal R_0$ is minimized for out-of-phase efficacies and maximized for in-phase efficacies in a special case where drug efficacies are ``bang-bang'' functions.
Optimal control of a perturbed sweeping process via discrete approximations
Tan H. Cao and  Boris S. Mordukhovich
2016, 21(10): 3331-3358 doi: 10.3934/dcdsb.2016100 +[Abstract](39) +[PDF](592.2KB)
The paper addresses an optimal control problem for a perturbed sweeping process of the rate-independent hysteresis type described by a controlled ``play-and stop" operator with separately controlled perturbations. This problem can be reduced to dynamic optimization of a state-constrained unbounded differential inclusion with highly irregular data that cannot be treated by means of known results in optimal control theory for differential inclusions. We develop the method of discrete approximations, which allows us to adequately replace the original optimal control problem by a sequence of well-posed finite-dimensional optimization problems whose optimal solutions strongly converge to that of the controlled perturbed sweeping process. To solve the discretized control systems, we derive effective necessary optimality conditions by using second-order generalized differential tools of variational analysis that explicitly calculated in terms of the given problem data.
Exponential integrability properties of Euler discretization schemes for the Cox--Ingersoll--Ross process
Andrei Cozma and  Christoph Reisinger
2016, 21(10): 3359-3377 doi: 10.3934/dcdsb.2016101 +[Abstract](33) +[PDF](517.2KB)
We study exponential integrability properties of the Cox--Ingersoll--Ross (CIR) process and its Euler--Maruyama discretizations with various types of truncation and reflection at $0$. These properties play a key role in establishing the finiteness of moments and the strong convergence of numerical approximations for a class of stochastic differential equations arising in finance. We prove that both implicit and explicit Euler--Maruyama discretizations for the CIR process preserve the exponential integrability of the exact solution for a wide range of parameters, and find lower bounds on the explosion time.
Dynamics of a networked connectivity model of epidemics
Cristina Cross , Alysse Edwards , Dayna Mercadante and  Jorge Rebaza
2016, 21(10): 3379-3390 doi: 10.3934/dcdsb.2016102 +[Abstract](79) +[PDF](606.6KB)
A networked connectivity model of waterborne disease epidemics on a site of $n$ communities is studied. Existence and local stability analysis for both the disease-free equilibrium and the endemic equilibrium are studied. Using an appropriate Lyapunov function and Lasalle invariance principle, global asymptotic stability of the disease-free equilibrium point is established. Existence of a transcritical bifurcation at the disease outbreak is also proved. This work extends previous research in networked connectivity models of epidemics.
Computational methods for asynchronous basins
Ian H. Dinwoodie
2016, 21(10): 3391-3405 doi: 10.3934/dcdsb.2016103 +[Abstract](37) +[PDF](429.1KB)
For a Boolean network we consider asynchronous updates and define the exclusive asynchronous basin of attraction for any steady state or cyclic attractor. An algorithm based on commutative algebra is presented to compute the exclusive basin. Finally its use for targeting desirable attractors by selective intervention on network nodes is illustrated with two examples, one cell signalling network and one sensor network measuring human mobility.
Infinitely many solutions of the nonlinear fractional Schrödinger equations
Miao Du and  Lixin Tian
2016, 21(10): 3407-3428 doi: 10.3934/dcdsb.2016104 +[Abstract](73) +[PDF](488.7KB)
In this paper, we study the fractional Schrödinger equation \begin{equation*} (-\Delta)^{s} u+V(x)u=f(x,u), \quad x\in\mathbb{R}^{N}, \end{equation*} where $0< s <1$, $(-\Delta)^{s}$ denotes the fractional Laplacian of order $s$ and the nonlinearity $f$ is sublinear or superlinear at infinity. Under certain assumptions on $V$ and $f$, we prove that this equation has infinitely many solutions via variational methods, which unifies and sharply improves the recent results of Teng (2015) [33]. Moreover, we also consider the above equation with concave and critical nonlinearities, and obtain the existence of infinitely many solutions.
Long-time behavior of an SIR model with perturbed disease transmission coefficient
Nguyen Huu Du and  Nguyen Thanh Dieu
2016, 21(10): 3429-3440 doi: 10.3934/dcdsb.2016105 +[Abstract](199) +[PDF](1217.7KB)
In this paper, we consider a stochastic SIR model with the perturbed disease transmission coefficient. We determine the threshold $\lambda$ that is used to classify the extinction and permanence of the disease. Precisely, $\lambda<0$ implies that the disease-free $(\frac{\alpha}{\mu}, 0, 0)$ is globally asymptotic stable, i.e., the disease will disappear and the entire population will become susceptible individuals. If $\lambda>0$ the epidemic takes place. In this case, we derive that the Markov process $(S(t), I(t))$ has a unique invariant probability measure. We also characterize the support of a unique invariant probability measure and prove that the transition probability converges to this invariant measures in total variation norm. Our result is considered as an significant improvement over the results in [6,7,11,18].
