# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

May 2006 , Volume 15 , Issue 2

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2006, 15(2): 367-377 doi: 10.3934/dcds.2006.15.367 +[Abstract](2105) +[PDF](239.0KB)
Abstract:
In this paper we consider the problem of finding a star-shaped compact hypersurface with prescribed $k$-th mean curvature in hyperbolic space. Under some sufficient conditions, we obtain an existence result by establishing a priori estimates and using degree theory arguments.
2006, 15(2): 379-394 doi: 10.3934/dcds.2006.15.379 +[Abstract](1728) +[PDF](297.3KB)
Abstract:
We consider entire transcendental functions $f$ with an invariant (or periodic) Baker domain $U$. First, we classify these domains into three types (hyperbolic, simply parabolic and doubly parabolic) according to the surface they induce when we take the quotient by the dynamics. Second, we study the space of quasiconformal deformations of an entire map with such a Baker domain by studying its Teichmüller space. More precisely, we show that the dimension of this set is infinite if the Baker domain is hyperbolic or simply parabolic, and from this we deduce that the quasiconformal deformation space of $f$ is infinite dimensional. Finally, we prove that the function $f(z)=z+e^{-z}$, which possesses infinitely many invariant Baker domains, is rigid, i.e., any quasiconformal deformation of $f$ is affinely conjugate to $f$.
2006, 15(2): 395-414 doi: 10.3934/dcds.2006.15.395 +[Abstract](1581) +[PDF](321.3KB)
Abstract:
We give a complete topological classification of three dimensional dynamical systems with a "non-return" configuration of multiple saddle-connections along the skeleton of a normal crossings divisor. Our classifying space has a geometrical description, is finite and depends only on the distribution of the eigenvalues.
2006, 15(2): 415-432 doi: 10.3934/dcds.2006.15.415 +[Abstract](1599) +[PDF](458.7KB)
Abstract:
Let $X$ and $Y$ be two smooth vector fields on $\R^2$, globally asymptotically stable at the origin, and consider the time-dependent nonlinear system $\dot q(t)=u(t)X(q(t))+(1-u(t))Y(q(t))$, where $u:[0,\infty)\to\{0,1\}$ is an arbitrary measurable function. Analyzing the topology of the set where $X$ and $Y$ are parallel, we give some sufficient and some necessary conditions for global asymptotic stability, uniform with respect to $u(.)$. Such conditions can be verified without any integration or construction of a Lyapunov function, and they do not change under small perturbations of the vector fields.
2006, 15(2): 433-446 doi: 10.3934/dcds.2006.15.433 +[Abstract](1497) +[PDF](252.0KB)
Abstract:
In this paper we study hyperbolic sets with nonempty interior. We prove the folklore theorem that every transitive hyperbolic set with interior is Anosov. We also show that on a compact surface every locally maximal hyperbolic set with nonempty interior is Anosov. Finally, we give examples of hyperbolic sets with nonempty interior for a non-Anosov diffeomorphism.
2006, 15(2): 447-479 doi: 10.3934/dcds.2006.15.447 +[Abstract](2048) +[PDF](441.3KB)
Abstract:
Using the definition of solution and the qualitative properties established in the recent paper [17], some existence results are obtained both for crossing radial solutions and for positive or compactly supported radial ground states in $\mathbb R^n$ of quasilinear singular or degenerate elliptic equations with weights and with non--linearities which can be possibly singular at $x=0$ and $u=0$, respectively. The technique used is based on the papers [1] and [12]. Furthermore we obtain a non--existence theorem for radial ground states using a technique of Ni and Serrin [13].
2006, 15(2): 481-504 doi: 10.3934/dcds.2006.15.481 +[Abstract](1273) +[PDF](269.5KB)
Abstract:
We prove that Hénon-like strange attractors of diffeomorphisms in any dimensions, such as considered in [2], [7], and [9] support a unique Sinai-Ruelle-Bowen (SRB) measure and have the no-hole property: Lebesgue almost every point in the basin of attraction is generic for the SRB measure. This extends two-dimensional results of Benedicks-Young [4] and Benedicks-Viana [3], respectively.
2006, 15(2): 505-528 doi: 10.3934/dcds.2006.15.505 +[Abstract](1719) +[PDF](437.5KB)
Abstract:
Equilibrium distributions of multicomponent systems minimize the free energy functional under the constraint of mass conservation of the components. However, since the free energy is not convex in general, usually one tries to characterize and to construct equilibrium distributions as steady states of an adequate evolution equation, for example, the nonlocal Cahn-Hilliard equation for binary alloys. In this work a direct descent method for nonconvex functionals is established and applied to phase separation problems in multicomponent systems and image segmentation.
2006, 15(2): 529-558 doi: 10.3934/dcds.2006.15.529 +[Abstract](1875) +[PDF](472.9KB)
Abstract:
