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Discrete & Continuous Dynamical Systems - B

2014 , Volume 19 , Issue 6

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Traveling wave in backward and forward parabolic equations from population dynamics
Lianzhang Bao and  Zhengfang Zhou
2014, 19(6): 1507-1522 doi: 10.3934/dcdsb.2014.19.1507 +[Abstract](37) +[PDF](459.6KB)
This work is concerned with the properties of the traveling wave of the backward and forward parabolic equation \begin{equation*} u_t= [ D(u)u_x]_x + g(u),\quad t\geq 0, x\in \mathbb{R}, \end{equation*} where $D(u)$ changes its sign once, from negative to positive value, in the interval $u\in [0,1]$ and $g(u)$ is a mono-stable nonlinear reaction term. The existence of infinitely many traveling wave solutions is proven. These traveling waves are parameterized by their wave speed and monotonically connect the stationary states $u\equiv0$ and $u\equiv 1$.
Symmetric periodic orbits in three sub-problems of the $N$-body problem
Nai-Chia Chen
2014, 19(6): 1523-1548 doi: 10.3934/dcdsb.2014.19.1523 +[Abstract](22) +[PDF](6479.6KB)
We study three sub-problems of the $N$-body problem that have two degrees of freedom, namely the $n-$pyramidal problem, the planar double-polygon problem, and the spatial double-polygon problem. We prove the existence of several families of symmetric periodic orbits, including ``Schubart-like" orbits and brake orbits, by using topological shooting arguments.
On strong causal binomial approximation for stochastic processes
Nikolai Dokuchaev
2014, 19(6): 1549-1562 doi: 10.3934/dcdsb.2014.19.1549 +[Abstract](26) +[PDF](360.9KB)
This paper considers binomial approximation of continuous time stochastic processes. It is shown that, under some mild integrability conditions, a process can be approximated in mean square sense and in other strong metrics by binomial processes, i.e., by processes with fixed size binary increments at sampling points. Moreover, this approximation can be causal, i.e., at every time it requires only past historical values of the underlying process. In addition, possibility of approximation of solutions of stochastic differential equations by solutions of ordinary equations with binary noise is established. Some consequences for the financial modelling and options pricing models are discussed.
Catalog of the optimal controls in cancer chemotherapy for the Gompertz model depending on PK/PD and the integral constraint
Luis A. Fernández and  Cecilia Pola
2014, 19(6): 1563-1588 doi: 10.3934/dcdsb.2014.19.1563 +[Abstract](29) +[PDF](1546.7KB)
We study three optimal control problems associated with Gompertz-type differential equations, including bound control and integral constraints. These problems can be interpreted in terms of planning anticancer therapies. Existence of optimal controls is proved and all their possible structures are determined in detail, by using the Pontryagin's Maximum Principle. The influence of the pharmacokinetics and pharmacodynamics variants, together with the integral constraint, is analyzed. Moreover, the numerical results of some illustrative examples and our conclusions are presented.
Exact wavefronts and periodic patterns in a competition system with nonlinear diffusion
M. Guedda , R. Kersner , M. Klincsik and  E. Logak
2014, 19(6): 1589-1600 doi: 10.3934/dcdsb.2014.19.1589 +[Abstract](30) +[PDF](644.3KB)
We consider a two-species competition system with nonlinear diffusion and exhibit exact solutions of the system. We first show the existence of spatially stationary solutions that are periodic patterns. In a particular case, we also provide a time-dependent solution that approximates this periodic solution. We also show that the system may sustain unbounded wavefronts above the coexistence equilibrium. In the case of equal intrinsic growth rates, we give a sharp wavefront solution with semi-finite support.
Analysis on the initial-boundary value problem of a full bipolar hydrodynamic model for semiconductors
Haifeng Hu and  Kaijun Zhang
2014, 19(6): 1601-1626 doi: 10.3934/dcdsb.2014.19.1601 +[Abstract](45) +[PDF](525.1KB)
We consider the initial boundary value problem of the one dimensional full bipolar hydrodynamic model for semiconductors. The existence and uniqueness of the stationary solution are established by the theory of strongly elliptic systems and the Banach fixed point theorem. The exponentially asymptotic stability of the stationary solution is given by means of the energy estimate method.
