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Discrete & Continuous Dynamical Systems - B

2017 , Volume 22 , Issue 6

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Stabilization of difference equations with noisy proportional feedback control
Elena Braverman and  Alexandra Rodkina
2017, 22(6): 2067-2088 doi: 10.3934/dcdsb.2017085 +[Abstract](38) +[HTML](2) +[PDF](1838.9KB)
Abstract:

Given a deterministic difference equation $x_{n+1}= f(x_n)$ with a continuous $f$ increasing on $[0, b]$, $f(0) \geq 0$, we would like to stabilize any point $x^{\ast}\in (f(0), f(b))$, by introducing the proportional feedback (PF) control. We assume that PF control contains either a multiplicative $x_{n+1}= f\left((\nu + \ell\chi_{n+1})x_n \right)$ or an additive noise $x_{n+1}=f(\lambda x_n) +\ell\chi_{n+1}$. We study conditions under which the solution eventually enters some interval, treated as a stochastic (blurred) equilibrium. In addition, we prove that, for each $\varepsilon>0$, when the noise level $\ell$ is sufficiently small, all solutions eventually belong to the interval $(x^{\ast}-\varepsilon, x^{\ast}+\varepsilon)$.

An analysis of functional curability on HIV infection models with Michaelis-Menten-type immune response and its generalization
Jeng-Huei Chen
2017, 22(6): 2089-2120 doi: 10.3934/dcdsb.2017086 +[Abstract](75) +[HTML](1) +[PDF](660.7KB)
Abstract:

Let HIV infection be modeled by a dynamical system with a Michaelis-Mente-type immune response. A functional cure refers to driving the system from a stable high-viral-load state to a stable low-viral-load state. This may occur only when at least two stable equilibrium states coexist in the system. This paper analyzes how the number of biologically meaningful equilibrium states varies with system parameters. Meanwhile, it investigates how patients' profiles of immune responses determine their clinical outcomes, with focus on functional curability. The analysis provides a criterion that a functional cure is possible only if the capability of immune stimulation starts to attenuate when the density of infected cells is below a threshold. From treatment viewpoints, such a criterion is crucial because it identifies which patients cannot use a low-viral-load state as a treatment endpoint. The deriving process also provides a method to study functional curability problems with a wider class of immune response functions and functional curability problems of similar virus infections such as chronic hepatitis B virus infection.

Synchronising and non-synchronising dynamics for a two-species aggregation model
Casimir Emako-Kazianou , Jie Liao and  Nicolas Vauchelet
2017, 22(6): 2121-2146 doi: 10.3934/dcdsb.2017088 +[Abstract](38) +[HTML](2) +[PDF](512.0KB)
Abstract:

This paper deals with analysis and numerical simulations of a one-dimensional two-species hyperbolic aggregation model. This model is formed by a system of transport equations with nonlocal velocities, which describes the aggregate dynamics of a two-species population in interaction appearing for instance in bacterial chemotaxis. Blow-up of classical solutions occurs in finite time. This raises the question to define measure-valued solutions for this system. To this aim, we use the duality method developed for transport equations with discontinuous velocity to prove the existence and uniqueness of measure-valued solutions. The proof relies on a stability result. In addition, this approach allows to study the hyperbolic limit of a kinetic chemotaxis model. Moreover, we propose a finite volume numerical scheme whose convergence towards measure-valued solutions is proved. It allows for numerical simulations capturing the behaviour after blow up. Finally, numerical simulations illustrate the complex dynamics of aggregates until the formation of a single aggregate: after blow-up of classical solutions, aggregates of different species are synchronising or nonsynchronising when collide, that is move together or separately, depending on the parameters of the model and masses of species involved.

Averaging principle for the Schrödinger equations
Peng Gao and  Yong Li
2017, 22(6): 2147-2168 doi: 10.3934/dcdsb.2017089 +[Abstract](38) +[HTML](1) +[PDF](400.8KB)
Abstract:

Averaging principle for the cubic nonlinear Schrödinger equations with rapidly oscillating potential and rapidly oscillating force are obtained, both on finite but large time intervals and on the entire time axis. This includes comparison estimate, stability estimate, and convergence result between nonlinear Schrödinger equation and its averaged equation. Furthermore, the existence of almost periodic solution for cubic nonlinear Schrödinger equations is also investigated.

