# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

January 2000 , Volume 6 , Issue 1

Millennium issue
Recent Development on Differential Equations & Dynamical Systems

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2000, 6(1): 1-20 doi: 10.3934/dcds.2000.6.1 +[Abstract](1361) +[PDF](374.6KB)
Abstract:
We study properties of solitary-wave solutions of three evolution equations arising in the modeling of internal waves. Our experiments indicate that broad classes of initial data resolve into solitary waves, but also suggest that solitary waves do not interact exactly, thus suggesting two of these equations are not integrable. In the course of our numerical simulations, interesting meta-stable quasi-periodic structures have also come to light.
2000, 6(1): 21-38 doi: 10.3934/dcds.2000.6.21 +[Abstract](1203) +[PDF](218.3KB)
Abstract:
We consider the Cauchy problem for a system of $2n$ balance laws which arises from the modelling of multi-component chromatography:

$u_t + u_x =-\frac{1}{\varepsilon} (F(u)-v),$    (1)

$v_t = \frac{1}{\varepsilon} (F(u)-v),$

This model describes a liquid flowing with unit speed over a solid bed. Several chemical substances are partly dissolved in the liquid, partly deposited on the solid bed. Their concentrations are represented respectively by the vectors $u = (u_1, ... , u_n)$ and $v = (v_1, ... , v_n)$.
We show that, if the initial data have small total variation, then the solution of (1) remains with small variation for all times $t \geq 0$. Moreover, using the $mathbf L^1$ distance, this solution depends Lipschitz continuously on the initial data, with a Lipschitz constant uniform w.r.t. $\varepsilon$. Finally we prove that as $\varepsilon\to 0$, the solutions of (1) converge to a limit described by the system

$(u + F(u))_t + u_x = 0,$    $v= F(u)$.   (2)

The proof of the uniform BV estimates relies on the application of probabilistic techniques. It is shown that the components of the gradients $v_x$, $u_x$ can be interpreted as densities of random particles travelling with speed $0$ or $1$. The amount of coupling between different components is estimated in terms of the expected number of crossing of these random particles. This provides a first example where BV estimates are proved for general solutions to a class of $2n \times 2n$ systems with relaxation.

