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Discrete & Continuous Dynamical Systems - A

2010 , Volume 27 , Issue 3

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Decoration invariants for horseshoe braids
André de Carvalho and Toby Hall
2010, 27(3): 863-906 doi: 10.3934/dcds.2010.27.863 +[Abstract](85) +[PDF](492.9KB)
The Decoration Conjecture describes the structure of the set of braid types of Smale's horseshoe map ordered by forcing, providing information about the order in which periodic orbits can appear when a horseshoe is created. A proof of this conjecture is given for the class of so-called lone decorations, and it is explained how to calculate associated braid conjugacy invariants which provide additional information about forcing for horseshoe braids.
A nonlinear partial integro-differential equation from mathematical finance
Frederic Abergel and Remi Tachet
2010, 27(3): 907-917 doi: 10.3934/dcds.2010.27.907 +[Abstract](91) +[PDF](160.4KB)
Consistently fitting vanilla option surfaces is an important issue when it comes to modeling in finance. As far as local and stochastic volatility models are concerned, this problem boils down to the resolution of a nonlinear integro-differential pde. The non-locality of this equation stems from the quotient of two integral terms and is not defined for all bounded continuous functions. In this paper, we use a fixed point argument and suitable a priori estimates to prove short-time existence of solutions for this equation.
Boundary stabilization of the wave and Schrödinger equations in exterior domains
Lassaad Aloui and Moez Khenissi
2010, 27(3): 919-934 doi: 10.3934/dcds.2010.27.919 +[Abstract](103) +[PDF](226.6KB)
In this paper we complete our works on the local energy decay for the evolution damping problem in exterior domains. We consider the wave and Schrödinger equations in an exterior domain with dissipative boundary condition. We study the distribution of resonances under some natural assumptions on the behavior of the geodesics in order to deduce the uniform local energy decay.
Baire category and extremely non-normal points of invariant sets of IFS's
In-Soo Baek and Lars Olsen
2010, 27(3): 935-943 doi: 10.3934/dcds.2010.27.935 +[Abstract](76) +[PDF](200.5KB)
We prove that if $K$ is the invariant set of an IFS in $\ R^{d}$ satisfying the Strong Open Set Condition, then the set of extremely non-normal points of $K$ is a comeagre subset of $K$.
Discrete orbits in topologically transitive cylindrical transformations
Jan Kwiatkowski and Artur Siemaszko
2010, 27(3): 945-961 doi: 10.3934/dcds.2010.27.945 +[Abstract](113) +[PDF](211.2KB)
In this paper we provide a few recipes how to construct a topologically transitive cocycle over an arbitrary odometer possessing discrete orbits. It is shown that for every odometer, there exists a topologically transitive cocycle such that the set of points with discrete orbits starting form zero level has the cardinality of the continuum.
Cyclicity of unbounded semi-hyperbolic 2-saddle cycles in polynomial Lienard systems
Magdalena Caubergh, Freddy Dumortier and Stijn Luca
2010, 27(3): 963-980 doi: 10.3934/dcds.2010.27.963 +[Abstract](168) +[PDF](850.1KB)
The paper deals with the cyclicity of unbounded semi-hyperbolic 2-saddle cycles in polynomial Liénard systems of type $(m,n)$ with $m<2n+1$, $m$ and $n$ odd. We generalize the results in [1] (case $m=1$), providing a substantially simpler and more transparant proof than the one used in [1].
Optimal interior partial regularity for nonlinear elliptic systems
Shuhong Chen and Zhong Tan
2010, 27(3): 981-993 doi: 10.3934/dcds.2010.27.981 +[Abstract](112) +[PDF](190.1KB)
We consider interior regularity for weak solutions of nonlinear elliptic systems with subquadratic under controllable growth condition. By $\mathcal{A}$-harmonic approximation technique, we obtain a general criterion for a weak solution to be regular in the neighborhood of a given point. In particularly, the regular result is optimal.
A note on two approaches to the thermodynamic formalism
Vaughn Climenhaga
2010, 27(3): 995-1005 doi: 10.3934/dcds.2010.27.995 +[Abstract](126) +[PDF](172.9KB)
Inducing schemes provide a means of using symbolic dynamics to study equilibrium states of non-uniformly hyperbolic maps, but necessitate a solution to the liftability problem. One approach, due to Pesin and Senti, places conditions on the induced potential under which a unique equilibrium state exists among liftable measures, and then solves the liftability problem separately. Another approach, due to Bruin and Todd, places conditions on the original potential under which both problems may be solved simultaneously. These conditions include a bounded range condition, first introduced by Hofbauer and Keller. We compare these two sets of conditions and show that for many inducing schemes of interest, the conditions from the second approach are strictly stronger than the conditions from the first. We also show that the bounded range condition can be used to obtain Pesin and Senti's conditions for any inducing scheme with sufficiently slow growth of basic elements.
Microdynamics for Nash maps
William Geller, Bruce Kitchens and Michał Misiurewicz
2010, 27(3): 1007-1024 doi: 10.3934/dcds.2010.27.1007 +[Abstract](84) +[PDF](284.2KB)
We investigate a family of maps that arises from a model in economics and game theory. It has some features similar to renormalization and some similar to intermittency. In a one-parameter family of maps in dimension 2, when the parameter goes to 0, the maps converge to the identity. Nevertheless, after a linear rescaling of both space and time, we get maps with attracting invariant closed curves. As the parameter goes to 0, those curves converge in a strong sense to a certain circle. We call those phenomena microdynamics. The model can be also understood as a family of discrete time approximations to a Brown-von Neumann differential equation.
Well-posedness and blow-up phenomena for the interacting system of the Camassa-Holm and Degasperis-Procesi equations
Ying Fu, Changzheng Qu and Yichen Ma
2010, 27(3): 1025-1035 doi: 10.3934/dcds.2010.27.1025 +[Abstract](90) +[PDF](170.5KB)
In this paper, the well-posedness and blow up phenomena for the interacting system of the Camassa-Holm and Degasperis-Procesi equations are studied. We first establish the local well-posedness of strong solutions for the system. Then the precise blow-up scenarios for the strong solutions to the system are derived.
On very weak solutions of semi-linear elliptic equations in the framework of weighted spaces with respect to the distance to the boundary
Jesus Idelfonso Díaz and Jean Michel Rakotoson
2010, 27(3): 1037-1058 doi: 10.3934/dcds.2010.27.1037 +[Abstract](98) +[PDF](285.0KB)
We prove the existence of an appropriate function (very weak solution) $u$ in the Lorentz space $L^{N',\infty}(\Omega), \ N'=\frac N{N-1}$ satisfying $Lu-Vu+g(x,u,\nabla u)=\mu$ in $\Omega$ an open bounded set of $\R^N$, and $u=0$ on $\partial\Omega$ in the sense that

