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Discrete & Continuous Dynamical Systems - A

2004 , Volume 11 , Issue 2/3

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External arguments and invariant measures for the quadratic family
Gamaliel Blé
2004, 11(2/3): 241-260 doi: 10.3934/dcds.2004.11.241 +[Abstract](110) +[PDF](298.9KB)
There is a correspondence between the boundary of the main hyperbolic component $W_0$ of the Mandelbrot set $M$ and $M \cap \mathbb R$ . It is induced by the map $T(\theta)=1/2+\theta/4$ defined on the set of external arguments of $W_0$.
If $c$ is a point of the boundary of $W_0$ with internal argument $\gamma$ and external argument $\theta$ then $T(\theta)$ is an external argument of the real parameter $c'\in M.$ We give a characterization, for the parameter $c'$ corresponding to $\gamma$ rational, in terms of the Hubbard trees. If $\gamma$ is irrational, we prove that $P_{c'}$ does not satisfy the $CE$ condition. We obtain an asymmetrical diophantine condition implying the existence of an absolutely continuous invariant measure (a.c.i.m.) for $P_{c'}$. We also show an arithmetic condition on $\gamma$ preventing the existence of an a.c.i.m.
Smale diffeomorphisms of surfaces: a classification algorithm
François Béguin
2004, 11(2/3): 261-310 doi: 10.3934/dcds.2004.11.261 +[Abstract](97) +[PDF](616.4KB)
We are concerned here with Smale (i.e. $C^1$-structurally stable) diffeomorphisms of compact surfaces. Bonatti and Langevin have produced some combinatorial descriptions of the dynamics of any such diffeomorphism ([2]). Actually, each diffeomorphism admits infinitely many different combinatorial descriptions. The aim of the present article is to describe an algorithm which decides whether two combinatorial descriptions correspond to the same diffeomorphism or not. This provides an algorithmic way to classify Smale diffeomorphisms of surfaces up to topological conjugacy (on canonical neighbourhoods of the basic pieces).
Controllability properties of a vibrating string with variable axial load
Alexander Khapalov
2004, 11(2/3): 311-324 doi: 10.3934/dcds.2004.11.311 +[Abstract](139) +[PDF](216.5KB)
We show that the set of equilibrium-like states $ (y_d, 0) $ of a vibrating string which can approximately be reached in the energy space $ H_0^1 (0,1) \times L^2 (0,1) $ from almost any non-zero initial datum by varying its axial load is dense in the subspace $ H_0^1 (0,1) \times $ {0} of this space. Our result is based on a constructive argument and makes use of piecewise constant-in-time control functions (loads) only, which enter the model equation as coefficients.
Coupled map lattices without cluster expansion
Gerhard Keller and Carlangelo Liverani
2004, 11(2/3): 325-335 doi: 10.3934/dcds.2004.11.325 +[Abstract](107) +[PDF](242.7KB)
We present an approach to the investigation of the statistical properties of weakly coupled map lattices that avoids completely cluster expansion techniques. Although here it is implemented on a simple case we expect similar strategies to be applicable in a much larger class of situations.
Some notes on periodic systems with linear part at resonance
D. Ruiz and J. R. Ward
2004, 11(2/3): 337-350 doi: 10.3934/dcds.2004.11.337 +[Abstract](95) +[PDF](248.7KB)
The existence of $T$-periodic solutions is obtained for second order systems of ordinary differential equations of the form

$u''(t) + g(u(t)) = p(t).$

Most of the results assume that $g\in C(\mathbb R^N, \mathbb R^N)$ is bounded or sublinear. The main theorem unifies previous results and implies several new ones.

Asymptotic regularity of solutions of singularly perturbed damped wave equations with supercritical nonlinearities
Sergey Zelik
2004, 11(2/3): 351-392 doi: 10.3934/dcds.2004.11.351 +[Abstract](134) +[PDF](353.9KB)
We study the asymptotic behavior of weak energy solutions of the following damped hyperbolic equation in a bounded domain $\Omega\subset\R^3$:

$\varepsilon\partial_t^2u+\gamma\partial_t u-\Delta_x u+f(u)=g,\quad u|_{\partial\Omega}=0,$

where $\gamma$ is a positive constant and $\varepsilon>0$ is a small parameter. We do not make any growth restrictions on the nonlinearity $f$ and, consequently, we do not have the uniqueness of weak solutions for this problem.
We prove that the trajectory dynamical system acting on the space of all properly defined weak energy solutions of this equation possesses a global attractor $\mathcal A_\varepsilon^{tr}$ and verify that this attractor consists of global strong regular solutions, if $\varepsilon>0$ is small enough. Moreover, we also establish that, generically, any weak energy solution converges exponentially to the attractor $\mathcal A_\varepsilon^{tr}$.

