# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems

July 2005 , Volume 13 , Issue 4

Special Issue
Recent Development on Differential Equations and Dynamical Systems: Part I

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2005, 13(4): 843-876 doi: 10.3934/dcds.2005.13.843 +[Abstract](2841) +[PDF](944.3KB)
Abstract:
Wave propagation governed by reaction-diffusion equations in homogeneous media has been studied extensively, and initiation and propagation are well understood in scalar equations such as Fisher's equation and the bistable equation. However, in many biological applications the medium is inhomogeneous, and in one space dimension a typical model is a series of cells, within each of which the dynamics obey a reaction-diffusion equation, and which are coupled by reaction-free gap junctions. If the cell and gap sizes scale correctly such systems can be homogenized and the lowest order equation is the equation for a homogeneous medium [11]. However this usually cannot be done, as evidenced by the fact that such averaged equations cannot predict a finite range of propagation in an excitable system; once a wave is fully developed it propagates indefinitely. However, recent experimental results on calcium waves in numerous systems show that waves propagate though a fixed number of cells and then stop. In this paper we show how this can be understood within the framework of a very simple model for excitable systems.
2005, 13(4): 877-899 doi: 10.3934/dcds.2005.13.877 +[Abstract](2278) +[PDF](411.9KB)
Abstract:
We study travelling wave profiles for discrete approximations to hyperbolic systems of conservation laws. A detailed example is constructed, showing that for the Lax-Friedrichs scheme the travelling profiles do not depend continuously on the wave speed, in the BV norm. Namely, taking a sequence of wave speeds $\lambda_n\to\lambda$, the corresponding profiles $\Psi_n$ converge to a limit $\Psi$ uniformly on the real line, but Tot.Var.{$\Psi_n-\Psi$}$\geq c_0>0$ for all $n$.
2005, 13(4): 901-920 doi: 10.3934/dcds.2005.13.901 +[Abstract](2168) +[PDF](291.4KB)
Abstract:
We present a technique for the rigorous computation of periodic orbits in certain ordinary differential equations. The method combines set oriented numerical techniques for the computation of invariant sets in dynamical systems with topological index arguments. It not only allows for the proof of existence of periodic orbits but also for a precise (and rigorous) approximation of these. As an example we compute a periodic orbit for a differential equation introduced in [2].
2005, 13(4): 921-940 doi: 10.3934/dcds.2005.13.921 +[Abstract](2384) +[PDF](294.7KB)
Abstract:
The focus of the present study is the BBM equation which models unidirectional propagation of small amplitude long waves in shallow water and other dispersive media. Interest will be turned to the two-point boundary value problem wherein the wave motion is specified at both ends of a finite stretch of the medium of propagation. The principal new result is an exact theory of convergence of the two-point boundary value problem to the quarter-plane boundary value problem in which a semi-infinite stretch of the medium is disturbed at its finite end. The latter problem has been featured in modeling waves generated by a wavemaker in a flume and in describing the evolution of long crested, deep water waves propagating into the near shore zone of large bodies of water. In addition to their intrinsic interest, our results provide justification for the use of the two-point boundary value problem in numerical studies of the quarter plane problem.
2005, 13(4): 941-960 doi: 10.3934/dcds.2005.13.941 +[Abstract](2080) +[PDF](1403.7KB)
Abstract:
This paper proposes a new upscaling method in the simulations of solute transport in heterogeneous media. We provide a detailed convergence analysis of the method under the assumption that the oscillating coefficients are periodic. While such a simplifying assumption is not required by our method, it allows us to use homogenization theory to obtain the asymptotic structure of the solutions. Numerical experiments are carried out for solute transport in a porous medium with a random log-normal relative permeability to demonstrate the efficiency and accuracy of the proposed method.
2005, 13(4): 961-983 doi: 10.3934/dcds.2005.13.961 +[Abstract](2589) +[PDF](317.9KB)
Abstract:
We study existence of minimizers for problems of the type

inf{$\int_\Omega f(Du(x)) dx:u=u_{\xi _0}$ on $\partial\Omega$ }

where $f$ is non quasiconvex and $u_{\xi_0}$ is an affine function. Applying some new results on differential inclusions, we get sufficient conditions. We also study necessary conditions. We then consider some examples.

2005, 13(4): 985-1005 doi: 10.3934/dcds.2005.13.985 +[Abstract](2281) +[PDF](279.5KB)
Abstract:
We give a proof of cocycle rigidity in Hölder and smooth categories for Cartan actions on $SL(n, \mathbb R)$/$\Gamma$ and $SL(n, \mathbb C)$/$\Gamma$ for $n\ge 3$ and $\Gamma$ cocompact lattice, and for restrictions of those actions to subspaces which contain a two-dimensional plane in general position. This proof does not use harmonic analysis, it relies completely on the structure of stable and unstable foliations of the action. The key new ingredient is the use of the description of generating relations in the group $SL_n$.
2005, 13(4): 1007-1034 doi: 10.3934/dcds.2005.13.1007 +[Abstract](2597) +[PDF](388.1KB)
Abstract:
The robustness of asymptotic stability properties of ordinary differential equations with respect to small constant time delays is investigated. First, a local robustness result is established for compact asymptotically stable sets of systems with nonlinearities which need be only continuous, so the solutions may even be non-unique. The proof is based on the total stability of the differential inclusion obtained by inflating the original system. Using this first result, it is shown that an exponentially asymptotically stable equilibrium of a nonlinear equation which is Lipschitz in a neighborhood of the equilibrium remains exponentially asymptotically stable under small time delays. Then a global result regarding robustness of exponential dissipativity to small time delays is established with the help of a Lyapunov function for nonlinear systems which satisfy a global Lipschitz condition. The extension of these results to variable time delays is indicated. Finally, conditions ensuring the continuous convergence of the delay system attractors to the attractor of the system without delays are presented.
2005, 13(4): 1035-1046 doi: 10.3934/dcds.2005.13.1035 +[Abstract](2514) +[PDF](254.8KB)
Abstract:
In this paper we prove the existence of a new type of Sierpinski curve Julia set for certain families of rational maps of the complex plane. In these families, the complementary domains consist of open sets that are preimages of the basin at $\infty$ as well as preimages of other basins of attracting cycles.
2005, 13(4): 1047-1056 doi: 10.3934/dcds.2005.13.1047 +[Abstract](2322) +[PDF](210.3KB)
Abstract:
In this paper we investigate periodic orbits near a fixed point of a holomorphic twist map.
2005, 13(4): 1057-1067 doi: 10.3934/dcds.2005.13.1057 +[Abstract](2486) +[PDF](352.5KB)
Abstract:
This paper deals with the long-time behaviour of numerical solutions of neutral delay differential equations that have stable hyperbolic periodic orbits. It is shown that Runge--Kutta discretizations of such equations have attractive invariant closed curves which approximate the periodic orbit with the full order of the method, in spite of the lack of a finite-time smoothing property of the flow.
2005, 13(4): 1069-1096 doi: 10.3934/dcds.2005.13.1069 +[Abstract](2441) +[PDF](419.1KB)
Abstract:
Travelling fronts with conical-shaped level sets are constructed for reaction-diffusion equations with bistable nonlinearities of positive mass. The construction is valid in space dimension 2, where two proofs are given, and in arbitrary space dimensions under the assumption of cylindrical symmetry. General qualitative properties are presented under various assumptions: conical conditions at infinity, existence of a sub-level set with globally Lipschitz boundary, monotonicity in a given direction.

2020 Impact Factor: 1.392
5 Year Impact Factor: 1.610
2020 CiteScore: 2.2