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Discrete & Continuous Dynamical Systems - A

2011 , Volume 31 , Issue 3

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Explicit formula for the solution of the Szegö equation on the real line and applications
Oana Pocovnicu
2011, 31(3): 607-649 doi: 10.3934/dcds.2011.31.607 +[Abstract](65) +[PDF](658.1KB)
Abstract:
We consider the cubic Szegö equation

$i\partial$$t$$u=$$\Pi$$(|u|^{2}u)$

in the Hardy space $L^2_+$$(\mathbb{R})$ on the upper half-plane, where $\Pi$ is the Szegö projector. It is a model for totally non-dispersive evolution equations and is completely integrable in the sense that it admits a Lax pair. We find an explicit formula for solutions of the Szegö equation. As an application, we prove soliton resolution in $H^s$ for all $s\geq 0$, for generic rational function data. As for non-generic data, we construct an example for which soliton resolution holds only in $H^s$, $0\leq s<1/2$, while the high Sobolev norms grow to infinity over time, i.e. $\lim_{t\to\pm\infty}\|u(t)\|_{H^s}=\infty,$ $s>1/2.$ As a second application, we construct explicit generalized action-angle coordinates by solving the inverse problem for the Hankel operator $H_u$ appearing in the Lax pair. In particular, we show that the trajectories of the Szegö equation with generic rational function data are spirals around Lagrangian toroidal cylinders $\mathbb{T}^N$$\times$$\mathbb{R}^N$.
Homeomorphisms of the annulus with a transitive lift II
Salvador Addas-Zanata and Fábio A. Tal
2011, 31(3): 651-668 doi: 10.3934/dcds.2011.31.651 +[Abstract](85) +[PDF](409.2KB)
Abstract:
Let $f$ be a homeomorphism of the closed annulus $A$ that preserves the orientation, the boundary components and that has a lift $\tilde f$ to the infinite strip $\tilde A$ which is transitive. We show that, if the rotation number of $\tilde f$ restricted to both boundary components of $A$ is strictly positive, then there exists a closed nonempty connected set $\Gamma\subset\tilde A$ such that $\Gamma\subset]-\infty,0]\times[0,1]$, $\Gamma$ is unbounded, the projection of $\Gamma$ to $A$ is dense, $\Gamma-(1,0)\subset\Gamma$ and $\tilde{f}(\Gamma)\subset \Gamma.$ Also, if $p_1$ is the projection on the first coordinate of $\tilde A$, then there exists $d>0$ such that, for any $\tilde z\in\Gamma,$ $$\limsup_{n\to\infty}\frac{p_1(\tilde f^n(\tilde z))-p_1(\tilde z)}{n}<-d.$$
Extensive escape rate in lattices of weakly coupled expanding maps
Jean-Baptiste Bardet and Bastien Fernandez
2011, 31(3): 669-684 doi: 10.3934/dcds.2011.31.669 +[Abstract](67) +[PDF](443.7KB)
Abstract:
In this paper, we study the escape rate of infinite lattices of weakly coupled maps with uniformly expanding repeller. In particular, it is proved that the escape rate of spatially periodic approximations is extensive and grows linearly with the period size. The proof relies on symbolic dynamics and is based on the control of cumulative effects of perturbations in cylinder sets with distinct spatial periods. A piecewise affine diffusive example is presented that exhibits monotonic decay of the escape rate with coupling intensity.
Minimal Følner foliations are amenable
Fernando Alcalde Cuesta and Ana Rechtman
2011, 31(3): 685-707 doi: 10.3934/dcds.2011.31.685 +[Abstract](105) +[PDF](415.4KB)
Abstract:
For finitely generated groups, amenability and Følner properties are equivalent. However, contrary to a widespread idea, Kaimanovich showed that Følner condition does not imply amenability for discrete measured equivalence relations. In this paper, we exhibit two examples of $C^\infty$ foliations of closed manifolds that are Følner and non amenable with respect to a finite transverse invariant measure and a transverse invariant volume, respectively. We also prove the equivalence between the two notions when the foliation is minimal, that is all the leaves are dense, giving a positive answer to a question of Kaimanovich. The equivalence is stated with respect to transverse invariant measures or some tangentially smooth measures. The latter include harmonic measures, and in this case the Følner condition has to be replaced by $\eta$-Følner (where the usual volume is modified by the modular form $\eta$ of the measure).
An example of rapid evolution of complex limit cycles
Nikolay Dimitrov
2011, 31(3): 709-735 doi: 10.3934/dcds.2011.31.709 +[Abstract](81) +[PDF](483.4KB)
Abstract:
In the current article we study complex cycles of higher multiplicity in a specific polynomial family of holomorphic foliations in the complex plane. The family in question is a perturbation of an exact polynomial one-form giving rise to a foliation by Riemann surfaces. In this setting, a complex cycle is defined as a nontrivial element of the fundamental group of a leaf from the foliation. In addition to that, we introduce the notion of a multi-fold cycle and show that in our example there exists a limit cycle of any multiplicity. Furthermore, such a cycle gives rise to a one-parameter family of cycles continuously depending on the perturbation parameter. As the parameter decreases in absolute value, the cycles from the continuous family escape from a very large subdomain of the complex plane.