Global classical solutions for mass-conserving, (super)-quadratic reaction-diffusion systems in three and higher space dimensions
Klemens Fellner , Evangelos Latos and  Takashi Suzuki
2016, 21(10): 3441-3462 doi: 10.3934/dcdsb.2016106 +[Abstract](36) +[PDF](484.4KB)
This paper considers quadratic and super-quadratic reaction-diffusion systems, for which all species satisfy uniform-in-time $L^1$ a-priori estimates, for instance, as a consequence of suitable mass conservation laws. A new result on the global existence of classical solutions is proved in three and higher space dimensions by combining regularity and interpolation arguments in Bochner spaces, a bootstrap scheme and a weak comparison argument. Moreover, provided that the considered system allows for entropy entropy-dissipation estimates proving exponential convergence to equilibrium, we are also able to prove that solutions are bounded uniformly-in-time.
Error estimates of the aggregation-diffusion splitting algorithms for the Keller-Segel equations
Hui Huang and  Jian-Guo Liu
2016, 21(10): 3463-3478 doi: 10.3934/dcdsb.2016107 +[Abstract](38) +[PDF](427.1KB)
In this paper, we discuss error estimates associated with three different aggregation-diffusion splitting schemes for the Keller-Segel equations. We start with one algorithm based on the Trotter product formula, and we show that the convergence rate is $C\Delta t$, where $\Delta t$ is the time-step size. Secondly, we prove the convergence rate $C\Delta t^2$ for the Strang's splitting. Lastly, we study a splitting scheme with the linear transport approximation, and prove the convergence rate $C\Delta t$.
The vanishing surface tension limit for the Hele-Shaw problem
Hyung Ju Hwang , Youngmin Oh and  Marco Antonio Fontelos
2016, 21(10): 3479-3514 doi: 10.3934/dcdsb.2016108 +[Abstract](30) +[PDF](552.6KB)
In this paper, we show the existence of solutions of the Hele-Shaw problem in two dimensions in the presence of surface tension and for a general class of initial data. The limit problem from nonzero to zero surface tension will also be investigated. In the case of injection and when volume conservation holds, for a sufficiently small surface tension, we prove the existence and uniqueness of perturbed solutions with nonzero surface tension near solutions with zero surface tension. We also show that solutions with nonzero surface tension exist up to a finite time before a possible singularity occurs in which solutions with zero surface tension are well defined. In addition, in the finite time interval, we prove that the solutions with nonzero surface tension approach the solutions with zero surface tension as the surface tension coefficient goes to zero. In the case of suction, for sufficiently small surface tension, we prove the existence of perturbed solutions near solutions with zero surface tension in any initially smooth domains. In this case, the local existence time depends on the surface tension coefficient.
A multi-group SIR epidemic model with age structure
Toshikazu Kuniya , Jinliang Wang and  Hisashi Inaba
2016, 21(10): 3515-3550 doi: 10.3934/dcdsb.2016109 +[Abstract](52) +[PDF](1041.2KB)
This paper provides the first detailed analysis of a multi-group SIR epidemic model with age structure, which is given by a nonlinear system of $3n$ partial differential equations. The basic reproduction number $\mathcal{R}_0$ is obtained as the spectral radius of the next generation operator, and it is shown that if $\mathcal{R}_0 < 1$, then the disease-free equilibrium is globally asymptotically stable, while if $\mathcal{R}_0 >1$, then an endemic equilibrium exists. The global asymptotic stability of the endemic equilibrium is also shown under additional assumptions such that the transmission coefficient is independent from the age of infective individuals and the mortality and removal rates are constant. To our knowledge, this is the first paper which applies the method of Lyapunov functional and graph theory to a multi-dimensional PDE system.
Global attracting set, exponential decay and stability in distribution of neutral SPDEs driven by additive $\alpha$-stable processes
Kai Liu and  Zhi Li
2016, 21(10): 3551-3573 doi: 10.3934/dcdsb.2016110 +[Abstract](36) +[PDF](476.0KB)
In this paper, we are concerned with a class of neutral stochastic partial differential equations driven by $\alpha$-stable processes. By combining some stochastic analysis techniques, tools from semigroup theory and delay integral inequalities, we identify the global attracting sets of the equations under investigation. Some sufficient conditions ensuring the exponential decay of mild solutions in the $p$-th moment to the stochastic systems are obtained. Subsequently, by employing a weak convergence approach, we try to establish some stability conditions in distribution of the segment processes of mild solutions to the stochastic systems under consideration. Last, an example is presented to illustrate our theory in the work.