The linear hyperbolic initial and boundary value problems in a domain with corners
Aimin Huang and  Roger Temam
2014, 19(6): 1627-1665 doi: 10.3934/dcdsb.2014.19.1627 +[Abstract](33) +[PDF](675.4KB)
In this article, we consider linear hyperbolic Initial and Boundary Value Problems (IBVP) in a rectangle (or possibly curvilinear polygonal domains) in both the constant and variable coefficients cases. We use semigroup method instead of Fourier analysis to achieve the well-posedness of the linear hyperbolic system, and we find by diagonalization that there are only two elementary modes in the system which we call hyperbolic and elliptic modes. The hyperbolic system in consideration is either symmetric or Friedrichs-symmetrizable.
Compact implicit integration factor methods for a family of semilinear fourth-order parabolic equations
Lili Ju , Xinfeng Liu and  Wei Leng
2014, 19(6): 1667-1687 doi: 10.3934/dcdsb.2014.19.1667 +[Abstract](45) +[PDF](693.4KB)
When developing efficient numerical methods for solving parabolic types of equations, severe temporal stability constraints on the time step are often required due to the high-order spatial derivatives and/or stiff reactions. The implicit integration factor (IIF) method, which treats spatial derivative terms explicitly and reaction terms implicitly, can provide excellent stability properties in time with nice accuracy. One major challenge for the IIF is the storage and calculation of the dense exponentials of the sparse discretization matrices resulted from the linear differential operators. The compact representation of the IIF (cIIF) can overcome this shortcoming and greatly save computational cost and storage. On the other hand, the cIIF is often hard to be directly applied to deal with problems involving cross derivatives. In this paper, by treating the discretization matrices in diagonalized forms, we develop an efficient cIIF method for solving a family of semilinear fourth-order parabolic equations, in which the bi-Laplace operator is explicitly handled and the computational cost and storage remain the same as to the classic cIIF for second-order problems. In particular, the proposed method can deal with not only stiff nonlinear reaction terms but also various types of homogeneous or inhomogeneous boundary conditions. Numerical experiments are finally presented to demonstrate effectiveness and accuracy of the proposed method.
Numerical study of blow-up in solutions to generalized Kadomtsev-Petviashvili equations
Christian Klein and  Ralf Peter
2014, 19(6): 1689-1717 doi: 10.3934/dcdsb.2014.19.1689 +[Abstract](26) +[PDF](5077.4KB)
We present a numerical study of solutions to the generalized Kadomtsev-Petviashvili equations with critical and supercritical nonlinearity for localized initial data with a single minimum and single maximum. In the cases with blow-up, we use a dynamic rescaling to identify the type of the singularity. We present the first discussion of the observed blow-up scenarios. We show that the blow-up in solutions to the $L_{2}$ critical generalized Kadomtsev-Petviashvili I case is similar to what is known for the $L_{2}$ critical generalized Korteweg-de Vries equation. No blow-up is observed for solutions to the generalized Kadomtsev-Petviashvili II equations for $n\leq2$.
Bifurcations and exact travelling wave solutions of the generalized two-component Hunter-Saxton system
Jibin Li
2014, 19(6): 1719-1729 doi: 10.3934/dcdsb.2014.19.1719 +[Abstract](33) +[PDF](4926.3KB)
In this paper, we apply the method of dynamical systems to a generalized two-component Hunter-Saxton system. Through qualitative analysis, we obtain bifurcations of phase portraits of the traveling system. Under different parameter conditions, exact explicit smooth solitary wave solutions, solitary cusp wave solutions, as well as periodic wave solutions are obtained. To guarantee the existence of these solutions, rigorous parametric conditions are given.
Zero-Hopf bifurcation for a class of Lorenz-type systems
Jaume Llibre and  Ernesto Pérez-Chavela
2014, 19(6): 1731-1736 doi: 10.3934/dcdsb.2014.19.1731 +[Abstract](45) +[PDF](275.2KB)
In this paper we apply the averaging theory to a class of three-dimensional autonomous quadratic polynomial differential systems of Lorenz-type, to show the existence of limit cycles bifurcating from a degenerate zero-Hopf equilibrium.
An unconditionally stable numerical method for the viscous Cahn--Hilliard equation
Jaemin Shin , Yongho Choi and  Junseok Kim
2014, 19(6): 1737-1747 doi: 10.3934/dcdsb.2014.19.1737 +[Abstract](38) +[PDF](740.8KB)
We present an unconditionally stable finite difference method for solving the viscous Cahn--Hilliard equation. We prove the unconditional stability of the proposed scheme by using the decrease of a discrete functional. We present numerical results that validate the convergence and unconditional stability properties of the method. Further, we present numerical experiments that highlight the different temporal evolutions of the Cahn--Hilliard and viscous Cahn--Hilliard equations.
