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Discrete & Continuous Dynamical Systems - B

2009 , Volume 12 , Issue 1

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A viscoelastic Timoshenko beam with dynamic frictionless impact
Jeongho Ahn and David E. Stewart
2009, 12(1): 1-22 doi: 10.3934/dcdsb.2009.12.1 +[Abstract](177) +[PDF](720.1KB)
Abstract:
Two coupled partial differential equations which describe the motion of a viscoelastic (Kelvin-Voigt type) Timoshenko beam are formulated with the complementarity conditions. This dynamic impact problem is considered a boundary thin obstacle problem. The existence of solutions is proved. A major concern is to pursue an investigation into conservation of energy (or energy balance), which is performed both theoretically and numerically.
Vibrations of a nonlinear dynamic beam between two stops
K. T. Andrews, M. F. M'Bengue and Meir Shillor
2009, 12(1): 23-38 doi: 10.3934/dcdsb.2009.12.23 +[Abstract](169) +[PDF](208.4KB)
Abstract:
This work extends the model developed by Gao (1996) for the vibrations of a nonlinear beam to the case when one of its ends is constrained to move between two reactive or rigid stops. Contact is modeled with the normal compliance condition for the deformable stops, and with the Signorini condition for the rigid stops. The existence of weak solutions to the problem with reactive stops is shown by using truncation and an abstract existence theorem involving pseudomonotone operators. The solution of the Signorini-type problem with rigid stops is obtained by passing to the limit when the normal compliance coefficient approaches infinity. This requires a continuity property for the beam operator similar to a continuity property for the wave operator that is a consequence of the so-called div-curl lemma of compensated compactness.
Stability of constant states and qualitative behavior of solutions to a one dimensional hyperbolic model of chemotaxis
Francesca Romana Guarguaglini, Corrado Mascia, Roberto Natalini and Magali Ribot
2009, 12(1): 39-76 doi: 10.3934/dcdsb.2009.12.39 +[Abstract](168) +[PDF](1394.1KB)
Abstract:
We consider a general model of chemotaxis with finite speed of propagation in one space dimension. For this model we establish a general result of stability of some constant states both for the Cauchy problem on the whole real line and for the Neumann problem on a bounded interval. These results are obtained using the linearized operators and the accurate analysis of their nonlinear perturbations. Numerical schemes are proposed to approximate these equations, and the expected qualitative behavior for large times is compared to several numerical tests.
Particle, kinetic and fluid models for phototaxis
Seung-Yeal Ha and Doron Levy
2009, 12(1): 77-108 doi: 10.3934/dcdsb.2009.12.77 +[Abstract](236) +[PDF](512.8KB)
Abstract:
In this work we derive a hierarchy of new mathematical models for describing the motion of phototactic bacteria, i.e., bacteria that move towards light. These models are based on recent experiments suggesting that the motion of such bacteria depends on the individual bacteria, on group dynamics, and on the interaction between bacteria and their environment. Our first model is a collisionless interacting particle system in which we follow the location of the bacteria, their velocity, and their internal excitation (a parameter whose role is assumed to be related to communication between bacteria). In this model, the light source acts as an external force. The resulting particle system is an extension of the Cucker-Smale flocking model. We prove that when all particles are fully excited, their asymptotic velocity tends to an identical (pre-determined) terminal velocity. Our second model is a kinetic model for the one-particle distribution function that includes an internal variable representing the excitation level. The kinetic model is a Vlasov-type equation that is derived from the particle system using the BBGKY hierarchy and molecular chaos assumption. Since bacteria tend to move in areas that were previously traveled by other bacteria, a surface memory effect is added to the kinetic model as a turning operator that accounts for the collisions between bacteria and the environment. The third and final model is derived as a formal macroscopic limit of the kinetic model. It is shown to be the Vlasov-McKean equation coupled with a reaction-diffusion equation.
Numerical computation of dichotomy rates and projectors in discrete time
Thorsten Hüls
2009, 12(1): 109-131 doi: 10.3934/dcdsb.2009.12.109 +[Abstract](217) +[PDF](602.8KB)
Abstract:
We introduce a characterization of exponential dichotomies for linear difference equations that can be tested numerically and enables the approximation of dichotomy rates and projectors with high accuracy. The test is based on computing the bounded solutions of a specific inhomogeneous difference equation. For this task a boundary value and a least squares approach is applied. The results are illustrated using Hénon's map. We compute approximations of dichotomy rates and projectors of the variational equation, along a homoclinic orbit and an orbit on the attractor as well as for an almost periodic example. For the boundary value and the least squares approach, we analyze in detail errors that occur, when restricting the infinite dimensional problem to a finite interval.
Global bifurcation for discrete competitive systems in the plane
M. R. S. Kulenović and Orlando Merino
2009, 12(1): 133-149 doi: 10.3934/dcdsb.2009.12.133 +[Abstract](299) +[PDF](799.7KB)
Abstract:
A global bifurcation result is obtained for families of competitive systems of difference equations