Well-posedness, blowup, and global existence for an integrable shallow water equation
Zhaoyang Yin
2004, 11(2/3): 393-411 doi: 10.3934/dcds.2004.11.393 +[Abstract](259) +[PDF](259.5KB)
We establish the local well-posedness for a recently derived model that combines the linear dispersion of Korteweg-de Veris equation with the nonlinear/nonlocal dispersion of the Camassa-Holm equation, and we prove that the equation has solutions that exist for indefinite times as well as solutions that blow up in finite time. We also derive an explosion criterion for the equation, and we give a sharp estimate of the existence time for solutions with smooth initial data.
On the double pendulum: An example of double resonant situations
Antonio Pumariño and Claudia Valls
2004, 11(2/3): 413-448 doi: 10.3934/dcds.2004.11.413 +[Abstract](84) +[PDF](343.5KB)
As a model of double resonant situations, we study fast periodic perturbations of a double pendulum. The associated dynamical system presents periodic orbits whose invariant manifolds split under the perturbation. The main purpose of this paper is to analytically show that this splitting is given, in first order, by the Melnikov function and give a lower bound for such splitting in terms of the perturbative parameter. Many results used in "simple pendulum cases" have to be adapted in order to give a description of the intricate dynamics exhibited by these periodic perturbations.
Recurrent dimensions of quasi-periodic solutions for nonlinear evolution equations II: Gaps of dimensions and Diophantine conditions
Koichiro Naito
2004, 11(2/3): 449-488 doi: 10.3934/dcds.2004.11.449 +[Abstract](122) +[PDF](314.5KB)
In our previous paper we introduced recurrent dimensions of discrete dynamical systems and we have estimated the upper and lower recurrent dimensions of discrete quasi-periodic orbits. In this paper, treating the case of 2-frequencies discrete quasi-periodic orbits, which correspond to the Poincaré sections of the 3-frequencies continuous quasi-periodic orbits, we estimate recurrent dimensions of the quasi-periodic orbits. Introducing some algebraic conditions between the two irrational frequencies, which are related to the Diophantine conditions of KAM theorem, we can estimate upper and lower recurrent dimensions of the orbits. We propose the gaps between the upper and the lower recurrent dimensions as the index parameters, which measure unpredictability levels of the orbits. Furthermore, we investigate these dimensions and their gaps for the quasi-periodic trajectories given by solutions of PDE with three periodic terms, the frequencies of which are rationally independent.
Polynomial growth of the derivative for diffeomorphisms on tori
Krzysztof Frączek
2004, 11(2/3): 489-516 doi: 10.3934/dcds.2004.11.489 +[Abstract](93) +[PDF](331.4KB)
We consider area--preserving zero entropy ergodic diffeomorphisms on tori. We classify such diffeomorphisms for which the sequence {$Df^n$} has a polynomial growth on the $3$-torus: they are necessary of the form

$\mathbb T^3\quad (x_1,x_2,x_3)\mapsto (x_1+\alpha,\varepsilon x_2+\beta(x_1),x_3+\gamma(x_1,x_2))\in\mathbb T^3,

where $\varepsilon =\pm 1$. We also indicate why there is no $4$-dimensional analogue of the above result. Random diffeomorphisms on the $2$-torus are studied as well.

Expanding interval maps with intermittent behaviour, physical measures and time scales
Xavier Bressaud
2004, 11(2/3): 517-546 doi: 10.3934/dcds.2004.11.517 +[Abstract](107) +[PDF](332.2KB)
We exhibit a new family of piecewise monotonic expanding interval maps with interesting intermittent-like statistical behaviours. Among these maps, there are uniformly expanding ones for which a Lebesgue-typical orbit spends most of the time close to an "indifferent Cantor set" which plays the role of the usual neutral fixed point. There are also examples with an indifferent fixed point and an infinite absolutely continuous invariant measure. Like in the classical case, the Dirac mass at $0$ describes the statistical behaviour at usual time scale while the infinite one tells about the statistical behaviour at larger scales. But, here, there is another invariant measure describing the statistical behaviour of the ergodic sums at a third natural (intermediate) time scale.
To try to understand this last phenomenon, we propose a more general construction that yields an example for which we conjecture there is an infinite number of natural time scales.
The construction of chaotic maps in the sense of Devaney on dendrites which commute to continuous maps on the unit interval
Tatsuya Arai and Naotsugu Chinen
2004, 11(2/3): 547-556 doi: 10.3934/dcds.2004.11.547 +[Abstract](87) +[PDF](261.6KB)
Let $f$ be a continuous map from the unit interval to itself. In this paper, it is shown that $f$ has positive topological entropy if and only if $f$ is pointwise $P$-expansive for some periodic orbit $P$ of $f$. And it is also proved that if $f$ has a periodic orbit with odd period, then there exists a chaotic map from a dendrite to itself in the sense of Devaney which is semiconjugate to $f$ and has positive topological entropy.
Dynamics of oscillations in a multi-dimensional delay differential system
Eugenii Shustin
2004, 11(2/3): 557-576 doi: 10.3934/dcds.2004.11.557 +[Abstract](106) +[PDF](275.5KB)
We consider a system of delay differential equations

