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Discrete & Continuous Dynamical Systems - B

2003 , Volume 3 , Issue 3

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Monte-Carlo and polyhedron-based simulations I: extremal states of the logarithmic N-body problem on a sphere
Chjan C. Lim , Joseph Nebus and  Syed M. Assad
2003, 3(3): 313-342 doi: 10.3934/dcdsb.2003.3.313 +[Abstract](38) +[PDF](632.3KB)
The problem of N bodies on the surface of the sphere interacting by a logarithmic potential is examined for selected N ranging from $4$ to $40,962$, comparing the energies found by placing points at the vertices of certain polyhedrons to the lowest energies found by a Monte Carlo algorithm. The polyhedron families are generated from simple polyhedrons through two triangular face splitting operations which are used iteratively to increase the number of vertices. The closest energy of these polyhedron vertex configurations to the Monte Carlo-generated minimum energy is identified and the two energies are found to agree well. Finally the energy per particle pair is found to asymptotically approach a mean field theory limit of $- 1/2 (log(2) - 1)$, approximately $0.153426$, for both the polyhedron and the Monte Carlo-generated energies. The deterministic algorithm of generating polyhedrons is shown to be a method able to generate consistently good approximations to the extremal energy configuration for a wide range of numbers of points.
T cell sensitivity and specificity - kinetic proofreading revisited
Cliburn Chan , Andrew J.T. George and  Jaroslav Stark
2003, 3(3): 343-360 doi: 10.3934/dcdsb.2003.3.343 +[Abstract](108) +[PDF](274.4KB)
T cells recognise foreign antigen presented by antigen presenting cells at extremely low concentrations, and are able to discriminate between different ligands with high specificity. McKeithan's kinetic proofreading model is often invoked to explain this sensitivity and specificity of the T cell. In this paper, we analyse the strengths and limitations of this model, and suggest that it does not seem adequate to explain the observed degree of T cell sensitivity, specificity and robustness.
Set oriented methods for computing reachable sets and control sets
Dietmar Szolnoki
2003, 3(3): 361-382 doi: 10.3934/dcdsb.2003.3.361 +[Abstract](53) +[PDF](480.3KB)
The computation of reachable sets and control sets is a difficult problem, since the objects to be computed have full dimension in the state space. In this paper we present suitably reformulated problems which in many cases allow to apply set oriented numerical methods for the computation of reachable sets and control sets.
Stability in thermoelasticity of type III
Ramon Quintanilla and  Reinhard Racke
2003, 3(3): 383-400 doi: 10.3934/dcdsb.2003.3.383 +[Abstract](49) +[PDF](187.6KB)
We consider initial-boundary value problems in a hyperbolic thermoelastic system, called thermoelasticity of type III. First, we prove the exponential stability in one space dimension for different boundary conditions with energy methods and spectral methods, respectively. Then the exponential stability in more two or three space dimensions is proved for radially symmetric situations. Finally, the equipartition of energy is investigated.
The asymptotic solution of the Cauchy problem for a generalized Boussinesq equation
Shaoyong Lai and  Yong Hong Wu
2003, 3(3): 401-408 doi: 10.3934/dcdsb.2003.3.401 +[Abstract](36) +[PDF](119.4KB)
In this paper, we consider the solution of an initial value problem for the generalized damped Boussinesq equation

$ u_{t t} - a u_{t t x x}- 2 b u_{t x x} = - c u_{x x x x}+ u_{x x} - p^2 u + \beta(u^2)_{x x}, $

where $x\in R^1,$ $t > 0,$ $a ,$ $b$ and $c $ are positive constants, $p \ne 0,$ and $\beta \in R^1$. For the case $a + c > b^2$ corresponding to damped oscillations with an infinite number of oscillation cycles, we establish the well-posedness theorem of the global solution to the problem and derive a large time asymptotic solution.

