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Volume 1, 2001

Discrete & Continuous Dynamical Systems - B

2001 , Volume 1 , Issue 1

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A spectral formulation of the Nekhoroshev theorem and its relevance for numerical and experimental data analysis
Massimiliano Guzzo and Giancarlo Benettin
2001, 1(1): 1-28 doi: 10.3934/dcdsb.2001.1.1 +[Abstract](113) +[PDF](1052.6KB)
In this paper we provide an analytical characterization of the Fourier spectrum of the solutions of quasi-integrable Hamiltonian systems, which completes the Nekhoroshev theorem and looks particularly suitable to describe resonant motions. We also discuss the application of the result to the analysis of numerical and experimental data. The comparison of the rigorous theoretical estimates with numerical results shows a quite good agreement. It turns out that an observation of the spectrum for a relatively short time scale (of order $1/\sqrt{\varepsilon}$, where $\varepsilon$ is a natural perturbative parameter) can provide information on the behavior of the system for the much larger Nekhoroshev times.
Structure of 2D incompressible flows with the Dirichlet boundary conditions
Tian Ma and Shouhong Wang
2001, 1(1): 29-41 doi: 10.3934/dcdsb.2001.1.29 +[Abstract](111) +[PDF](165.9KB)
We study in this article the structure and its stability of 2-D divergence-free vector fields with the Dirichlet boundary conditions. First we classify boundary points into two new categories: $\partial$−singular points and $\partial$−regular points, and establish an explicit formulation of divergence-free vector fields near the boundary. Second, local orbit structure near the boundary is classified. Then a structural stability theorem for divergence-free vector fields with the Dirichlet boundary conditions is obtained, providing necessary and sufficient conditions of a divergence-free vector fields. These structurally stability conditions are extremely easy to verify, and examples on stability of typical flow patterns are given.
The main motivation of this article is to provide an important step for a forthcoming paper, where, for the first time, we are able to establish precise rigorous criteria on boundary layer separations of incompressible fluid flows, a long standing problem in fluid mechanics.
Cell death and the maintenance of immunological memory
Andrew Yates and Robin Callard
2001, 1(1): 43-59 doi: 10.3934/dcdsb.2001.1.43 +[Abstract](101) +[PDF](279.1KB)
Immunological memory is found in diverse populations of a class of lymphocytes called T cells, that are held at roughly constant numbers. Its composition is in continuous flux as we encounter new pathogens and cells are lost. The mechanisms which preserve the memory T cell population in the face of these uncertain factors are largely unknown. We propose a mechanism for homeostasis, driven by density-dependent cell death, that both fits experimental data and naturally preserves the clonal composition of the T cell pool with fluctuating cell numbers. It also provides clues as to the source of differences in diversity between T cell memory subpopulations.
Non-persistence of roll-waves under viscous perturbations
Pascal Noble and Sebastien Travadel
2001, 1(1): 61-70 doi: 10.3934/dcdsb.2001.1.61 +[Abstract](57) +[PDF](145.7KB)
In this paper, we study the existence of periodic traveling waves for the equations of the Shallow-water theory with a small viscosity. This small viscosity leads to a singularly perturbed problem. The slow-fast system involves a point where normal hyperbolicity breaks down. We first prove the existence of a slow manifold around this point. Then we show that there are no periodic solutions for small viscosity and describe completely the structure of a travelling wave.
On the box method for a non-local parabolic variational inequality
Walter Allegretto, Yanping Lin and Shuqing Ma
2001, 1(1): 71-88 doi: 10.3934/dcdsb.2001.1.71 +[Abstract](92) +[PDF](400.0KB)
In this paper we study a box scheme (or finite volume element method) for a non-local nonlinear parabolic variational inequality arising in the study of thermistor problems. Under some assumptions on the data and regularity of the solution, optimal error estimates in the $H^1$-norm are attained.
Derivation of viscous Saint-Venant system for laminar shallow water; Numerical validation
Jean-Frédéric Gerbeau and Benoit Perthame
2001, 1(1): 89-102 doi: 10.3934/dcdsb.2001.1.89 +[Abstract](333) +[PDF](239.9KB)
We derive the Saint-Venant system for the shallow waters including small friction, viscosity and Coriolis-Boussinesq factor departing from the Navier-Stokes system with a free moving boundary. This derivation relies on the hydrostatic approximation where we follow the role of viscosity and friction on the bottom. Numerical comparisons between the limiting Saint-Venant system and direct Navier-Stokes simulation allow to validate this derivation.
Analysis of IVGTT glucose-insulin interaction models with time delay
Jiaxu Li, Yang Kuang and Bingtuan Li
2001, 1(1): 103-124 doi: 10.3934/dcdsb.2001.1.103 +[Abstract](91) +[PDF](297.3KB)
In the last three decades, several models on the interaction of glucose and insulin have appeared in the literature, the mostly used one is generally known as the "minimal model" which was first published in 1979 and modified in 1986. Recently, this minimal model has been questioned by De Gaetano and Arino [4] from both physiological and modeling aspects. Instead, they proposed a new and mathematically more reasonable model, called "dynamic model". Their model makes use of certain simple and specific functions and introduces time delay in a particular way. The outcome is that the model always admits a globally asymptotically stable steady state. The objective of this paper is to find out if and how this outcome depends on the specific choice of functions and the way delay is incorporated. To this end, we generalize the dynamical model to allow more general functions and an alternative way of incorporating time delay. Our findings show that in theory, such models can possess unstable positive steady states. However, for all conceivable realistic data, such unstable steady states do not exist. Hence, our work indicates that the dynamic model does provide qualitatively robust dynamics for the purpose of clinic application. We also perform simulations based on data from a clinic study and point out some plausible but important implications.
The numerical detection of connecting orbits
Michael Dellnitz, O. Junge and B Thiere
2001, 1(1): 125-135 doi: 10.3934/dcdsb.2001.1.125 +[Abstract](85) +[PDF](615.8KB)
We present a new technique for the numerical detection and localization of connecting orbits between hyperbolic invariant sets in parameter dependent dynamical systems. This method is based on set-oriented multilevel methods for the computation of invariant manifolds and it can be applied to systems of moderate dimension. The main idea of the algorithm is to detect intersections of coverings of the stable and unstable manifolds of the invariant sets on different levels of the approximation. We demonstrate the applicability of the new method by three examples.

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