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Discrete & Continuous Dynamical Systems - B

2011 , Volume 15 , Issue 3

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Boundary stabilizability of the linearized viscous Saint-Venant system
Hassen Arfaoui , Faker Ben Belgacem , Henda El Fekih and  Jean-Pierre Raymond
2011, 15(3): 491-511 doi: 10.3934/dcdsb.2011.15.491 +[Abstract](40) +[PDF](338.8KB)
Abstract:
We consider a shallow water flow in a channel modeled by the Saint-Venant equations with a viscous term. We are interested in the stabilization of the flow at a steady state. We establish that the semi-group of the linearized system is exponentially stable. However when the convection coefficient is dominant, the natural stabilization turns out to be very slow. One way to enhance the stabilization of the system is to use boundary controls by means of a moving device located at the extremities of the channel. We determine, by an extension method due to Fursikov, boundary Dirichlet controls able to accelerate the stabilization of the flow. Numerical experiments illustrate the efficiency of the control.
Mathematical models for strongly magnetized plasmas with mass disparate particles
Mihai Bostan and  Claudia Negulescu
2011, 15(3): 513-544 doi: 10.3934/dcdsb.2011.15.513 +[Abstract](29) +[PDF](371.7KB)
Abstract:
The controlled fusion is achieved by magnetic confinement : the plasma is confined into toroidal devices called tokamaks, under the action of strong magnetic fields. The particle motion reduces to advection along the magnetic lines combined to rotation around the magnetic lines. The rotation around the magnetic lines is much faster than the parallel motion and efficient numerical resolution requires homogenization procedures. Moreover the rotation period, being proportional to the particle mass, introduces very different time scales in the case when the plasma contains disparate particles; the electrons turn much faster than the ions, the ratio between their cyclotronic periods being the mass ratio of the electrons with respect to the ions. The subject matter of this paper concerns the mathematical study of such plasmas, under the action of strong magnetic fields. In particular, we are interested in the limit models when the small parameter, representing the mass ratio as we ll as the fast cyclotronic motion, tends to zero.
Heterogeneous viral environment in a HIV spatial model
Claude-Michel Brauner , Danaelle Jolly , Luca Lorenzi and  Rodolphe Thiebaut
2011, 15(3): 545-572 doi: 10.3934/dcdsb.2011.15.545 +[Abstract](39) +[PDF](1785.9KB)
Abstract:
We consider a basic model of virus dynamics in the modeling of Human Immunodeficiency Virus (HIV), in a two-dimensional heterogenous environment. It consists of two ODEs for the uninfected and infected CD4$^+$ T lymphocytes, $T$ and $I$, and a parabolic PDE for the free virus particles $V$. We introduce a new parameter $\lambda_0$ which is the largest eigenvalue of some Sturm-Liouville problem and takes the heterogenous reproductive ratio into account. For $\lambda_0<0$ the uninfected steady state is the only equilibrium. When $\lambda_0>0$, it becomes unstable and there is a unique positive infected equilibrium. Considering the model as a dynamical system, we prove the existence of a positively invariant region. Finally, in the case of an alternating structure of viral sources, we define a homogenized limiting environment which justifies the classical approach via ODE systems.
Well-posedness and stability analysis for a moving boundary problem modelling the growth of nonnecrotic tumors
Joachim Escher and  Anca-Voichita Matioc
2011, 15(3): 573-596 doi: 10.3934/dcdsb.2011.15.573 +[Abstract](60) +[PDF](498.7KB)
Abstract:
We study a moving boundary problem describing the growth of nonnecrotic tumors in different regimes of vascularisation. This model consists of two decoupled Dirichlet problem, one for the rate at which nutrient is added to the tumor domain and one for the pressure inside the tumor. These variables are coupled by a relation which describes the dynamic of the boundary. By re-expressing the problem as an abstract evolution equation, we prove local well-posedness in the small Hölder spaces context. Further on, we use the principle of linearised stability to characterise the stability properties of the unique radially symmetric equilibrium of the problem.
Optimal transmission through a randomly perturbed waveguide in the localization regime
Josselin Garnier
2011, 15(3): 597-621 doi: 10.3934/dcdsb.2011.15.597 +[Abstract](43) +[PDF](481.0KB)
Abstract:
We demonstrate that increased power transmission through a random single-mode or multi-mode channel can be obtained in the localization regime by optimizing the spatial wave front or the time pulse profile of the source. The idea is to select and excite the few modes or the few frequencies whose transmission coefficients are anomalously large compared to the typical exponentially small value. We prove that time reversal is optimal for maximizing the transmitted intensity at a given time or space, while iterated time reversal is optimal for maximizing the total transmitted energy. The statistical stability of the optimal transmitted intensity and energy is also obtained.