Homoclinic standing waves in focusing DNLS equations
Michael Herrmann
2011, 31(3): 737-752 doi: 10.3934/dcds.2011.31.737 +[Abstract](105) +[PDF](5208.8KB)
Abstract:
We study focusing discrete nonlinear Schrödinger equations and present a novel variational existence proof for homoclinic standing waves (bright solitons). Our approach relies on the constrained maximization of an energy functional and provides the existence of two one-parameter families of waves with unimodal and even profile function for a wide class of nonlinearities. Finally, we illustrate our results by numerical simulations.
On piecewise affine interval maps with countably many laps
Jozef Bobok and Martin Soukenka
2011, 31(3): 753-762 doi: 10.3934/dcds.2011.31.753 +[Abstract](89) +[PDF](370.6KB)
Abstract:
We study a special conjugacy class $\mathcal F$ of continuous piecewise monotone interval maps: with countably many laps, which are locally eventually onto and have common topological entropy $\log9$. We show that $\mathcal F$ contains a piecewise affine map $f_{\lambda}$ with a constant slope $\lambda$ if and only if $\lambda\ge 9$. Our result specifies the known fact that for piecewise affine interval leo maps with countably many pieces of monotonicity and a constant slope $\pm\lambda$, the topological (measure-theoretical) entropy is not determined by $\lambda$. We also consider maps from the class $\mathcal F$ preserving the Lebesgue measure. We show that some of them have a knot point (a point $x$ where Dini's derivatives satisfy $D^{+}f(x)=D^{-}f(x)= \infty$ and $D_{+}f(x)=D_{-}f(x)= -\infty$) in its fixed point $1/2$.
On the birth of minimal sets for perturbed reversible vector fields
Jaume Llibre, Ricardo Miranda Martins and Marco Antonio Teixeira
2011, 31(3): 763-777 doi: 10.3934/dcds.2011.31.763 +[Abstract](69) +[PDF](384.2KB)
Abstract:
The results in this paper fit into a program to study the existence of periodic orbits, invariant cylinders and tori filled with periodic orbits in perturbed reversible systems. Here we focus on bifurcations of one-parameter families of periodic orbits for reversible vector fields in $\mathbb{R}^4$. The main used tools are normal forms theory, Lyapunov-Schmidt method and averaging theory.
Pullback attractors for globally modified Navier-Stokes equations with infinite delays
Pedro Marín-Rubio, Antonio M. Márquez-Durán and José Real
2011, 31(3): 779-796 doi: 10.3934/dcds.2011.31.779 +[Abstract](72) +[PDF](422.8KB)
Abstract:
We establish the existence of pullback attractors for the dynamical system associated to a globally modified model of the Navier-Stokes equations containing delay operators with infinite delay in a suitable weighted space. Actually, we are able to prove the existence of attractors in different classes of universes, one is the classical of fixed bounded sets, and the other is given by a tempered condition. Relationship between these two kind of objects is also analyzed.
Coherent lists and chaotic sets
Piotr Oprocha
2011, 31(3): 797-825 doi: 10.3934/dcds.2011.31.797 +[Abstract](82) +[PDF](567.6KB)
Abstract:
In this article we apply (recently extended by Kato and Akin) an elegant method of Iwanik (which adopts independence relations of Kuratowski and Mycielski) in the construction of various chaotic sets. We provide ''easy to track'' proofs of some known facts and establish new results as well. The main advantage of the presented approach is that it is easy to verify each step of the proof, when previously it was almost impossible to go into all the details of the construction (usually performed as an inductive procedure). Furthermore, we are able extend known results on chaotic sets in an elegant way. Scrambled, distributionally scrambled and chaotic sets with relation to various notions of mixing are considered.
Hamiltonian formalism for models of rotating shallow water in semigeostrophic scaling
Marcel Oliver and Sergiy Vasylkevych
2011, 31(3): 827-846 doi: 10.3934/dcds.2011.31.827 +[Abstract](77) +[PDF](459.4KB)
Abstract:
This paper presents a first rigorous study of the so-called large-scale semigeostrophic equations which were first introduced by R. Salmon in 1985 and later generalized by the first author. We show that these models are Hamiltonian on the group of $H^s$ diffeomorphisms for $s>2$. Notably, in the Hamiltonian setting an apparent topological restriction on the Coriolis parameter disappears. We then derive the corresponding Hamiltonian formulation in Eulerian variables via Poisson reduction and give a simple argument for the existence of $H^s$ solutions locally in time.
Frequency locking of modulated waves
Lutz Recke, Anatoly Samoilenko, Alexey Teplinsky, Viktor Tkachenko and Serhiy Yanchuk
2011, 31(3): 847-875 doi: 10.3934/dcds.2011.31.847 +[Abstract](98) +[PDF](690.0KB)
