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Discrete & Continuous Dynamical Systems - A

2002 , Volume 8 , Issue 1

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On random Schrödinger operators on $\mathbb Z^2$
Jean Bourgain
2002, 8(1): 1-15 doi: 10.3934/dcds.2002.8.1 +[Abstract](97) +[PDF](225.7KB)
This paper is concerned with random lattice operators on $\mathbb Z^2$ of the form $H_\omega = \Delta + V_\omega$ where $\Delta$ is the lattice Laplacian and $V_\omega$ a random potential $V_\omega(n) = \omega_nv_n, \{\omega_n\}$ independent Bernoulli or Gaussian variables and $\{v_n\}$ satisfying the condition sup$_n |v_n| |n|^\rho < \infty$ for some $\rho>\frac{1}{2}$. In this setting and restricting the spectrum away from the edges and 0, existence and completeness of the wave operators is shown. This leads to statements on the a.c. spectrum of $H_\omega$. It should be pointed out that, although we do consider here only a specific (and classical) model, the core of our analysis does apply in greater generality.
Global inversion via the Palais-Smale condition
Scott Nollet and Frederico Xavier
2002, 8(1): 17-28 doi: 10.3934/dcds.2002.8.17 +[Abstract](118) +[PDF](209.8KB)
Fixing a complete Riemannian metric g on $\mathbb R^n$, we show that a local diffeomorphism $f : \mathbb R^n\to \mathbb R^n$ is bijective if the height function $f\cdot v$ (standard inner product) satisfies the Palais-Smale condition relative to $g$ for each for each nonzero $v\in \mathbb R^n$. Our method substantially improves a global inverse function theorem of Hadamard. In the context of polynomial maps, we obtain new criteria for invertibility in terms of Lojasiewicz exponents and tameness of polynomials.
Inverse shadowing by continuous methods
S. Yu. Pilyugin
2002, 8(1): 29-38 doi: 10.3934/dcds.2002.8.29 +[Abstract](123) +[PDF](185.4KB)
We show that a structurally stable diffeomorphism has the inverse shadowingproperty with respect to classes of continuous methods. We also show thatany diffeomorphism belonging to the $C^1$-interior of the set of diffeomorphisms withthe above-mentioned property is structurally stable.
Guiding-like functions for periodic or bounded solutions of ordinary differential equations
Jean Mawhin and James R. Ward Jr
2002, 8(1): 39-54 doi: 10.3934/dcds.2002.8.39 +[Abstract](105) +[PDF](222.0KB)
We state and prove some extensions of the fundamental theorem of the method of guiding functions for periodic and for bounded solutions of ordinary differential systems. Those results unify and generalize previous results of Krasnosel'skii, Perov, Mawhin, Walter and Gossez.
On the dynamics of a class of quadratic polynomial automorphisms of $\mathbb C^3$
Dan Coman
2002, 8(1): 55-67 doi: 10.3934/dcds.2002.8.55 +[Abstract](68) +[PDF](244.0KB)
Fornæss and Wu classified the quadratic polynomial automorphisms of $\mathbb C^3$ into 7 classes. We describe the dynamics of the irregular maps in one of these classes.
Large time decay estimates of solutions of nonlinear parabolic equations
Huijiang Zhao
2002, 8(1): 69-114 doi: 10.3934/dcds.2002.8.69 +[Abstract](47) +[PDF](340.5KB)
We are concerned with the large time decay estimates of solutions to the Cauchy problem of nonlinear parabolic equations. Under the optimal growth conditions on the smooth nonlinear function $F(u,D_x u)$ as $(u,D_x u) \to (0,0)$, a global existence result is obtained and the influence of the nonlinear function $F(u,D_x u)$ on the large time behavior of the corresponding global smooth solution is also established.
A linearized instability result for solitary waves
Orlando Lopes
2002, 8(1): 115-119 doi: 10.3934/dcds.2002.8.115 +[Abstract](79) +[PDF](143.1KB)
In this paper we prove an abstract instability result for the linearized hamiltonian

$\dot u = JEu$

in a Hilbert space $X$ without assuming that $J$ is onto.