$x_{n+1} = f_\alpha(x_n,y_n) $
$y_{n+1} = g_\alpha(x_n,y_n)$

where $\alpha$ is a parameter, $f_\alpha$ and $g_\alpha$ are continuous real valued functions on a rectangular domain $\mathcal{R}_\alpha \subset \mathbb{R}^2$ such that $f_\alpha(x,y)$ is non-decreasing in $x$ and non-increasing in $y$, and $g_\alpha(x, y)$ is non-increasing in $x$ and non-decreasing in $y$. A unique interior fixed point is assumed for all values of the parameter $\alpha$.
    As an application of the main result for competitive systems a global period-doubling bifurcation result is obtained for families of second order difference equations of the type

$x_{n+1} = F_\alpha(x_n, x_{n-1}), \quad n=0,1, \ldots $

where $\alpha$ is a parameter, $F_\alpha:\mathcal{I_\alpha}\times \mathcal{I_\alpha} \rightarrow \mathcal{I_\alpha}$ is a decreasing function in the first variable and increasing in the second variable, and $\mathcal{I_\alpha}$ is a interval in $\mathbb{R}$, and there is a unique interior equilibrium point. Examples of application of the main results are also given.

Bifurcation analysis in models of tumor and immune system interactions
Dan Liu, Shigui Ruan and Deming Zhu
2009, 12(1): 151-168 doi: 10.3934/dcdsb.2009.12.151 +[Abstract](203) +[PDF](2061.4KB)
Abstract:
The purpose of this paper is to present qualitative and bifurcation analysis near the degenerate equilibrium in models of interactions between lymphocyte cells and solid tumor and to understand the development of tumor growth. Theoretical analysis shows that these cancer models can exhibit Bogdanov-Takens bifurcation under sufficiently small perturbation of the system parameters whether it is vascularized or not. Periodic oscillation behavior and coexistence of the immune system and the tumor in the host are found to be influenced significantly by the choice of bifurcation parameters. It is also confirmed that bifurcations of codimension higher than 2 cannot occur at this equilibrium in both cases. The analytic bifurcation diagrams and numerical simulations are given. Some anomalous properties are discovered from comparing the vascularized case with the avascular case.
Threshold dynamics in a time-delayed periodic SIS epidemic model
Yijun Lou and Xiao-Qiang Zhao
2009, 12(1): 169-186 doi: 10.3934/dcdsb.2009.12.169 +[Abstract](373) +[PDF](681.8KB)
Abstract:
The global dynamics of a periodic SIS epidemic model with maturation delay is investigated. We first obtain sufficient conditions for the single population growth equation to admit a globally attractive positive periodic solution. Then we introduce the basic reproduction ratio $\mathcal{R}_0$ for the epidemic model, and show that the disease dies out when $\mathcal{R}_0<1$, and the disease remains endemic when $\mathcal{R}_0>1$. Numerical simulations are also provided to confirm our analytic results.
Lyapunov exponents and persistence in discrete dynamical systems
Paul L. Salceanu and H. L. Smith
2009, 12(1): 187-203 doi: 10.3934/dcdsb.2009.12.187 +[Abstract](268) +[PDF](270.7KB)
Abstract:
The theory of Lyapunov exponents and methods from ergodic theory have been employed by several authors in order to study persistence properties of dynamical systems generated by ODEs or by maps. Here we derive sufficient conditions for uniform persistence, formulated in the language of Lyapunov exponents, for a large class of dissipative discrete-time dynamical systems on the positive orthant of $\mathbb{R}^m$, having the property that a nontrivial compact invariant set exists on a bounding hyperplane. We require that all so-called normal Lyapunov exponents be positive on such invariant sets. We apply the results to a plant-herbivore model, showing that both plant and herbivore persist, and to a model of a fungal disease in a stage-structured host, showing that the host persists and the disease is endemic.