Global stability and backward bifurcation of a general viral infection model with virus-driven proliferation of target cells
Hongying Shu and  Lin Wang
2014, 19(6): 1749-1768 doi: 10.3934/dcdsb.2014.19.1749 +[Abstract](44) +[PDF](685.1KB)
In this paper, a general viral model with virus-driven proliferation of target cells is studied. Global stability results are established by employing the Lyapunov method and a geometric approach developed by Li and Muldowney. It is shown that under certain conditions, the model exhibits a global threshold dynamics, while if these conditions are not met, then backward bifurcation and bistability are possible. An example is presented to provide some insights on how the virus-driven proliferation of target cells influences the virus dynamics and the drug therapy strategies.
On the minimal speed of front propagation in a model of the Belousov-Zhabotinsky reaction
Elena Trofimchuk , Manuel Pinto and  Sergei Trofimchuk
2014, 19(6): 1769-1781 doi: 10.3934/dcdsb.2014.19.1769 +[Abstract](36) +[PDF](416.9KB)
In this paper, we answer the question about the existence of the minimal speed of front propagation in a delayed version of the Murray model of the Belousov-Zhabotinsky (BZ) chemical reaction. It is assumed that the key parameter $r$ of this model satisfies $0< r \leq 1$ that makes it formally monostable. By proving that the set of all admissible speeds of propagation has the form $[c_*,+\infty)$, we show here that the BZ system with $r \in (0,1]$ is actually of the monostable type (in general, $c_*$ is not linearly determined). We also establish the monotonicity of wavefronts and present the principal terms of their asymptotic expansions at infinity (in the critical case $r=1$ inclusive).
Permanence and extinction of a non-autonomous HIV-1 model with time delays
Xia Wang , Shengqiang Liu and  Libin Rong
2014, 19(6): 1783-1800 doi: 10.3934/dcdsb.2014.19.1783 +[Abstract](28) +[PDF](1468.9KB)
The environment of HIV-1 infection and treatment could be non-periodically time-varying. The purposes of this paper are to investigate the effects of time-dependent coefficients on the dynamics of a non-autonomous and non-periodic HIV-1 infection model with two delays, and to provide explicit estimates of the lower and upper bounds of the viral load. We established sufficient conditions for the permanence and extinction of the non-autonomous system based on two positive constants $R^{\ast}$ and $R_{\ast}$ ($R^{\ast}\geq R_{\ast}$) that could be precisely expressed by the coefficients of the system: (i) If $R^{\ast}<1$, then the infection-free steady state is globally attracting; (ii) if $R_{\ast}>1$, then the system is permanent. When the system is permanent, we further obtained detailed estimates of both the lower and upper bounds of the viral load. The results show that both $R^{\ast}$ and $R_{\ast}$ reduce to the basic reproduction ratio of the corresponding autonomous model when all the coefficients become constants. Numerical simulations have been performed to verify/extend our analytical results. We also provided some numerical results showing that both permanence and extinction are possible when $R_{\ast }< 1 < R^{\ast}$ holds.
Pullback attractors for the non-autonomous quasi-linear complex Ginzburg-Landau equation with $p$-Laplacian
Bo You , Yanren Hou , Fang Li and  Jinping Jiang
2014, 19(6): 1801-1814 doi: 10.3934/dcdsb.2014.19.1801 +[Abstract](29) +[PDF](435.1KB)
In this paper, we are concerned with the long-time behavior of the following non-autonomous quasi-linear complex Ginzburg-Landau equation with $p$-Laplacian \begin{align*} \frac{\partial u}{\partial t}-(\lambda+i\alpha)\Delta_p u+(\kappa+i\beta)|u|^{q-2}u-\gamma u=g(x,t) \end{align*} without any restriction on $q>2$ under additional assumptions. We first prove the existence of a pullback absorbing set in $L^2(\Omega) \cap W^{1,p}_0(\Omega)\cap L^q(\Omega)$ for the process $\{U(t,\tau)\}_{t\geq \tau}$ corresponding to the non-autonomous quasi-linear complex Ginzburg-Landau equation (1)-(3) with $p$-Laplacian. Next, the existence of a pullback attractor in $L^2(\Omega)$ is established by the Sobolev compactness embedding theorem. Finally, we prove the existence of a pullback attractor in $W^{1,p}_0(\Omega)$ for the process $\{U(t,\tau)\}_{t\geq \tau}$ associated with the non-autonomous quasi-linear complex Ginzburg-Landau equation (1)-(3) with $p$-Laplacian by asymptotic a priori estimates.

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