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Discrete & Continuous Dynamical Systems - B

2007 , Volume 7 , Issue 1

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On Bloch waves for the Stokes equations
Grégoire Allaire, Carlos Conca, Luis Friz and Jaime H. Ortega
2007, 7(1): 1-28 doi: 10.3934/dcdsb.2007.7.1 +[Abstract](69) +[PDF](638.9KB)
Abstract:
In this work, we study the Bloch wave decomposition for the Stokes equations in a periodic media in $\R^d$. We prove that, because of the incompressibility constraint, the Bloch eigenvalues, as functions of the Bloch frequency $\xi$, are not continuous at the origin. Nevertheless, when $\xi$ goes to zero in a fixed direction, we exhibit a new limit spectral problem for which the eigenvalues are directionally differentiable. Finally, we present an analogous study for the Bloch wave decomposition for a periodic perforated domain.
Well-posedness of the modified Crank-Nicholson difference schemes in Bochner spaces
Allaberen Ashyralyev
2007, 7(1): 29-51 doi: 10.3934/dcdsb.2007.7.29 +[Abstract](87) +[PDF](260.7KB)
Abstract:
The nonlocal boundary value problem $v$’$(t)+Av(t)=f(t)(0\leq t\leq 1),v(0)=v(\lambda )+\mu ,0<\lambda \leq 1$ for differential equations in an arbitrary Banach space $E$ with the strongly positive operator $A$ is considered. The well-posedness of the modified Crank-Nicholson difference schemes of the second order of accuracy for the approximate solutions of this problem in Bochner spaces is established. In applications, the almost coercive stability and the coercive stability estimates for the solutions of difference schemes for the approximate solutions of the nonlocal boundary value problem for parabolic equation are obtained.
Galerkin method and approximate tracking in a non-minimum phase bilinear system
E. Fossas and J. M. Olm
2007, 7(1): 53-76 doi: 10.3934/dcdsb.2007.7.53 +[Abstract](73) +[PDF](294.8KB)
Abstract:
The tracking control of non-minimum phase systems is nowadays an open and challenging field, because a general theory is still not available. This article proposes an indirect control strategy in which a key role is played by the inverse problem that arises and their approximate solutions. These are obtained with the Galerkin method, a standard functional analysis tool. A detailed study of the effect on the output caused by the use of an approximate input is performed. Error bounds are also provided. The technique is motivated through its implementation in basic, DC-to-DC nonlinear power converters that are intended to be used as DC-to-AC voltage sources.
Pulse vaccination strategy in a delayed sir epidemic model with vertical transmission
Shujing Gao, Dehui Xie and Lansun Chen
2007, 7(1): 77-86 doi: 10.3934/dcdsb.2007.7.77 +[Abstract](77) +[PDF](144.1KB)
Abstract:
Pulse vaccination is an important and effective strategy for the elimination of infectious diseases. In this paper, a delayed SIR epidemic model with pulse vaccination and vertical transmission is proposed. Using the discrete dynamical system determined by the stroboscopic map, we obtain the exact periodic infection-free solution of the impulsive epidemic system and prove that the infection-free periodic solution is globally attractive if $R^$*$<1$. Moreover, we obtain that the disease is uniformly persistent if . Our results indicate that a large pulse vaccination rate is sufficient condition for the extinction of the disease.
Spin-polarized transport: Existence of weak solutions
Carlos J. Garcia-Cervera and Xiao-Ping Wang
2007, 7(1): 87-100 doi: 10.3934/dcdsb.2007.7.87 +[Abstract](109) +[PDF](195.4KB)
Abstract:
A system modeling spin-polarized transport in ferromagnetic multilayers is considered. In this model, the spin accumulation is described by a quasilinear diffusion equation with discontinuous, measurable coefficients. This equation is coupled to the Landau-Lifshitz-Gilbert equation, a nonlinear, nonlocal equation describing the precession of the magnetization in the ferromagnetic layers. The global existence of weak solutions is proved.
Construction of a global Lyapunov function using radial basis functions with a single operator
Peter Giesl
2007, 7(1): 101-124 doi: 10.3934/dcdsb.2007.7.101 +[Abstract](109) +[PDF](329.9KB)
Abstract:
We study the basin of attraction of an asymptotically stable equilibrium of a general autonomous ordinary differential equation. Sublevel sets of Lyapunov functions provide subsets of the basin of attraction. In this paper we construct a Lyapunov function by approximation via radial basis functions. We show the existence and the smoothness of a Lyapunov function with certain, given orbital derivative. By approximation of this Lyapunov function via its orbital derivative using radial basis functions we obtain a global Lyapunov function and can thus determine each compact subset of the basin of attraction.
Global existence for chemotaxis with finite sampling radius
T. Hillen, K. Painter and Christian Schmeiser
2007, 7(1): 125-144 doi: 10.3934/dcdsb.2007.7.125 +[Abstract](130) +[PDF](2484.5KB)
Abstract:
Migrating cells measure the external environment through receptor-binding of specific chemicals at their outer cell membrane. In this paper this non-local sampling is incorporated into a chemotactic model. The existence of global bounded solutions of the non-local model is proven for bounded and unbounded domains in any space dimension. According to a recent classification of spikes and plateaus, it is shown that steady state solutions cannot be of spike-type. Finally, numerical simulations support the theoretical results, illustrating the ability of the model to give rise to pattern formation. Some biologically relevant extensions of the model are also considered.
Uniform a priori estimates for elliptic and static Maxwell interface problems
Jianguo Huang and Jun Zou
2007, 7(1): 145-170 doi: 10.3934/dcdsb.2007.7.145 +[Abstract](77) +[PDF](351.8KB)
Abstract:
We present some new a priori estimates of the solutions to three-dimensional elliptic interface problems and static Maxwell interface system with variable coefficients. Different from the classical a priori estimates, the physical coefficients of the interface problems appear in these new estimates explicitly.
Wave equations and reaction-diffusion equations with several nonlinear source terms of different sign
Yacheng Liu and Runzhang Xu
2007, 7(1): 171-189 doi: 10.3934/dcdsb.2007.7.171 +[Abstract](163) +[PDF](265.6KB)
Abstract:
We study the initial boundary value problem of wave equations and reaction-diffusion equations with several nonlinear source terms of different sign. By introducing a family of potential wells $W_\delta$ and corresponding outside sets $V_\delta$ of $W_\delta$ we first obtain the invariant sets and vacuum isolating of solutions. Then we get the threshold result of global existence and nonexistence of solutions. Finally we prove the global existence of solutions for the problem with critical initial conditions $I(u_0)\ge 0$, $E(0)=d$ (or $J(u_0)=d$).
Local stability implies global stability in some one-dimensional discrete single-species models
Eduardo Liz
2007, 7(1): 191-199 doi: 10.3934/dcdsb.2007.7.191 +[Abstract](93) +[PDF](162.3KB)
Abstract:
We prove a criterion for the global stability of the positive equilibrium in discrete-time single-species population models of the form $x_{n+1}=x_nF(x_n)$. This allows us to demonstrate analytically (and easily) the conjecture that local stability implies global stability in some well-known models, including the Ricker difference equation and a combination of the models by Hassel and Maynard Smith. Our approach combines the use of linear fractional functions (Möbius transformations) and the Schwarzian derivative.

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