Abstract:
We consider the behavior of a modulated wave solution to an $\mathbb{S}^1$-equivariant autonomous system of differential equations under an external forcing of modulated wave type. The modulation frequency of the forcing is assumed to be close to the modulation frequency of the modulated wave solution, while the wave frequency of the forcing is supposed to be far from that of the modulated wave solution. We describe the domain in the three-dimensional control parameter space (of frequencies and amplitude of the forcing) where stable locking of the modulation frequencies of the forcing and the modulated wave solution occurs.
    Our system is a simplest case scenario for the behavior of self-pulsating lasers under the influence of external periodically modulated optical signals.
Typical points for one-parameter families of piecewise expanding maps of the interval
Daniel Schnellmann
2011, 31(3): 877-911 doi: 10.3934/dcds.2011.31.877 +[Abstract](82) +[PDF](600.1KB)
Abstract:
For one-parameter families of piecewise expanding maps of the interval we establish sufficient conditions such that a given point in the interval is typical for the absolutely continuous invariant measure for a full Lebesgue measure set of parameters. In particular, we consider $C^{1,1}(L)$-versions of $\beta$-transformations, piecewise expanding unimodal maps, and Markov structure preserving one-parameter families. For families of piecewise expanding unimodal maps we show that the turning point is almost surely typical whenever the family is transversal.
On the index problem of $C^1$-generic wild homoclinic classes in dimension three
Katsutoshi Shinohara
2011, 31(3): 913-940 doi: 10.3934/dcds.2011.31.913 +[Abstract](78) +[PDF](535.7KB)
Abstract:
We study the dynamics of homoclinic classes on three dimensional manifolds under the robust absence of dominated splittings. We prove that, $C^1$-generically, if such a homoclinic class contains a volume-expanding periodic point, then it contains a hyperbolic periodic point whose index (dimension of the unstable manifold) is equal to two.
Nonautonomous bifurcation of bounded solutions II: A Shovel-Bifurcation pattern
Christian Pötzsche
2011, 31(3): 941-973 doi: 10.3934/dcds.2011.31.941 +[Abstract](109) +[PDF](1085.3KB)
Abstract:
This paper continues our work on local bifurcations for nonautonomous difference and ordinary differential equations. Here, it is our premise that constant or periodic solutions are replaced by bounded entire solutions as bifurcating objects in order to encounter right-hand sides with an arbitrary time dependence.
    We introduce a bifurcation pattern caused by a dominant spectral interval (of the dichotomy spectrum) crossing the stability boundary. As a result, differing from the classical autonomous (or periodic) situation, the change of stability appears in two steps from uniformly asymptotically stable to asymptotically stable and finally to unstable. During the asymptotically stable regime, a whole family of bounded entire solutions occurs (a so-called "shovel"). Our basic tools are exponential trichotomies and a quantitative version of the surjective implicit function theorem yielding the existence of strongly center manifolds.
A Harnack inequality for fractional Laplace equations with lower order terms
Jinggang Tan and Jingang Xiong
2011, 31(3): 975-983 doi: 10.3934/dcds.2011.31.975 +[Abstract](118) +[PDF](321.0KB)
Abstract:
We establish a Harnack inequality of fractional Laplace equations without imposing sign condition on the coefficient of zero order term via the Moser's iteration and John-Nirenberg inequality.
Weighted Green functions of nondegenerate polynomial skew products on $\mathbb{C}^2$
Kohei Ueno
2011, 31(3): 985-996 doi: 10.3934/dcds.2011.31.985 +[Abstract](61) +[PDF](355.2KB)
Abstract:
We consider the dynamics of nondegenerate polynomial skew products on $\mathbb{C}^{2}$. The paper includes investigations of the existence of the Green and fiberwise Green functions of the maps, which induce generalized Green functions that are well-behaved on $\mathbb{C}^{2}$, and examples of the Green functions which are not defined on some curves in $\mathbb{C}^{2}$. Moreover, we consider the dynamics of the extensions of the maps to holomorphic or rational maps on weighted projective spaces.
Almost periodic solutions for a class of semilinear quantum harmonic oscillators
Jian Wu and Jiansheng Geng
2011, 31(3): 997-1015 doi: 10.3934/dcds.2011.31.997 +[Abstract](81) +[PDF](459.2KB)
Abstract:
In this paper, we show that there are many almost periodic solutions corresponding to full dimensional invariant tori for the semilinear quantum harmonic oscillators with Hermite multiplier $${\rm i}{u}_{t}-u_{xx}+x^2u + M_\xi u+\varepsilon |u|^{2m}u=0,\quad u\in C^1(\Bbb R,L^2(\Bbb R)),$$ where $m \geq 1$ is an integer. The proof is based on an abstract infinite dimensional KAM theorem.

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