2000, 6(1): 39-50 doi: 10.3934/dcds.2000.6.39 +[Abstract](1057) +[PDF](235.2KB)
Abstract:
In this paper, we construct multipeak solutions with all the peaks close to a designated saddle point or strict local minimum point of the function $Q(y)$ for the nonlinear field equations. We also show that there is no multipeak solution with all its peaks near a strict local maximum point of $Q(y)$.
2000, 6(1): 51-60 doi: 10.3934/dcds.2000.6.51 +[Abstract](1038) +[PDF](234.3KB)
Abstract:
We investigate complex dynamics in infinite dimensions. Such systems can be described via quasiconjugacies with finite dimensional systems. Natural examples can be found within the field of quantum chaos. We show that the dynamics localizes.
2000, 6(1): 61-88 doi: 10.3934/dcds.2000.6.61 +[Abstract](1133) +[PDF](385.5KB)
Abstract:
We describe in detail a construction of weakly mixing $C^\infty$ diffeomorphisms preserving a smooth measure and a measurable Riemannian metric as well as ${\mathbb} Z^k$ actions with similar properties. We construct those as a perturbation of elements of a nontrivial non-transitive circle action. Our construction works on all compact manifolds admitting a nontrivial circle action.
It is shown in the appendix that a Riemannian metric preserved by a weakly mixing diffeomorphism can not be square integrable.
2000, 6(1): 89-120 doi: 10.3934/dcds.2000.6.89 +[Abstract](1397) +[PDF](473.1KB)
Abstract:
We introduce notions of expansiveness, conjugation, and specification for random bundle transformations and derive the uniqueness of equilibrium states for a large class of functions. We consider both invertible and noninvertible cases and discuss the results in the random subshifts case. As an example of such systems we introduce random sofic shifts which can be described both via random graphs and as factors of random subshifts of finite type. Based on the random graph description we discuss large deviation results for random sofic shifts.
2000, 6(1): 121-142 doi: 10.3934/dcds.2000.6.121 +[Abstract](1587) +[PDF](319.5KB)
Abstract:
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2000, 6(1): 143-145 doi: 10.3934/dcds.2000.6.143 +[Abstract](1465) +[PDF](121.3KB)
Abstract:
We survey some of the recent works relating the study of ideas from dynamics systems to the theory of hyperbolic conservation laws.
2000, 6(1): 147-154 doi: 10.3934/dcds.2000.6.147 +[Abstract](1285) +[PDF](227.9KB)
Abstract:
Let $M$ be the unit tangent bundle of a compact manifold with negative sectional curvatures and let $\hat M$ be a $\mathbb Z^d$ cover for $M$. Let $\mu$ be the measure of maximal entropy for the associated geodesic flow on $M$ and let $\hat\mu$ be the lift of $\mu$ to $\hat M$.
We show that the foliation $\hat{M^{s s}}$ is ergodic with respect to $\hat\mu$. (This was proved in the special case of surfaces by Babillot and Ledrappier by a different method.) Our method extends to certain Anosov and hyperbolic flows.
2000, 6(1): 155-164 doi: 10.3934/dcds.2000.6.155 +[Abstract](1601) +[PDF](266.4KB)
Abstract:
We study heteroclinic connections in a nonlinear heat equation that involves blow-up. More precisely we discuss the existence of $L^1$ connections among equilibrium solutions. By an $L^1$-connection from an equilibrium $\phi^{-1}$ to an equilibrium $\phi^+$ we mean a function $u$($.,t$) which is a classical solution on the interval $(-\infty,T)$ for some $T\in \mathbb R$ and blows up at $t=T$ but continues to exist in the space $L^1$ in a certain weak sense for $t\in [T,\infty)$ and satisfies $u$($.,t$)$\to \phi^\pm$ as $t\to\pm\infty$ in a suitable sense. The main tool in our analysis is the zero number argument; namely to count the number of intersections between the graph of a given solution and that of various specific solutions.
2000, 6(1): 165-174 doi: 10.3934/dcds.2000.6.165 +[Abstract](1384) +[PDF](229.9KB)
Abstract:
This paper is devoted to the computation of the index at infinity for some asymptotically linear completely continuous vector fields $x-T(x)$, when the principal linear part $x-Ax$ is degenerate ($1$ is an eigenvalue of $A$), and the sublinear part is not asymptotically homogeneous (in particular do not satisfy Landesman-Lazer conditions). In this work we consider only the case of a one-dimensional degeneration of the linear part, i.e.s $1$ is a simple eigenvalue of $A$. For this case we formulate an abstract theorem and give some general examples for vector fields of Hammerstein type and for a two point boundary value problem.
2000, 6(1): 175-190 doi: 10.3934/dcds.2000.6.175 +[Abstract](1362) +[PDF](270.9KB)
Abstract:
In this paper we investigate the role of cross-diffusion in the $3\times 3$ Lotka-Volterra competition model. Of particular interest is the existence of non-constant steady states created by cross-diffusion in $3\times 3$ systems. A comparison with $2\times 2$ systems is also included.
2000, 6(1): 191-210 doi: 10.3934/dcds.2000.6.191 +[Abstract](1676) +[PDF](302.6KB)
Abstract:
The aim of this article is to present a rather unusual and partly heuristic application of the renormalization group (RG) theory to the Navier-Stokes equations with space periodic boundary conditions. We obtain in this way a new nonlinear renormalized equation with a nonlinear term which is invariant under the Stokes operator. Its relation to the Navier-Stokes equations is investigated for non-resonant domains.
2000, 6(1): 211-220 doi: 10.3934/dcds.2000.6.211 +[Abstract](1221) +[PDF](240.6KB)
Abstract:
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2000, 6(1): 221-236 doi: 10.3934/dcds.2000.6.221 +[Abstract](1093) +[PDF](290.3KB)
Abstract:
A model of phenotype evolution incorporating mutation, selection, and recombination is investigated. The model consists of a partial differential equation for population density with respect to a continuous variable representing phenotype diversity. Mutation is modeled by diffusion, selection is modeled by differential phenotype fitness, and genetic recombination is modeled by an averaging process. It is proved that if the recombination process is suffciently weak, then there is a unique globally asymptotically stable attractor.
2000, 6(1): 237-241 doi: 10.3934/dcds.2000.6.237 +[Abstract](1156) +[PDF](167.9KB)
Abstract:
A point is called $C^r$ preperiodic if it can be made periodic via arbitrarily small $C^r$ perturbation. We discuss some general properties of the $C^r$ preperiodic set, and prove that the $C^1$ preperiodic set contains no obstruction points if and only if the system is Axiom A plus no-cycle.
2000, 6(1): 243-253 doi: 10.3934/dcds.2000.6.243 +[Abstract](1119) +[PDF](240.5KB)
Abstract:
In this paper, we prove certain persistence properties of the homoclinic points in Hamiltonian systems and symplectic diffeomorphisms. We show that, under some general conditions, stable and unstable manifolds of hyperbolic periodic points intersect in a very persistent way and we also give some simple criteria for positive topological entropy. The method used is the intersection theory of Lagrangian submanifolds of symplectic manifolds.

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