Limit of the infinite horizon discounted Hamilton-Jacobi equation
Renato Iturriaga and  Héctor Sánchez-Morgado
2011, 15(3): 623-635 doi: 10.3934/dcdsb.2011.15.623 +[Abstract](37) +[PDF](359.1KB)
Abstract:
Motivated by the infinite horizon discounted problem, we study the convergence of solutions of the Hamilton Jacobi equation when the discount vanishes. If the Aubry set consists in a finite number of hyperbolic critical points, we give an explicit expression for the limit. Additionaly, we give a new characterization of Mañé's critical value as for wich the set of viscosity solutions is equibounded.
Lipschitz continuous data dependence of sweeping processes in BV spaces
Pavel Krejčí and  Thomas Roche
2011, 15(3): 637-650 doi: 10.3934/dcdsb.2011.15.637 +[Abstract](41) +[PDF](374.0KB)
Abstract:
For a rate independent sweeping process with a time dependent smooth convex constraint, we prove that the Kurzweil solution for possibly discontinuous inputs depends locally Lipschitz continuously on the data in terms of the $BV$-norm.
Adaptive method for spike solutions of Gierer-Meinhardt system on irregular domain
Siu-Long Lei
2011, 15(3): 651-668 doi: 10.3934/dcdsb.2011.15.651 +[Abstract](55) +[PDF](1296.9KB)
Abstract:
In this paper, an adaptive numerical method is proposed to solve the Gierer-Meinhardt (GM) system on irregular domain. The method works for domains defined by level sets of implicit functions and the generated mesh is of high quality. The method is shown to be effective by comparing with asymptotic result. Boundary spike solutions of the GM system are obtained and studied numerically, including stability of boundary spike and spike motion along the boundary.
On the effects of migration and inter-specific competitions in steady state of some Lotka-Volterra model
Fang Li , Liping Wang and  Yang Wang
2011, 15(3): 669-686 doi: 10.3934/dcdsb.2011.15.669 +[Abstract](45) +[PDF](414.3KB)
Abstract:
Shadow systems are often used to approximate reaction-diffusion systems when one of the diffusion rates is large. In this paper, we investigate in a shadow system the effects of migration and interspecific competition coefficients on the existence of positive solutions. Our study shows that for any given migration, if the interspecific competition coefficient of the invader is small, then the shadow system has coexistence state; otherwise we can always find some migration such that it has no coexistence state. Moreover, these findings can be applied to steady state of a two-species Lotka-Volterra competition-diffusion model. Particularly, we show that if the interspecific competition coefficient of the invader is sufficiently small, then rapid diffusion of the invader can drive to coexistence state.
Analysis of a frictional contact problem for viscoelastic materials with long memory
Stanisław Migórski , Anna Ochal and  Mircea Sofonea
2011, 15(3): 687-705 doi: 10.3934/dcdsb.2011.15.687 +[Abstract](37) +[PDF](418.4KB)
Abstract:
We consider a mathematical model which describes the frictional contact between a deformable body and a foundation. The process is time-dependent, the material behavior is described with a viscoelastic constitutive law with long memory and the contact is modeled with subdifferential boundary conditions. We derive the variational formulation of the problem which is of the form of a hemivariational inequality with Volterra integral term for the displacement field. Then we prove existence and uniqueness results in the study of abstract inclusions as well as in the study of abstract hemivariational inequalities with Volterra integral term. The proofs are based on arguments on pseudomonotone operators, compactness and fixed point. We use the abstract results to prove the unique solvability of the frictional contact problem. Finally, we present examples of contact and frictional boundary conditions for which our results work.
An epidemiological approach to the spread of political third parties
Daniel M. Romero , Christopher M. Kribs-Zaleta , Anuj Mubayi and  Clara Orbe
2011, 15(3): 707-738 doi: 10.3934/dcdsb.2011.15.707 +[Abstract](31) +[PDF](550.7KB)
Abstract:
Third political parties are influential in shaping American politics. In this work we study the spread of a third party ideology in a voting population where we assume that party members/activists are more influential in recruiting new third party voters than non-member third party voters. The study uses an epidemiological metaphor to develop a theoretical model with nonlinear ordinary differential equations as applied to a case study, the Green Party. Considering long-term behavior, we identify three threshold parameters in our model that describe the different possible scenarios for the political party and its spread. We also apply the model to the study of the Green Party's growth using voting and registration data in six states and the District of Columbia to identify and explain trends over the past decade. Our system produces a backward bifurcation that helps identify conditions under which a sufficiently dedicated activist core can enable a third party to thrive, under conditions which would not normally allow it to arise. Our results explain the critical role activists play in sustaining grassroots movements under adverse conditions.