We describe a rigorous and efficient computer algorithm for building a model of the dynamics of a polynomial diffeomorphism of C2 on its chain recurrent set, $R$, and for sorting points into approximate chain transitive components. Further, we give explicit estimates which quantify how well this algorithm approximates the chain recurrent set and distinguishes the chain transitive components. We also discuss our implementation for the family of Hénon mappings, $f_{a,c}(x,y) = (x^2 + c - ay, x)$, into a computer program called Hypatia, and give several examples of running Hypatia on Hénon mappings.
2006, 15(2): 559-578 doi: 10.3934/dcds.2006.15.559 +[Abstract](1488) +[PDF](313.3KB)
Abstract:
We analyze the asymptotic behaviour of a 3D Lagrangian averaged Navier-Stokes $\alpha$-model (3D LANS$-\alpha$) with delays. In fact, we apply the theory of pullback attractors to ensure the existence of a pullback attractor, and at the same time, we also prove the existence of a uniform (forward) attractor in the sense of Chepyzhov and Vishik. Instead of working directly with the 3D LANS$-\alpha$ model, we establish a general theory for an abstract delay model and then we apply the general results to our particular situation.
2006, 15(2): 579-596 doi: 10.3934/dcds.2006.15.579 +[Abstract](2375) +[PDF](467.6KB)
Abstract:
We discuss the relationship between invariant manifolds of nonautonomous differential equations and pullback attractors. This relationship is essential, e.g., for the numerical approximation of these manifolds. In the first step, we show that the unstable manifold is the pullback attractor of the differential equation. The main result says that every (hyperbolic or nonhyperbolic) invariant manifold is the pullback attractor of a related system which we construct explicitly using spectral transformations. To illustrate our theorem, we present an application to the Lorenz system and approximate numerically the stable as well as the strong stable manifold of the origin.
2006, 15(2): 597-609 doi: 10.3934/dcds.2006.15.597 +[Abstract](1934) +[PDF](182.1KB)
Abstract:
In this paper, we consider the inverse generalized Riemann problem for quasilinear hyperbolic systems of conservation laws on the whole domain $t\geq 0$ and obtain that, under suitable conditions, the initial data on the positive (resp. negative) $x$-axis can be uniquely determined by the position of $n$ non-degenerate shocks and the initial data on the negative (resp. positive) $x$-axis.
2006, 15(2): 611-640 doi: 10.3934/dcds.2006.15.611 +[Abstract](1739) +[PDF](363.9KB)
Abstract:
Let $M$ be a compact manifold of dimension three with a non-degenerate volume form $\Omega$ and Diff$^r_\Omega(M)$ be the space of $C^r$-smooth ($\Omega$-) volume-preserving diffeomorphisms of $M$ with $2\le r< \infty$. In this paper we prove two results. One of them provides the existence of a Newhouse domain $\mathcal N$ in Diff$^r_\Omega(M)$. The proof is based on the theory of normal forms [13], construction of certain renormalization limits, and results from [23], [26], [28], [32]. To formulate the second one, associate to each diffeomorphism a sequence $P_n(f)$ which gives for each $n$ the number of isolated periodic points of $f$ of period $n$. The main result of this paper states that for a Baire generic diffeomorphism $f$ in $\mathcal N$, the number of periodic points $P_n(f)$ grows with $n$ faster than any prescribed sequence of numbers $\{a_n\}_{n \in \mathbb Z_+}$ along a subsequence, i.e., $P_{n_i}(f)>a_{n_i}$ for some $n_i\to \infty$ with $i\to \infty$. The strategy of the proof is similar to the one of the corresponding $2$-dimensional non volume-preserving result [16]. The latter one is, in its turn, based on the Gonchenko-Shilnikov-Turaev Theorem [8], [9].
2006, 15(2): 641-656 doi: 10.3934/dcds.2006.15.641 +[Abstract](2102) +[PDF](277.3KB)
Abstract:
In this paper we give a new proof of the local analytic linearization of flows on T2 with a Brjuno rotation number, using renormalization techniques.
2006, 15(2): 657-668 doi: 10.3934/dcds.2006.15.657 +[Abstract](1636) +[PDF](249.2KB)
Abstract:
We prove global well-posedness in Sobolev spaces with weighted low frequencies for a class of non local dispersive wave equations.
2006, 15(2): 669-680 doi: 10.3934/dcds.2006.15.669 +[Abstract](1659) +[PDF](253.5KB)
Abstract:
In this paper we establish the existence of positive and multiple solutions for the quasilinear elliptic problem

$-\Delta_p u = g(x,u)$  in  $\Omega$
$u = 0$  on  $\partial \Omega$,

where $\Omega \subset \mathbb{R}^N$ is an open bounded domain with smooth boundary $\partial \Omega$, $g:\Omega\times\mathbb{R}\to \mathbb{R}$ is a Carathéodory function such that $g(x,0)=0$ and which is asymptotically linear. We suppose that $g(x,t)/t$ tends to an $L^r$-function, $r>N/p$ if 1 < p ≤ N and $r=1$ if $p>N$, which can change sign. We consider both the resonant and the nonresonant cases.

Xiaojie Hou and
2006, 15(2): 681-701 doi: 10.3934/dcds.2006.15.681 +[Abstract](2200) +[PDF](307.9KB)
Abstract:
We investigate the local stability of traveling-wave solutions of the nonlinear reaction-diffusion equations in various weighted Banach spaces. New methods are used in analyzing the location of the spectrum. The result covers the stability results of the traveling-wave solutions of reaction-diffusion equations including the well known Fisher-KPP-type nonlinearity.

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