Deterministic and random aspects of porosities
Esa Järvenpää, Maarit Järvenpää and R. Daniel Mauldin
2002, 8(1): 121-136 doi: 10.3934/dcds.2002.8.121 +[Abstract](79) +[PDF](240.3KB)
We study porosities of limit sets of finite conformal iterated function systems and certain random fractals. We characterize systems with positive porosity and prove that porosity is continuous within a special class of one dimensional systems. We also show that for certain typical random recursive constructions related to fractal percolation both 0-porous and 1/2-porous points are dense, that is, porosity obtains its minimum and maximum values in a dense set.
Dynamically defined recurrence dimension
Petr Kůrka, Vincent Penné and Sandro Vaienti
2002, 8(1): 137-146 doi: 10.3934/dcds.2002.8.137 +[Abstract](62) +[PDF](203.3KB)
We modify the idea of a previous article [8] and introduce polynomial and exponential dynamically defined recurrence dimensions, topological invariants which express how the Poincaré recurrence time of a set grows when the diameter of the set shrinks. We introduce also the concept of polynomial entropy which applies in the case that topological entropy is zero and complexity function is polynomial. We compare recurrence dimensions with topological and polynomial entropies, evaluate recurrence dimensions of Sturmian subshifts and show some examples with Toeplitz subshifts.
Global bifurcation structure of stationary solutions for a Lotka-Volterra competition model
Yukio Kan-On
2002, 8(1): 147-162 doi: 10.3934/dcds.2002.8.147 +[Abstract](66) +[PDF](243.0KB)
In this paper, we establish the global bifurcation structure of positive stationary solutions for a certain Lotka-Volterra competition model with diffusion. To do this, the comparison principle and the bifurcation theory are employed.
Existence of semi-discrete shocks
Sylvie Benzoni-Gavage and Pierre Huot
2002, 8(1): 163-190 doi: 10.3934/dcds.2002.8.163 +[Abstract](67) +[PDF](294.2KB)
The approximation of shock waves by finite difference schemes is considered. This question has been investigated by many authors, but mainly under some restrictions on discrete wave speeds. Basic works are due to Majda and Ralston (rational speed) and, more recently, to Liu and Yu (Diophantine speed). The main purpose of the present work is to obtain shock profiles of arbitrary speed for rather general schemes. As a first step, we deal with semi-discretizations in space. For dissipative and non-resonant schemes, using the terminology of Majda and Ralston, we show the existence of shock profiles of small strength. For this we prove a center manifold theorem for a functional differential equation of mixed type (with both delay and advance). An additional invariance principle enables us to find semi-discrete shocks as heteroclinic orbits on the center manifold exhibited. This result generalizes a previous one of the first author, dealing with the special "upwind" scheme. In particular, it holds for the Godunov scheme and for a semi-discrete version of the Lax-Friedrichs scheme also known as the Rusanov scheme.
Topological entropy of shift function on the sequences space induced by expanding piecewise linear transformations
Prof. Dr.rer.nat Widodo
2002, 8(1): 191-208 doi: 10.3934/dcds.2002.8.191 +[Abstract](80) +[PDF](230.3KB)
Let $I = [0, 1]$. The topological entropy of shift function on the sequences space induced by a piecewise linear transformation from $I$ into itself is studied. The main goal of the paper is to investigate the relation between the topological entropy of piecewise linear transformations which in general are not continuous, and the topological entropy of shift function which the transformation induces on a space of symbol sequences. The main result is that for a class of piecewise linear (possibly discontinuous) self-maps of $I$, the topological entropy coincides with the topological entropy of shift function which the map induces on a space of symbol sequences.
Stable subharmonic solutions of reaction-diffusion equations on an arbitrary domain
Peter Poláčik and Eiji Yanagida
2002, 8(1): 209-218 doi: 10.3934/dcds.2002.8.209 +[Abstract](60) +[PDF](181.2KB)