Computation of symbolic dynamics for two-dimensional piecewise-affine maps
Lorenzo Sella and  Pieter Collins
2011, 15(3): 739-767 doi: 10.3934/dcdsb.2011.15.739 +[Abstract](44) +[PDF](394.1KB)
Abstract:
In this paper we design and implement an algorithm for computing symbolic dynamics for two dimensional piecewise-affine maps. The algorithm is based on detection of periodic orbits using the Conley index and Szymczak decomposition of Conley index pair. The algorithm is also extended to deal with discontinuous maps. We compare the algorithm with the algorithm based on tangle of fixed points. We apply the algorithms to compute the symbolic dynamics and entropy bounds for the Lozi map.
Robust control problems for primitive equations of the ocean
T. Tachim Medjo
2011, 15(3): 769-788 doi: 10.3934/dcdsb.2011.15.769 +[Abstract](44) +[PDF](397.4KB)
Abstract:
In this article, we study some robust control problems associated with the primitive equations of the ocean and related to data assimilation in oceanography. We prove the existence and uniqueness of solutions to these control problems.
On the Hamiltonian dynamics and geometry of the Rabinovich system
Răzvan M. Tudoran and  Anania Gîrban
2011, 15(3): 789-823 doi: 10.3934/dcdsb.2011.15.789 +[Abstract](51) +[PDF](5975.6KB)
Abstract:
In this paper, we describe some relevant dynamical and geometrical properties of the Rabinovich system from the Poisson geometry and the dynamics point of view. Starting with a Lie-Poisson realization of the Rabinovich system, we determine and then completely analyze the Lyapunov stability of all the equilibrium states of the system, study the existence of periodic orbits, and then using a new geometrical approach, we provide the complete dynamical behavior of the Rabinovich system, in terms of the geometric semialgebraic properties of a two-dimensional geometric figure, associated with the problem. Moreover, in tight connection with the dynamical behavior, by using this approach, we also recover all the dynamical objects of the system (e.g. equilibrium states, periodic orbits, homoclinic and heteroclinic connections). Next, we integrate the Rabinovich system by Jacoby elliptic functions, and give some Lax formulations of the system. The last part of the article discusses some numerics associated with the Poisson geometrical structure of the Rabinovich system.
On a generalized Boussinesq model around a rotating obstacle: Existence of strong solutions
Elder J. Villamizar-Roa and  Elva E. Ortega-Torres
2011, 15(3): 825-847 doi: 10.3934/dcdsb.2011.15.825 +[Abstract](28) +[PDF](472.3KB)
Abstract:
The aim of this work is to prove the existence of strong solutions for a generalized Boussinesq model, with nonlinear diffusion for the equations of velocity and temperature, occupying a domain $\Omega,$ exterior to a rigid body that rotates with constant angular velocity $\omega.$
Stability of positive constant steady states and their bifurcation in a biological depletion model
Yan'e Wang and  Jianhua Wu
2011, 15(3): 849-865 doi: 10.3934/dcdsb.2011.15.849 +[Abstract](41) +[PDF](312.4KB)
Abstract:
This paper is concerned with a biological depletion model in a bounded domain. The stability of the positive constant steady states is discussed. In one dimensional case, we make a detailed description for the global bifurcation structure from two positive constant solutions. The result indicates that if $d$ is properly small, the system has at least one non-constant positive steady-state. The main tools used here include the stability theory, bifurcation theory and simulations. From extensive numerical simulations, the predictions from linear theory are confirmed and the influence of parameters $d,D,\sigma$ on these patterns is depicted.
Asymptotic speed of propagation and traveling wavefronts for a SIR epidemic model
Chufen Wu and  Peixuan Weng
2011, 15(3): 867-892 doi: 10.3934/dcdsb.2011.15.867 +[Abstract](63) +[PDF](471.3KB)
Abstract:
In this article, the well-posedness of the initial value problem, the existence of traveling wavefronts and the asymptotic speed of propagation for a SIR epidemic model with stage structure and nonlocal response are studied. We further show that the minimum wave speed in fact coincides with the asymptotic speed of propagation.
Oscillations in a plasmid turbidostat model with delayed feedback control
Sanling Yuan , Yongli Song and  Junhui Li
2011, 15(3): 893-914 doi: 10.3934/dcdsb.2011.15.893 +[Abstract](37) +[PDF](1251.0KB)
Abstract:
A model of competition between plasmid-bearing and plasmid-free organisms in a turbidostat with delayed feedback control is investigated. By choosing the delay in the measurement of the optical sensor to the turbidity of the fluid as a bifurcation parameter, we show that Hopf bifurcations can occur as the delay crosses some critical values. The direction and stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem. Computer simulations illustrate the results.
Addendum
Zhenguo Bai and  Yicang Zhou
2011, 15(3): 915-916 doi: 10.3934/dcdsb.2011.15.915 +[Abstract](45) +[PDF](173.1KB)
Abstract:
N/A.

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