A computational method for an inverse problem in a parabolic system
Mohsen Tadi
2009, 12(1): 205-218 doi: 10.3934/dcdsb.2009.12.205 +[Abstract](156) +[PDF](156.6KB)
Abstract:
This note is concerned with the identification of the absorption coefficient in a parabolic system. It introduces an algorithm that can be used to recover the unknown function. The algorithm is iterative in nature. It assumes an initial value for the unknown function and updates it at each iteration. Using the assumed value, the algorithm obtains a background field and computes the equation for the error at each iteration. The error equation includes the correction to the assumed value of the unknown function. Using the measurements obtained at the boundaries, the algorithm introduces two formulations for the error dynamics. By equating the responses of these two formulations it is then possible to obtain an equation for the unknown correction term. A number of numerical examples are also used to study the performance of the algorithm.
A remark on exponential stability of time-delayed Burgers equation
Yanbin Tang and Ming Wang
2009, 12(1): 219-225 doi: 10.3934/dcdsb.2009.12.219 +[Abstract](235) +[PDF](127.9KB)
Abstract:
In this paper, we consider the initial-boundary value problem of Burgers equation with a time delay. Using a fixed point theorem and a comparison principle, we show that the time-delayed Burgers equation is exponentially stable under small delays. The result is more explicit, but also complements, the result given by Weijiu Liu [Discrete and Continuous Dynamical Systems-Series B, 2:1(2002),47-56], which was based on the Liapunov function approach.
Normal and slow growth states of microbial populations in essential resource-based chemostat
Yi Wang and Chengmin Zheng
2009, 12(1): 227-250 doi: 10.3934/dcdsb.2009.12.227 +[Abstract](179) +[PDF](470.1KB)
Abstract:
To mimic the striking capability of microbial culture for growth adaptation after the onset of the novel environmental conditions, a modified heterogeneous microbial population model in the chemostat with essential resources is proposed which considers adaptation by spontaneously phenotype-switching between normally growing cells and persister cells having reduced growth rate. A basic reproductive number $R_0$ is introduced so that the population dies out when $R_0<1$, and when $R_0>1$ the population will be asymptotic to a steady state of persister cells, or a steady state of only normal cells, or a steady state corresponding to a heterogeneous population of both normal and persister cells. Our analysis confirms that inherent heterogeneity of bacterial populations is important in adaption to fluctuating environments and in the persistence of bacterial infections.
Some existence results on periodic and subharmonic solutions of ordinary $P$-Laplacian systems
Yuxiang Zhang and Shiwang Ma
2009, 12(1): 251-260 doi: 10.3934/dcdsb.2009.12.251 +[Abstract](222) +[PDF](143.2KB)
Abstract:
Some existence theorems are obtained for periodic and subharmonic solutions of ordinary $P$-Laplacian systems by the minimax methods in critical point theory.

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