Dynamics of vertical delay endomorphisms
N. Romero , A. Rovella and  F. Vilamajó
2003, 3(3): 409-422 doi: 10.3934/dcdsb.2003.3.409 +[Abstract](27) +[PDF](223.9KB)
A vertical delay endomorphism $F$ on $\mathbb{R}^k$, with $k\ge 2$, is the endomorphism associated to the difference equation $x_{n+k}=f(x_n,\cdots,x_{n+k-1})$, where the function $f$ is $C^2$ and its partial derivative of second order with respect to the first variable is bigger than every other partial derivative of second order. The main goal of this paper is to describe the dynamical behaviour of a huge class $\mathcal{F}$ of one-parameter families of vertical delay endomorphisms. We will prove that for any $\{F_\mu\}_{\mu\in\mathbb{R}}$ in $\mathcal{F}$ and every $|\mu|$ large enough, the nonwandering set $\Omega(F_\mu)$ of $F_\mu$, is either the empty set or an expanding Cantor set and the restriction of $F_{\mu}$ to $\Omega(F_\mu)$ is conjugated to the unilateral shift on two symbols.
Degenerate resonances in forced oscillators
Kazuyuki Yagasaki
2003, 3(3): 423-438 doi: 10.3934/dcdsb.2003.3.423 +[Abstract](29) +[PDF](229.1KB)
Using an extended version of the subharmonic Melnikov method, we discuss resonance behavior in a class of forced nonlinear oscillators when the resonance is degenerate in the following meaning: The frequency of the resonant periodic orbit has a null derivative with respect to the energy level in the unperturbed system without forcing and damping terms. Such an appropriate treatment of degenerate resonances in systems of physical or engineering meaning as performed here was not previously presented. In particular, we show that the degenerate resonances can generally give rise to cusp bifurcations. Moreover, we describe a numerical strategy for the necessary computations for application of the theory. To illustrate our technique, two examples are presented for a nonsymmetric oscillator and a feedback controlled pendulum. Direct numerical bifurcation analysis results are also given and compared with the theoretical results.
Asymptotic behavior of a singular transport equation modelling cell division
Mostafa Adimy and  Laurent Pujo-Menjouet
2003, 3(3): 439-456 doi: 10.3934/dcdsb.2003.3.439 +[Abstract](31) +[PDF](169.1KB)
This paper analyses the behavior of the solutions of a model of cells that are capable of simultaneous proliferation and maturation. This model is described by a first-order singular partial differential system with a retardation of the maturation variable and a time delay. Both delays are due to cell replication. We prove that uniqueness and asymptotic behavior of solutions depend only on cells with small maturity (stem cells).
Characterizing attraction probabilities via the stochastic Zubov equation
Fabio Camilli and  Lars Grüne
2003, 3(3): 457-468 doi: 10.3934/dcdsb.2003.3.457 +[Abstract](36) +[PDF](639.9KB)
A stochastic differential equation with an a.s. locally stable compact set is considered. The attraction probabilities to the set are characterized by the sublevel sets of the limit of a sequence of solutions to $2^{nd}$ order partial differential equations. Two numerical examples illustrating the method are presented.
Western boundary currents versus vanishing depth
Didier Bresch and  Jacques Simon
2003, 3(3): 469-477 doi: 10.3934/dcdsb.2003.3.469 +[Abstract](32) +[PDF](220.6KB)
In the case of a constant depth, western intensification of currents in oceanic basins was mathematically recovered in various models (such as Stommel, Munk or quasi-geostrophic ones) as a boundary layer appearing when the solution of equations converges to the solution of a pure transport equation. This convergence is linked to the fact that any characteristic line of the transport vector field included in the equations crosses the boundary, and the boundary layer is located at outgoing points.
Here we recover such a boundary layer for the vertical-geostrophic model with a general bathymetry. More precisely, we allow depth to vanish on the shore in which case the above mentioned characteristic lines no longer cross the boundary. However a boundary layer still appears because the transport vector field $a$ (which is tangential to the boundary) locally converges to a vector field $\overline{a}$ with characteristic lines crossing the boundary.

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