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Discrete & Continuous Dynamical Systems - A

2001 , Volume 7 , Issue 2

Select all articles


Composition in fractional Sobolev spaces
Haim Brezis and Petru Mironescu
2001, 7(2): 241-246 doi: 10.3934/dcds.2001.7.241 +[Abstract](96) +[PDF](169.9KB)
To be added shortly.
Topological equivalence of some variational problems involving distances
Giuseppe Buttazzo, Luigi De Pascale and Ilaria Fragalà
2001, 7(2): 247-258 doi: 10.3934/dcds.2001.7.247 +[Abstract](189) +[PDF](172.7KB)
To every distance $d$ on a given open set $\Omega\subseteq\mathbb R^n$, we may associate several kinds of variational problems. We show that, on the class of all geodesic distances $d$ on $\Omega$ which are bounded from above and from below by fixed multiples of the Euclidean one, the uniform convergence on compact sets turns out to be equivalent to the $\Gamma$-convergence of each of the corresponding variational problems under consideration.
Contracting return maps for monotone delayed feedback
Hans-Otto Walther
2001, 7(2): 259-274 doi: 10.3934/dcds.2001.7.259 +[Abstract](119) +[PDF](176.4KB)
For equations

$\dot{x}(t)=-\mu x(t)+f(x(t-1))$

with continuous odd nonlinearities close to a step function $f_a(\xi)=-a$sign$\xi$, $a>0,$ we find sets of initial data to which solutions return. For Lipschitz nonlinearities the associated return maps become Lipschitz continuous. Monotonicity of $f$ close to 0 permits sharp estimates of the Lipschitz constants of the return map. If outside a neighbourhood of 0 the Lipschitz constant of $f$ is sufficiently small then the return map becomes a contraction, whose fixed point defines an attracting periodic orbit of the differential equation. Applications include the cases $f(x)=g(\gamma x)$ with $g'(x)=-(1+x^r)^{-1}$ for $x>0$, $r>\frac{3}{2}$ and $\gamma>0$ sufficiently large.

A note on a class of higher order conformally covariant equations
Sun-Yung Alice Chang and Wenxiong Chen
2001, 7(2): 275-281 doi: 10.3934/dcds.2001.7.275 +[Abstract](99) +[PDF](118.3KB)
In this paper, we study the higher order conformally covariant equation

$(- \Delta )^{\frac{n}{2}} w = (n -1)! e^{n w} x \in R^n$

for all even dimensions n.

$\alpha = \frac{1}{|S^n|} \int_{R^n} e^{n w} dx .$

We prove, for every $0 < \alpha < 1$, the existence of at least one solution. In particular, for $ n = 4$, we obtain the existence of radial solutions.