Hopf bifurcation in a model of TGF-$\beta$ in regulation of the Th 17 phenotype
Jisun Lim , Seongwon Lee and  Yangjin Kim
2016, 21(10): 3575-3602 doi: 10.3934/dcdsb.2016111 +[Abstract](55) +[PDF](2106.9KB)
Airway exposure of lipopolysaccharide (LPS) is shown to regulate type I and type II helper T cell induced asthma. While high doses of LPS derive Th1- or Th17-immune responses, low LPS levels lead to Th2 responses. In this paper, we analyze a mathematical model of Th1/Th2/Th17 asthma regulation suggested by Lee (S. Lee, H.J. Hwang, and Y. Kim, Modeling the role of TGF-$\beta$ in regulation of the Th17 phenotype in the LPS-driven immune system, Bull Math Biol., 76 (5), 1045-1080, 2014) and show that the system can undergo a Hopf bifurcation at a steady state of the Th17 phenotype for high LPS levels in the presence of time delays in inhibition pathways of two key regulators: IL-4/Th2 activities ($H$) and TGF-$\beta$ levels ($G$). The time delays affect the phenotypic switches among the Th1, Th2, and Th17 phenotypes in response to time-dependent LPS doses via nonlinear crosstalk between $H$ and $G$. An extended reaction-diffusion model also predicts coexistence of these phenotypes under various biochemical and bio-mechanical conditions in the heterogeneous microenvironment.
Optimal contraception control for a nonlinear population model with size structure and a separable mortality
Rong Liu , Feng-Qin Zhang and  Yuming Chen
2016, 21(10): 3603-3618 doi: 10.3934/dcdsb.2016112 +[Abstract](57) +[PDF](424.7KB)
This paper is concerned with the problem of optimal contraception control for a nonlinear population model with size structure. First, the existence of separable solutions is established, which is crucial in obtaining the optimal control strategy. Moreover, it is shown that the population density depends continuously on control parameters. Then, the existence of an optimal control strategy is proved via compactness and extremal sequence. Finally, the conditions of the optimal strategy are derived by means of normal cones and adjoint systems.
Blow-up phenomena for a nonlocal quasilinear parabolic equation with time-dependent coefficients under nonlinear boundary flux
Zhiqing Liu and  Zhong Bo Fang
2016, 21(10): 3619-3635 doi: 10.3934/dcdsb.2016113 +[Abstract](83) +[PDF](400.2KB)
This paper deals with blow-up phenomena for an initial boundary value problem of a nonlocal quasilinear parabolic equation with time-dependent coefficients in a bounded star-shaped region under nonlinear boundary flux. Using the auxiliary function method and modified differential inequality technique, we establish some conditions on time-dependent coefficients and nonlinearities to guarantee that the solution $u(x,t)$ exists globally or blows up at some finite time $t^{\ast}$. Moreover, upper and lower bounds of $t^{\ast}$ are obtained under suitable measure in high-dimensional spaces. Finally, some application examples are presented.
Weak stability for integro-differential inclusions of diffusion-wave type involving infinite delays
Thanh-Anh Nguyen , Dinh-Ke Tran and  Nhu-Quan Nguyen
2016, 21(10): 3637-3654 doi: 10.3934/dcdsb.2016114 +[Abstract](121) +[PDF](434.5KB)
We deal with the Cauchy problem associated with integro-differential inclusions of diffusion-wave type involving infinite delays. Based on the behavior of resolvent operator associated with the linear part, an explicit estimate for solutions will be established. As a consequence, the weak stability of zero solution is proved in case the resolvent operator is asymptotically stable.
Finite-time synchronization of competitive neural networks with mixed delays
Tingting Su and  Xinsong Yang
2016, 21(10): 3655-3667 doi: 10.3934/dcdsb.2016115 +[Abstract](50) +[PDF](948.2KB)
In this paper, finite-time synchronization of competitive neural networks (CNNs) with bounded time-varying discrete and distributed delays (mixed delays) is investigated. A simple controller is added to response (slave) system such that it can be synchronized with the driving (master) CNN in a setting time. By introducing a suitable Lyapunov-Krasovskii's functional and utilizing some inequalities, several sufficient conditions are obtained to ensure the control object. Moreover, the setting time is explicitly given. Different from previous results, the setting is related to both the initial value of error system and the time delays. Finally, numerical examples are given to show the effectiveness of the theoretical results.