The paper is concerned with stable subharmonic solutions of timeperiodic spatially inhomogeneous reaction-diffusion equations. We show that such solutions exist on any spatial domain, provided the nonlinearity is chosen suitably. This contrasts with our previous results on spatially homogeneous equations that admit stable subharmonic solutions on some, but not on arbitrary domains.
The existence of the heat flow of H-systems
Y. Chen and S. Levine
2002, 8(1): 219-236 doi: 10.3934/dcds.2002.8.219 +[Abstract](88) +[PDF](232.2KB)
In this paper, we show the existence of a unique, regular solution to the flow of the H-system with Dirichlet boundary condition. The solution exists at least up until the time of energy concentration. If this solution satisfies a certain energy inequality, then it can be continued to a global solution with the exception of at most finitely many singularities. The behavior of the singularities also are discussed.
On the reduction of the modified Benjamin-Ono equation to the cubic derivative nonlinear Schrödinger equation
Nakao Hayashi and Pavel Naumkin
2002, 8(1): 237-255 doi: 10.3934/dcds.2002.8.237 +[Abstract](110) +[PDF](255.5KB)
We study the asymptotic behavior for large time of small solutions to theCauchy problem for the modified Benjamin-Ono equation: $u_t + (u^3)_x + \mathcal H u_{x x} = 0$,where $\mathcal H$ is the Hilbert transformation, $x, t\in \mathbf R$. We investigate the reduction ofthe modified Benjamin-Ono equation to the cubic derivative nonlinear Shrödingerequation and then apply techniques developed in [11] - [14] to the resulting cubicnonlocal nonlinear Schrödinger equation. Our method is simpler than that usedin [10] because we can use the factorization of the free Schrödinger group. Ourpurpose in this paper is to show that solutions have the same $L^infty$ time decay rateas in the corresponding linear Benjamin-Ono equation and to prove the existence ofmodified scattering states, when the initial data are sufficiently small in the weightedSobolev spaces $\mathbf H^{2,0} \cap \mathbf H^{1,1}$, where $\mathbf H^{m, s} = \{ \phi\in S' : ||\phi||_{m,s} =||(1 + x^2)^{s/2}(1-\partial^2_x)^{m/2}\phi||_{\mathbf L^2}<\infty \}, m, s\in \mathbf R$. This is an improvement of the previous result [10],where we considered small initial data from the space $\mathbf H^{3,0}\cap \mathbf H^{1,2}$. Our method isbased on a certain gauge transformation and an appropriate phase function.
A uniform $C^1$ connecting lemma
Lan Wen
2002, 8(1): 257-265 doi: 10.3934/dcds.2002.8.257 +[Abstract](80) +[PDF](181.7KB)
We prove there are uniform bounds for quantities that guarantee C1 connecting of orbits. Here the uniformity is with respect to all systems in a $C^1$ neighborhood of the given system, and with respect to certain set of accumulation points.
Deformation from symmetry and multiplicity of solutions in non-homogeneous problems
Christine Chambers and Nassif Ghoussoub
2002, 8(1): 267-281 doi: 10.3934/dcds.2002.8.267 +[Abstract](81) +[PDF](233.8KB)
A general theorem on the multiplicity of critical points for non-invariant deformations of symmetric functionals is established, using a method introduced by Bolle [5]. This result is used to find conditions sufficient for the existence of multiple solutions of semi-linear elliptic partial differential equations of the form

$-\Delta u = p(x, u) + f(\theta, x, u)\quad $ on $\Omega$

$u = 0\quad$ on $\partial \Omega$

where $p(x, \cdot)$ is odd and $f$ is a perturbative term. An application of this result is the problem

$-\Delta u = \lambda |u|^{q-1}u + |u|^{p-1}u + f\quad$ on $\Omega$

$u = u_0\quad$ on $\partial \Omega$

where $\Omega$ is a smooth, bounded, open subset of $\mathbf R^n (n \geq 3), \lambda > 0, 1\leq q < p, f \in C(\bar \Omega, \mathbf R)$ and $u_0\in C^2(\partial \Omega, \mathbf R)$. It is proven that this equation has an infinite number of solutions for $p < \frac{n+1}{n-1}$ and that for any sub-critical $p$ i.e., $p < \frac{n+2}{n-2}$, there are as many solutions as we like, provided $||f||_{frac{p+1}{p}}$ and $||u_0||_{p+1}$ are small enough.

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