Exact/approximate controllability of thermoelastic plates with variable thermal coefficients
M. Eller and Roberto Triggiani
2001, 7(2): 283-302 doi: 10.3934/dcds.2001.7.283 +[Abstract](152) +[PDF](233.8KB)
We study a controllability problem (exact in the mechanical variables {$w,w_t$} and, simultaneously, approximate in the thermal variable $\theta$) of thermoelastic plates by means of boundary controls, in the clamped/Dirichlet B.C. case, when the 'thermal expansion' term is variable in space.
Periodic solutions of twist type of an earth satellite equation
Daniel Núñez and Pedro J. Torres
2001, 7(2): 303-306 doi: 10.3934/dcds.2001.7.303 +[Abstract](76) +[PDF](89.8KB)
We study Lyapunov stability for a given equation modelling the motion of an earth satellite. The proof combines bilateral bounds of the solution with the theory of twist solutions.
On liquid crystal flows with free-slip boundary conditions
Chun Liu and Jie Shen
2001, 7(2): 307-318 doi: 10.3934/dcds.2001.7.307 +[Abstract](180) +[PDF](147.9KB)
In this paper, we study a coupled dynamic system describing nematic liquid crystal flows. The system was motivated by the Ericksen-Leslie equations modeling the flow of nematic liquid crystals. The purpose of studying the simplified system is to understand the flow properties of more complicated materials, the material configurations , as well as the interactions between them. Unlike in the previous studies where the Dirichlet boundary conditions are prescribed, we consider here the free-slip boundary conditions which possess a number of distinct advantages. The results in this paper form the analytical background for the forthcoming numerical simulations of the system.
A note on stabilization with saturating feedback
Thomas I. Seidman and Houshi Li
2001, 7(2): 319-328 doi: 10.3934/dcds.2001.7.319 +[Abstract](67) +[PDF](155.4KB)
We assume, for a distributed parameter control system, that a linear stabilizing is available. We then seek a stabilizing, necessarily nonlinear, subject to an a priori bound on the control.
Positivity of Paneitz operators
Xingwang Xu and Paul C. Yang
2001, 7(2): 329-342 doi: 10.3934/dcds.2001.7.329 +[Abstract](120) +[PDF](151.0KB)
In this paper, we give two results concerning the positivity property of the Paneitz operator-- a fourth order conformally covariant elliptic operator. We prove that the Paneitz operator is positive for a compact Riemannian manifold without boundary of dimension at least six if it has positve scalar curvature as well as nonnegative $Q-$curvature. We also show that the positivity of the Paneitz operator is preserved in dimensions greater than four in taking a connected sum.
Maximum principles for the primitive equations of the atmosphere
Brian D. Ewald and Roger Témam
2001, 7(2): 343-362 doi: 10.3934/dcds.2001.7.343 +[Abstract](81) +[PDF](175.8KB)
In this article, maximum principles are derived for a suitably modified form of the equation of temperature for the primitive equations of the atmosphere; we consider both the limited domain case in Cartesian coordinates and the ow of the whole atmosphere in spherical coordinates.
Non-uniformly expanding dynamics: Stability from a probabilistic viewpoint
José F. Alves
2001, 7(2): 363-375 doi: 10.3934/dcds.2001.7.363 +[Abstract](68) +[PDF](165.2KB)
We present some recent developments in the theory of smooth dynamical systems exhibiting non-uniformly expanding behavior in the sense of [2]. In particular, we show that these systems have a finite number of SRB measures whose basins cover the whole manifold, and that under some uniform fast approach on the rates of expansion, their dynamics is statistical and stochastically stable.
Modeling chemical reactions in rivers: A three component reaction
Congming Li and Eric S. Wright
2001, 7(2): 377-384 doi: 10.3934/dcds.2001.7.373 +[Abstract](104) +[PDF](204.9KB)
What follows is the analysis of a model for dynamics of chemical reactions in a river. Dominant forces to be considered include diffusion, advection, and rates of creation or destruction of participating species (due to chemical reactions). In light of this, the model we will be using will be based upon a nonlinear system of reaction-advection-diffusion equations. The nonlinearity comes solely from the influences of the chemical reactions.
First, we will establish some general results for reaction diffusion systems. In particular, we will illustrate a class of reaction diffusion systems whose solutions are bounded from below by zero. We will also provide a local existence result for this class of problems. Afterwards, we will focus on the dynamics of an equidiffusive three component reaction system. Specifically, we will provide conditions under which one could be guaranteed the existence of global solutions. We will also discuss the qualities of the $\omega$-limit set for this system.
3D wave equations in sphere-symmetric domain with periodically oscillating boundaries
Masaru Yamaguchi
2001, 7(2): 385-396 doi: 10.3934/dcds.2001.7.385 +[Abstract](52) +[PDF](149.8KB)
IBVP for linear nonhomogeneous three dimensional wave equations is considered in sphere-symmetric domain with periodically moving boundaries. The unknown function and all data are assumed to be sphere-symmetric. The boundary data and the nonhomogeneous term are periodic. We shall define one dimensional dynamical system $A$, using the boundary functions. Then we shall show that under some Diophantine conditions on periods of the given data and the rotation number of $A$, every solution of IBVP is quasiperiodic in $t$.
Resynchronization of delayed neural networks
Ying Sue Huang
2001, 7(2): 397-401 doi: 10.3934/dcds.2001.7.397 +[Abstract](62) +[PDF](97.6KB)
We consider a network of identical neurons whose dynamics is governed by a system of differential delay equations. We study the stability of the slowly oscillatory synchronous periodic solution for such systems. We obtain sufficient conditions on the interconnection matrix under which the linearized system has a Floquet multiplier greater than 1, and thus the slowly oscillatory synchronous periodic solution is unstable.
The Lorenz equation as a metaphor for the Navier-Stokes equations
C. Foias, M. S Jolly, I. Kukavica and E. S. Titi
2001, 7(2): 403-429 doi: 10.3934/dcds.2001.7.403 +[Abstract](210) +[PDF](282.3KB)
Three approaches for the rigorous study of the 2D Navier-Stokes equations (NSE) are applied to the Lorenz system. Analysis of time averaged solutions leads to a description of invariant probability measures on the Lorenz attractor which is much more complete than what is known for the NSE. As is the case for the NSE, solutions on the Lorenz attractor are analytic in a strip about the real time axis. Rigorous estimates are combined with numerical computation of Taylor coefficients to estimate the width of this strip. Approximate inertial forms originally developed for the NSE are analyzed for the Lorenz system, and the dynamics for the latter are completely described.
Global structure of 2-D incompressible flows
Tian Ma and Shouhong Wang
2001, 7(2): 431-445 doi: 10.3934/dcds.2001.7.431 +[Abstract](92) +[PDF](187.9KB)
The main objective of this article is to classify the structure of divergence-free vector fields on general two-dimensional compact manifold with or without boundaries. First we prove a Limit Set Theorem, Theorem 2.1, a generalized version of the Poincaré-Bendixson to divergence-free vector fields on 2-manifolds of nonzero genus. Namely, the $\omega$ (or $\alpha$) limit set of a regular point of a regular divergence-free vector field is either a saddle point, or a closed orbit, or a closed domain with boundaries consisting of saddle connections. We call the closed domain ergodic set. Then the ergodic set is fully characterized in Theorem 4.1 and Theorem 5.1. Finally, we obtain a global structural classification theorem (Theorem 3.1), which amounts to saying that the phase structure of a regular divergence-free vector field consists of finite union of circle cells, circle bands, ergodic sets and saddle connections.

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