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Discrete & Continuous Dynamical Systems - B

2001 , Volume 1 , Issue 2

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A restricted four-body model for the dynamics near the Lagrangian points of the Sun-Jupiter system
Frederic Gabern and  Àngel Jorba
2001, 1(2): 143-182 doi: 10.3934/dcdsb.2001.1.143 +[Abstract](36) +[PDF](2454.8KB)
We focus on the dynamics of a small particle near the Lagrangian points of the Sun-Jupiter system. To try to account for the effect of Saturn, we develop a specific model based on the computation of a true solution of the planar three-body problem for Sun, Jupiter and Saturn, close to the real motion of these three bodies. Then, we can write the equations of motion of a fourth infinitesimal particle moving under the attraction of these three masses. Using suitable coordinates, the model is written as a time-dependent perturbation of the well-known spatial Restricted Three-Body Problem.
Then, we study the dynamics of this model near the triangular points. The tools are based on computing, up to high order, suitable normal forms and first integrals. From these expansions, it is not difficult to derive approximations to invariant tori (of dimensions 2, 3 and 4) as well as bounds on the speed of diffusion on suitable domains. We have also included some comparisons with the motion of a few Trojan asteroids in the real Solar system.
Competitive exclusion in a discrete-time, size-structured chemostat model
H. L. Smith and  X. Q. Zhao
2001, 1(2): 183-191 doi: 10.3934/dcdsb.2001.1.183 +[Abstract](33) +[PDF](165.3KB)
Competitive exclusion is proved for a discrete-time, size-structured, nonlinear matrix model of m-species competition in the chemostat. The winner is the population able to grow at the lowest nutrient concentration. This extends the results of earlier work of the first author [11] where the case $m = 2$ was treated.
Existence of travelling wave solutions in a combustion-radiation model
Claude-Michael Brauner , Josephus Hulshof and  J.-F. Ripoll
2001, 1(2): 193-208 doi: 10.3934/dcdsb.2001.1.193 +[Abstract](45) +[PDF](190.5KB)
We consider a simple model of premixed flames propagating in a gaseous mixture containing inert dust. The radiation field is modelled by the classical Eddington equation. The main parameters are the dimensionless opacity and the Boltzmann number. We prove the existence of travelling solutions with increased speed w.r.t. the adiabatic case. Several singular limiting cases (including a modification involving an ignition temperature) of the parameter values are discussed.
Dynamical systems and operations research: A basic model
Leonid A. Bunimovich
2001, 1(2): 209-218 doi: 10.3934/dcdsb.2001.1.209 +[Abstract](32) +[PDF](177.4KB)
Operations Research and Logistics are the areas, where traditionally only stochastic models were applied. However, recently this situation started to change, and dynamical systems are becoming to be recognized as the relevant models in manufacturing, managing supply chains, conditioned based maintenance, etc. We discuss the simplest basic model for these processes and prove some results on its global dynamics. The general approach to a management of such processes (Stabilization of a Target Regime or STR method) is outlined and illustrated.
Hysteresis in layered spring magnets
J. Samuel Jiang , Hans G. Kaper and  Gary K Leaf
2001, 1(2): 219-232 doi: 10.3934/dcdsb.2001.1.219 +[Abstract](27) +[PDF](339.8KB)
This article addresses a problem of micromagnetics: the reversal of magnetic moments in layered spring magnets. A one-dimensional model is used of a film consisting of several atomic layers of a soft material on top of several atomic layers of a hard material. Each atomic layer is taken to be uniformly magnetized, and spatial inhomogeneities within an atomic layer are neglected. The state of such a system is described by a chain of magnetic spin vectors. Each spin vector behaves like a spinning top driven locally by the effective magnetic field and subject to damping (Landau–Lifshitz–Gilbert equation). A numerical integration scheme for the LLG equation is presented that is unconditionally stable and preserves the magnitude of the magnetization vector at all times. The results of numerical investigations for a bilayer in a rotating in-plane magnetic field show hysteresis with a basic period of $2\pi$ at moderate fields and hysteresis with a basic period of $\pi$ at strong fields.
Sufficient conditions for stability of linear differential equations with distributed delay
Samuel Bernard , Jacques Bélair and  Michael C Mackey
2001, 1(2): 233-256 doi: 10.3934/dcdsb.2001.1.233 +[Abstract](68) +[PDF](303.3KB)
We develop conditions for the stability of the constant (steady state) solutions oflinear delay differential equations with distributed delay when only information about the moments of the density of delays is available. We use Laplace transforms to investigate the properties of different distributions of delay. We give a method to parametrically determine the boundary of the region of stability, and sufficient conditions for stability based on the expectation of the distribution of the delay. We also obtain a result based on the skewness of the distribution. These results are illustrated on a recent model of peripheral neutrophil regulatory system which include a distribution of delays. The goal of this paper is to give a simple criterion for the stability when little is known about the distribution of the delay.
Necessary and sufficient conditions for strong ellipticity of isotropic functions in any dimension
Bernard Dacorogna
2001, 1(2): 257-263 doi: 10.3934/dcdsb.2001.1.257 +[Abstract](40) +[PDF](161.4KB)
We consider hyperelastic stored energy functions in $\mathbb{R}^{n\times n}$ that are isotropic. We give necessary and sufficient conditions for the ellipticity of such functions. The present article is essentially a review of recent results on the subject.

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