Domain control of nonlinear networked systems and applications to complex disease networks
Suoqin Jin , Fang-Xiang Wu and  Xiufen Zou
2017, 22(6): 2169-2206 doi: 10.3934/dcdsb.2017091 +[Abstract](141) +[HTML](16) +[PDF](5226.6KB)
Abstract:

The control of complex nonlinear dynamical networks is an ongoing challenge in diverse contexts ranging from biology to social sciences. To explore a practical framework for controlling nonlinear dynamical networks based on meaningful physical and experimental considerations, we propose a new concept of the domain control for nonlinear dynamical networks, i.e., the control of a nonlinear network in transition from the domain of attraction of an undesired state (attractor) to the domain of attraction of a desired state. We theoretically prove the existence of a domain control. In particular, we offer an approach for identifying the driver nodes that need to be controlled and design a general form of control functions for realizing domain controllability. In addition, we demonstrate the effectiveness of our theory and approaches in three realistic disease-related networks: the epithelial-mesenchymal transition (EMT) core network, the T helper (Th) differentiation cellular network and the cancer network. Moreover, we reveal certain genes that are critical to phenotype transitions of these systems. Therefore, the approach described here not only offers a practical control scheme for nonlinear dynamical networks but also helps the development of new strategies for the prevention and treatment of complex diseases.

Bility and traveling wavefronts for a convolution model of mistletoes and birds with nonlocal diffusion
Huimin Liang , Peixuan Weng and  Yanling Tian
2017, 22(6): 2207-2231 doi: 10.3934/dcdsb.2017093 +[Abstract](38) +[HTML](1) +[PDF](1034.7KB)
Abstract:

A convolution model of mistletoes and birds with nonlocal diffusion is considered in this paper. We first consider the stability of the constant steady states of the model by linearized method, and then the existence of traveling solutions. The main aim of this article is to challenge the hardness lying in the construction of upper-lowers for wave profile system. With the help of an additional condition, we at last obtain a pair of upper-lower solutions. A constant $c_{*}>0$ is obtained such that traveling wavefronts exist for $c\geq c_{*}$. Amongst the construction, we take advantage of the relation between two components of principle eigenvector for the linearized system to control the two components of upper solution. The method seems novel. Some simulations and discussions are given to illustrate the applications of our main results and the effect of parameters on $c_{*}$. A comparison for $c_{*}$ is also given with two different kernel functions.

Convergence of global and bounded solutions of a two-species chemotaxis model with a logistic source
Ke Lin and  Chunlai Mu
2017, 22(6): 2233-2260 doi: 10.3934/dcdsb.2017094 +[Abstract](44) +[HTML](12) +[PDF](575.0KB)
Abstract:

In this paper, we consider a system of three parabolic equations in high-dimensional smoothly bounded domain

which describes the mutual competition between two populations on account of the Lotka-Volterra dynamics.

For any cross-diffusivities $\chi_1>0$ and $\chi_2>0$ and the rates $a_1>0$ and $a_2>0$, it is proved that the global classical bounded solutions exist for sufficiently regular initial data when the parameters $\mu_1$ and $\mu_2$ are sufficiently large. In deriving the convergence of solutions to this system, we need to distinguish two cases $a_1, a_2\in[0, 1)$ and $a_1>1$ and $0\leq a_2 < 1$ to prove globally asymptotic stability.

Seasonal forcing and exponential threshold incidence in cholera dynamics
Jinhuo Luo , Jin Wang and  Hao Wang
2017, 22(6): 2261-2290 doi: 10.3934/dcdsb.2017095 +[Abstract](35) +[HTML](2) +[PDF](878.3KB)
Abstract:

We propose a seasonal forcing iSIR (indirectly transmitted SIR) model with a modified incidence function, due to the fact that the seasonal fluctuations can be the main culprit for cholera outbreaks. For this nonautonomous system, we provide a sufficient condition for the persistence and the existence of a periodic solution. Furthermore, we provide a sufficient condition for the global stability of the periodic solution. Finally, we present some simulation examples for both autonomous and nonautonomous systems. Simulation results exhibit dynamical complexities, including the bistability of the autonomous system, an unexpected outbreak of cholera for the nonautonomous system, and possible outcomes induced by sudden weather events. Comparatively the nonautonomous system is more realistic in describing the indirect transmission of cholera. Our study reveals that the relative difference between the value of immunological threshold and the peak value of bacterial biomass is critical in determining the dynamical behaviors of the system.