$\dot x_i(t)=F_i(x_1(t),\ldots,x_n(t),t)-$ sign $x_i(t-h_i),\quad i=1,\ldots,n,$

with positive constant delays $h_1,...,h_n$ and perturbations $F_1,...,F_n$ absolutely bounded by a constant less than 1. This is a model of a negative feedback controller of relay type intended to bring the system to the origin. Non-zero delays do not allow such a stabilization, but cause oscillations around zero level in any variable. We introduce integral-valued relative frequencies of zeroes of the solution components, and show that they always decrease to some limit values. Moreover, for any prescribed limit relative frequencies, there exists at least an $n$-parametric family of solutions realizing these oscillation frequencies. We also find sufficient conditions for the stability of slow oscillations, and show that in this case there exist absolute frequencies of oscillations.

Topological entropy of a magnetic flow and the growth of the number of trajectories
César J. Niche
2004, 11(2/3): 577-580 doi: 10.3934/dcds.2004.11.577 +[Abstract](110) +[PDF](155.1KB)
We prove formulae relating the topological entropy of a magnetic flow to the growth rate of the average number of trajectories connecting two points.
Birkhoff cones, symbolic dynamics and spectrum of transfer operators
Frédéric Naud
2004, 11(2/3): 581-598 doi: 10.3934/dcds.2004.11.581 +[Abstract](160) +[PDF](283.0KB)
In this paper, we give an alternative proof of the classical Ruelle-Perron-Frobenius theorem in the general setup of subshifts of finite type using Birkhoff cones and Hilbert metrics. This approach yields an explicit estimate of the spectral gap of transfer operators and can be applied to compute estimates of the rate of decay of correlations and the analytic continuation of zeta functions.
Sharkovsky orderings of higher order difference equations
Uwe an der Heiden and Mann-Lin Liang
2004, 11(2/3): 599-614 doi: 10.3934/dcds.2004.11.599 +[Abstract](113) +[PDF](261.6KB)
Given an arbitrary continuous function $f:I\rightarrow I$, $I$ some interval, the well known Sharkovsky ordering

$3\vartriangleright 5\rhd \cdots \rhd 2\cdot 3\rhd 2\cdot 5 \triangleright \cdots \rhd 2^2\cdot 3\rhd 2^2\cdot 5\rhd \cdots \rhd \cdots \rhd 2^2 \rhd 2\rhd 1 $

tells that if the difference equation

$x_n=f(x_{n-1}),\quad n=1,2,\ldots $     (1)

has a periodic solution with period $p$ then it has also periodic solutions of period $p'$ for all $p'$ to the right of $p$ in the Sharkovsky ordering. Here we generalize this result to the difference equation of $k-$th order

$x_n=f(x_{n-k}),\quad n=1,2,\ldots $    (2)

for arbitrary $k\in \mathbb N$. It turns out that for each $k$ there is an individual ordering. In these orderings the prime number decomposition of $k$ plays an important role. In particular each number in the set

$S_k(p')=${ $l\cdot p'$ where $l$ divides $k$ and the pair $(k/l,p')$ is coprime}

is a period of (2) if $p\trianglerighteq p'$ and $p$ is a period of (1). Thus, for different values of $k$ there are generally different bifurcation schemes.
We also prove theorems about the number of periodic solutions and of attractive cycles of $x_n=f(x_{n-k})$.
We suggest that the $k-$th order difference equation (2) may give important insight to the behavior of delay--differential equations of the type $\varepsilon \dot x(t)+x(t)=f(x(t-1))$ by considering the parameter $\varepsilon \rightarrow 0$ in a singular perturbation problem.