On the upper semicontinuity of pullback attractors for multi-valued noncompact random dynamical systems
Yejuan Wang
2016, 21(10): 3669-3708 doi: 10.3934/dcdsb.2016116 +[Abstract](101) +[PDF](641.4KB)
In this paper, we first present some conditions for the upper semicontinuity of pullback attractors for multi-valued noncompact random dynamical systems with small perturbations. Then we establish the upper semicontinuity of pullback attractors for nonautonomous and stochastic reaction-diffusion delay equations defined on $\mathbb{R}^n$ with small time delay perturbation for which uniqueness of solutions need not hold. Finally, we prove the existence and upper semicontinuity of pullback attractors for nonautonomous and stochastic nonclassical diffusion equations with polynomial growth nonlinearity of arbitrary order and without the uniqueness of solutions.
The steady state solutions to thermohaline circulation equations
Chao Xing , Ping Zhou and  Hong Luo
2016, 21(10): 3709-3722 doi: 10.3934/dcdsb.2016117 +[Abstract](35) +[PDF](365.3KB)
In the article, we study the existence and the regularity of the steady state solutions to thermohaline circulation equations. Firstly, we obtain a sufficient condition of the existence of weak solutions to the equations by acute angle theory of weakly continuous operator. Secondly, we prove the existence of strong solutions to the equations by ADN theory and iteration procedure. Furthermore, we study the generic property of the solutions by Sard-Smale theorem and the existence of classical solutions by ADN theorem.
Traveling waves in an SEIR epidemic model with the variable total population
Zhiting Xu
2016, 21(10): 3723-3742 doi: 10.3934/dcdsb.2016118 +[Abstract](107) +[PDF](439.4KB)
In the present paper, we propose a simple diffusive SEIR epidemic model where the total population is variable. We first give the explicit formula of the basic reproduction number $\mathcal{R}_0$ for the model. And hence, we show that if $\mathcal{R}_0>1$, then there exists a constant $c^*>0$ such that for any $c>c^*$, the model admits a nontrivial traveling wave solution, and if $\mathcal{R}_0<1$ and $c>0$ (or, $\mathcal{R}_0>1$ and $c\in(0,c^*)$), then the model has no nontrivial traveling wave solution. Consequently, we obtain the full information about the existence and non-existence of traveling wave solutions of the model by determined by the constants $\mathcal{R}_0$ and $c^*$. The proof of the main results is mainly based on Schauder fixed point theorem and Laplace transform.
Permanence and ergodicity of stochastic Gilpin-Ayala population model with regime switching
Hongfu Yang , Xiaoyue Li and  George Yin
2016, 21(10): 3743-3766 doi: 10.3934/dcdsb.2016119 +[Abstract](33) +[PDF](805.0KB)
This work is concerned with permanence and ergodicity of stochastic Gilpin-Ayala models involve continuous states as well as discrete events. A distinct feature is that the Gilpin-Ayala parameter and its corresponding perturbation parameter are allowed to be varying randomly in accordance with a random switching process. Necessary and sufficient conditions of the stochastic permanence and extinction are established, which are much weaker than the previous results. The existence of the unique stationary distribution is also established. Our approach treats much wider class of systems, uses much weaker conditions, and substantially generalizes previous results. It is shown that regime switching can suppress the impermanence. Furthermore, several examples and simulations are given to illustrate our main results.
Dynamics for the complex Ginzburg-Landau equation on non-cylindrical domains I: The diffeomorphism case
Feng Zhou and  Chunyou Sun
2016, 21(10): 3767-3792 doi: 10.3934/dcdsb.2016120 +[Abstract](59) +[PDF](520.7KB)
The purpose of this article is to analyze the dynamics of the following complex Ginzburg-Landau equation \begin{align*} \partial_{t}u-(\lambda+i\alpha)\Delta u+(\kappa+i\beta)|u|^{p-2}u-\gamma u=f(t) \end{align*} on non-cylindrical domains, which are obtained by diffeomorphic transformation of a bounded base domain, without any upper restriction on $p>2$, only with some restriction on $\beta/\kappa$. We establish the existence and uniqueness of strong and weak solutions as well as some energy inequalities for this equation on variable domains. Moreover the existence of a $\mathscr{D}$-pullback attractor is established for the process generated by the weak solutions under a slightly weaker condition that the measure of the spatial domains in the past is uniformly bounded above.
The periodic solutions bifurcated from a homoclinic solution for parabolic differential equations
Changrong Zhu and  Bin Long
2016, 21(10): 3793-3808 doi: 10.3934/dcdsb.2016121 +[Abstract](39) +[PDF](400.9KB)
Assume that the unperturbed parabolic equation has a degenerate homoclinic orbit. Under $T$-periodic perturbations, the periodic solutions bifurcated from the homoclinic solution are studied. By Fredholm alternative and Lyapunov-Schmidt reduction, the bifurcation functions defined between two finite-dimensional spaces are obtained. Some solvable conditions for the bifurcation functions are given. It is shown that, for any large $n>0$, the perturbed parabolic differential equation has a periodic solution with period $nT$.

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