Blow-up phenomena for nonlinear pseudo-parabolic equations with gradient term
Monica Marras , Stella Vernier-Piro and  Giuseppe Viglialoro
2017, 22(6): 2291-2300 doi: 10.3934/dcdsb.2017096 +[Abstract](37) +[HTML](2) +[PDF](375.4KB)
Abstract:

This paper is concerned with the pseudo-parabolic problem

where \begin{document}$\Omega$\end{document} is a bounded domain in \begin{document}$\mathbb{R}^n, \ n\geq 2$\end{document}, with smooth boundary \begin{document}$ \partial \Omega$\end{document}, \begin{document}$ k$\end{document} is a positive constant or in general positive derivable function of \begin{document}$t$\end{document}. The solution \begin{document}$u(x,t)$\end{document} may or may not blow up in finite time. Under suitable conditions on data, a lower bound for \begin{document}$t^*$\end{document} is derived, where \begin{document}$[0,t^*)$\end{document} is the time interval of existence of \begin{document}$u(x,t).$\end{document} We indicate how some of our results can be extended to a class of nonlinear pseudo-parabolic systems.

Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity
Masaaki Mizukami
2017, 22(6): 2301-2319 doi: 10.3934/dcdsb.2017097 +[Abstract](115) +[HTML](65) +[PDF](484.6KB)
Abstract:

This paper deals with the two-species chemotaxis-competition system

where \begin{document}$\Omega$\end{document} is a bounded domain in \begin{document}$\mathbb{R}^n$\end{document} with smooth boundary \begin{document}$\partial \Omega$\end{document}, \begin{document}$n\in \mathbb{N}$\end{document}; \begin{document}$h$\end{document}, \begin{document}$\chi_i$\end{document} are functions satisfying some conditions. In the case that \begin{document}$\chi_i(w)=\chi_i$\end{document}, Bai–Winkler [1] proved asymptotic behavior of solutions to the above system under some conditions which roughly mean largeness of \begin{document}$\mu_1, \mu_2$\end{document}. The main purpose of this paper is to extend the previous method for obtaining asymptotic stability. As a result, the present paper improves the conditions assumed in [1], i.e., the ranges of \begin{document}$\mu_1, \mu_2$\end{document} are extended.

Asymptotic behaviors of Green-Sch potentials at infinity and its applications
Lei Qiao
2017, 22(6): 2321-2338 doi: 10.3934/dcdsb.2017099 +[Abstract](33) +[HTML](1) +[PDF](423.6KB)
Abstract:

The first aim in this paper is to deal with asymptotic behaviors of Green-Sch potentials in a cylinder. As an application we prove the integral representation of nonnegative weak solutions of the stationary Schrödinger equation in a cylinder. Next we give asymptotic behaviors of them outside an exceptional set. Finally we obtain a quantitative property of rarefied sets with respect to the stationary Schrödinger operator at \begin{document}$+\infty$\end{document} in a cylinder. Meanwhile we show that the reverse of this property is not true.

On the Kolmogorov entropy of the weak global attractor of 3D Navier-Stokes equations:Ⅰ
Yong Yang and  Bingsheng Zhang
2017, 22(6): 2339-2350 doi: 10.3934/dcdsb.2017101 +[Abstract](38) +[HTML](1) +[PDF](389.4KB)
Abstract:

One particular metric that generates the weak topology on the weak global attractor \begin{document}$\mathcal{A}_w$\end{document} of three dimensional incompressible Navier-Stokes equations is introduced and used to obtain an upper bound for the Kolmogorov entropy of \begin{document}$\mathcal{A}_w$\end{document}. This bound is expressed explicitly in terms of the physical parameters of the fluid flow.