A note on periodic orbits for singular-hyperbolic flows
Carlos Arnoldo Morales
2004, 11(2/3): 615-619 doi: 10.3934/dcds.2004.11.615 +[Abstract](127) +[PDF](173.3KB)
A singular-hyperbolic set for flows is a partially hyperbolic set with singularities (hyperbolic ones) and volume expanding central direction [7]. Several properties of hyperbolic systems have been conjectured for the singular-hyperbolic sets [8, p. 335]. Related to these conjectures we shall prove the existence of transitive, isolated, singular-hyperbolic set without periodic orbits on any $3$-manifold. In particular, the periodic orbits are not necessarily dense in the limit set of a isolated singular-hyperbolic set.
Symbolic dynamics from signed matrices
Jim Wiseman
2004, 11(2/3): 621-638 doi: 10.3934/dcds.2004.11.621 +[Abstract](94) +[PDF](321.1KB)
We consider a method for assigning a sofic shift to a (not necessarily nonnegative integer) matrix by associating to it a directed graph with some vertices labelled 1 and the rest 2 (the decomposition of the vertices is arbitrary - in applications the choice should be natural). We can detect positive topological entropy for this sofic shift by comparing the characteristic polynomial of the original matrix to those for the matrices for the restrictions of the shifts to each piece (1 and 2). Our main application is to the use of the Conley index to detect symbolic dynamics in isolated invariant sets, and is an extension of a result by Carbinatto, Kwapisz, and Mischaikow.
Entropies of a semigroup of maps
Andrzej Biś
2004, 11(2/3): 639-648 doi: 10.3934/dcds.2004.11.639 +[Abstract](168) +[PDF](174.3KB)
Four entropy-like invariants, introduced for a single map, were generalized to the case of semigroup. Relations between those entropies of semigroup are given.
On the profile of solutions for an elliptic problem arising in nonlinear optics
Daomin Cao, Ezzat S. Noussair and Shusen Yan
2004, 11(2/3): 649-666 doi: 10.3934/dcds.2004.11.649 +[Abstract](94) +[PDF](270.2KB)
We study the profile of solutions of

$-\Delta u + (\lambda - h(x)) u = g(x) (u^{p-1} + f(u))$ in $\ \mathbb R^N,$

$u > 0$ in $\mathbb R^N,$

$u \in H^1(\mathbb R^N),$

where $\lambda > 0$ is a parameter, $h$ and $g$ are nonnegative functions in $L^\infty(\mathbb R^N).$ We obtain the asymptotic behaviour of the least energy solutions or solutions obtained by the minimax principle. From the asymptotic behaviour we conclude that those solutions are asymmetric for $\lambda$ large even if $h$ and $g$ are radially symmetric.

Periodic solutions for three-dimensional non-monotone cyclic systems with time delays
Anatoli F. Ivanov and Bernhard Lani-Wayda
2004, 11(2/3): 667-692 doi: 10.3934/dcds.2004.11.667 +[Abstract](91) +[PDF](313.2KB)
We study a model for three cyclically coupled neurons with eventually negative delayed feedback, and without symmetry or monotonicity properties. Periodic solutions are obtained from the Schauder fixed point theorem. It turns out that, contrary to lower dimensional cases, instability at zero does not exclude monotonously decaying solutions.
Non-collision periodic solutions of forced dynamical systems with weak singularities
Pedro J. Torres
2004, 11(2/3): 693-698 doi: 10.3934/dcds.2004.11.693 +[Abstract](98) +[PDF](166.1KB)
We prove the existence of periodic solutions in a second order differential system with a singular potential of attractive or repulsive type and forced periodically. The proof is based on a Krasnoselskii fixed point theorem for absolutely continuous operators on a Banach space, and this makes possible to avoid any kind of "strong force" condition.
Null controllability of a cascade system of parabolic-hyperbolic equations
Enrique Fernández-Cara and Luz de Teresa
2004, 11(2/3): 699-714 doi: 10.3934/dcds.2004.11.699 +[Abstract](121) +[PDF](231.9KB)
This paper is concerned with the null controllability of a cascade linear system formed by a heat and a wave equation in a cylinder $\Omega \times (0,T)$. The control acts only on the heat equation and is supported by a set of the form $\omega \times (0,T)$, where $\omega \subset \Omega$. In the wave equation, only the restriction of the solution to the heat equation to another set $\mathcal O \times (0,T)$ appears. In the main result in this paper, we show that, under appropriate assumptions on $T$, $\omega$ and $\mathcal O$, the system is null controllable.
Attractors from one dimensional Lorenz-like maps
Youngna Choi
2004, 11(2/3): 715-730 doi: 10.3934/dcds.2004.11.715 +[Abstract](106) +[PDF](222.9KB)
In this paper we study the properties of expanding maps with a single discontinuity on a closed interval and the resultant dynamics. For such a map, there exists a compact invariant subset which shares a lot of common properties with classical attractors such as the topological transitivity of the restricted map and the density of the periodic points. The invariant set, with more conditions on the boundary, can be shown to have an isolating neighborhood, hence is a chaotic attractor in the strong sense. Not all such maps derive trapping regions, yet by perturbation, those non-attractors can be made to have a trapping region.
Existence problem for the Kirchhoff type wave equation with a localized weakly nonlinear dissipation in exterior domains
Jeong Ja Bae and Mitsuhiro Nakao
2004, 11(2/3): 731-743 doi: 10.3934/dcds.2004.11.731 +[Abstract](100) +[PDF](176.1KB)
We prove the existence of global solutions to the initial-boundary value problem for the Kirchhoff type quasilinear wave equations in exterior domains with a localized weakly nonlinear dissipation.

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