Oscillation theorems for impulsive parabolic differential system of neutral type
Min Zou , An-Ping Liu and  Zhimin Zhang
2017, 22(6): 2351-2363 doi: 10.3934/dcdsb.2017103 +[Abstract](48) +[HTML](1) +[PDF](352.0KB)
Abstract:

In this paper, oscillatory properties of solutions to a nonlinear impulsive parabolic differential system of neutral type are investigated. A series of sufficient conditions are established for problems with Robin and Dirichlet boundary conditions. Examples are provided to confirm the validity of the analysis.

Stability and Hopf bifurcation of an HIV infection model with saturation incidence and two delays
Hui Miao , Zhidong Teng and  Chengjun Kang
2017, 22(6): 2365-2387 doi: 10.3934/dcdsb.2017121 +[Abstract](78) +[HTML](1) +[PDF](1486.6KB)
Abstract:

In this paper, the dynamical behaviors of a viral infection model with cytotoxic T-lymphocyte (CTL) immune response, immune response delay and production delay are investigated. The threshold values for virus infection and immune response are established. By means of Lyapunov functionals methods and LaSalle's invariance principle, sufficient conditions for the global stability of the infection-free and CTL-absent equilibria are established. Global stability of the CTL-present infection equilibrium is also studied when there is no immune delay in the model. Furthermore, to deal with the local stability of the CTL-present infection equilibrium in a general case with two delays being positive, we extend an existing geometric method to treat the associated characteristic equation. When the two delays are positive, we show some conditions for Hopf bifurcation at the CTL-present infection equilibrium by using the immune delay as a bifurcation parameter. Numerical simulations are performed in order to illustrate the dynamical behaviors of the model.

Numerical solutions of viscoelastic bending wave equations with two term time kernels by Runge-Kutta convolution quadrature
Da Xu
2017, 22(6): 2389-2416 doi: 10.3934/dcdsb.2017122 +[Abstract](155) +[HTML](1) +[PDF](509.9KB)
Abstract:

In this paper, we study the numerical solutions of viscoelastic bending wave equations

for \begin{document}$ 0<x<1,~ 0<t\leq T $\end{document}, with self-adjoint boundary and initial value conditions, in which the functions \begin{document}$ \beta_{1}(t) $\end{document} and \begin{document}$ \beta_{2}(t) $\end{document} are completely monotonic on \begin{document}$ (0,~\infty) $\end{document} and locally integrable, but not constant. The equations are discretised in space by the finite difference method and in time by the Runge-Kutta convolution quadrature. The stability and convergence of the schemes are analyzed by the frequency domain and energy methods. Numerical experiments are provided to illustrate the accuracy and efficiency of the proposed schemes.

Limit cycle bifurcations for piecewise smooth integrable differential systems
Jihua Yang and  Liqin Zhao
2017, 22(6): 2417-2425 doi: 10.3934/dcdsb.2017123 +[Abstract](51) +[HTML](2) +[PDF](385.5KB)
Abstract:

In this paper, we study a class of piecewise smooth integrable non-Hamiltonian systems, which has a center. By using the first order Melnikov function, we give an exact number of limit cycles which bifurcate from the above periodic annulus under the polynomial perturbation of degree n.

Invariant measures for complex-valued dissipative dynamical systems and applications
Xin Li , Wenxian Shen and  Chunyou Sun
2017, 22(6): 2427-2446 doi: 10.3934/dcdsb.2017124 +[Abstract](70) +[HTML](1) +[PDF](486.9KB)
Abstract:

In this work, we extend the classical real-valued framework to deal with complex-valued dissipative dynamical systems. With our new complex-valued framework and using generalized complex Banach limits, we construct invariant measures for continuous complex semigroups possessing global attractors. In particular, for any given complex Banach limit and initial data \begin{document}$u_{0}$\end{document}, we construct a unique complex invariant measure \begin{document}$\mu$\end{document} on a metric space which is acted by a continuous semigroup \begin{document}$\{S(t)\}_{t\geq 0}$\end{document} possessing a global attractor \begin{document}$\mathcal{A}$\end{document}. Moreover, it is shown that the support of \begin{document}$\mu$\end{document} is not only contained in global attractor \begin{document}$\mathcal{A}$\end{document} but also in \begin{document}$\omega(u_{0})$\end{document}. Next, the structure of the measure \begin{document}$\mu$\end{document} is studied. It is shown that both the real and imaginary parts of a complex invariant measure are invariant signed measures and that both the positive and negative variations of a signed measure are invariant measures. Finally, we illustrate the main results of this article on the model examples of a complex Ginzburg-Landau equation and a nonlinear Schrödinger equation and construct complex invariant measures for these two complex-valued equations.

Nonsmooth frameworks for an extended Budyko model
Anna M. Barry , Esther WIdiasih and  Richard Mcgehee
2017, 22(6): 2447-2463 doi: 10.3934/dcdsb.2017125 +[Abstract](150) +[HTML](1) +[PDF](635.0KB)
Abstract:

In latitude-dependent energy balance models, ice-free and ice-covered conditions form physical boundaries of the system. With carbon dioxide treated as a bifurcation parameter, the resulting bifurcation diagram is nonsmooth with curves of equilibria and boundaries forming corners at points of intersection. Over long time scales, atmospheric carbon dioxide varies dynamically and the nonsmooth diagram becomes a set of quasi-equilibria. However, when introducing carbon dynamics, care must be taken with the physical boundaries and appropriate boundary motion specified. In this article, we extend an energy balance model to include slowly varying carbon dioxide and develop nonsmooth frameworks based on physically relevant boundary dynamics. Within these frameworks, we prove existence and uniqueness of solutions, as well as invariance of the region of phase space bounded by ice-free and ice-covered states.

Chiellini integrability condition, planar isochronous systems and Hamiltonian structures of Liénard equation
A. Ghose Choudhury and  Partha Guha
2017, 22(6): 2465-2478 doi: 10.3934/dcdsb.2017126 +[Abstract](35) +[HTML](2) +[PDF](373.9KB)
Abstract:

Using a novel transformation involving the Jacobi Last Multiplier (JLM) we derive an old integrability criterion due to Chiellini for the Liénard equation. By combining the Chiellini condition for integrability and Jacobi's Last Multiplier the Lagrangian and Hamiltonian of the Liénard equation is derived. We also show that the Kukles equation is the only equation in the Liénard family which satisfies both the Chiellini integrability and the Sabatini criterion for isochronicity conditions. In addition we examine this result by mapping the Liénard equation to a harmonic oscillator equation using tacitly Chiellini's condition. Finally we provide a metriplectic and complex Hamiltonian formulation of the Liénard equation through the use of Chiellini condition for integrability.

Stationarity and periodicity of positive solutions to stochastic SEIR epidemic models with distributed delay
Qun Liu , Daqing Jiang , Ningzhong Shi , Tasawar Hayat and  Ahmed Alsaedi
2017, 22(6): 2479-2500 doi: 10.3934/dcdsb.2017127 +[Abstract](32) +[HTML](1) +[PDF](495.0KB)
Abstract:

In this paper, we consider two SEIR epidemic models with distributed delay in random environments. First of all, by constructing a suitable stochastic Lyapunov function, we obtain the existence of stationarity of the positive solution to the stochastic autonomous system. Then we establish sufficient conditions for extinction of the disease. Finally, by using Khasminskii's theory of periodic solutions, we prove that the stochastic nonautonomous epidemic model admits at least one nontrivial positive T-periodic solution under a simple condition.

Quasi-periodic solutions of generalized Boussinesq equation with quasi-periodic forcing
Yanling Shi , Junxiang Xu and  Xindong Xu
2017, 22(6): 2501-2519 doi: 10.3934/dcdsb.2017104 +[Abstract](39) +[HTML](1) +[PDF](509.2KB)
Abstract:

In this paper, one-dimensional quasi-periodically forced generalized Boussinesq equation

with hinged boundary conditions is considered, where \begin{document}$\varepsilon$\end{document} is a small positive parameter, \begin{document}$\phi(t)$\end{document} is a real analytic quasi-periodic function in \begin{document}$t$\end{document} with frequency vector \begin{document}$\omega=( \omega_1,\omega_2,\cdots,\omega_m ).$\end{document} It is proved that, under a suitable hypothesis on \begin{document}$\phi(t),$\end{document} there are many quasi-periodic solutions for the above equation via KAM theory.

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