$(u,L\varphi)_0-(Vu,\varphi)_0+(g(\cdot,u,\nabla u),\varphi)_0=\mu(\varphi),\quad\forall\varphi\in C^2_c(\Omega).$

The potential $V \le \lambda < \lambda_1$ is assumed to be in the weighted Lorentz space $L^{N,1}(\Omega,\delta)$, where $\delta(x)= dist(x,\partial\Omega),\ \mu\in M^1(\Omega,\delta)$, the set of weighted Radon measures containing $L^1(\Omega,\delta)$, $L$ is an elliptic linear self adjoint second order operator, and $\lambda_1$ is the first eigenvalue of $L$ with zero Dirichlet boundary conditions.
    If $\mu\in L^1(\Omega,\delta)$ we only assume that for the potential $V$ is in L1loc$(\Omega)$, $V \le \lambda<\lambda_1$. If $\mu\in M^1(\Omega,\delta^\alpha),\ \alpha\in$[$0,1[$[, then we prove that the very weak solution $|\nabla u|$ is in the Lorentz space $L^{\frac N{N-1+\alpha},\infty}(\Omega)$. We apply those results to the existence of the so called large solutions with a right hand side data in $L^1(\Omega,\delta)$. Finally, we prove some rearrangement comparison results.

Countable inverse limits of postcritical $w$-limit sets of unimodal maps
Chris Good, Robin Knight and Brian Raines
2010, 27(3): 1059-1078 doi: 10.3934/dcds.2010.27.1059 +[Abstract](100) +[PDF](263.7KB)
Let $f$ be a unimodal map of the interval with critical point $c$. If the orbit of $c$ is not dense then most points in lim{[0, 1], f} have neighborhoods that are homeomorphic with the product of a Cantor set and an open arc. The points without this property are called inhomogeneities, and the set, I, of inhomogeneities is equal to lim {w(c), f|w(c)}. In this paper we consider the relationship between the limit complexity of $w(c)$ and the limit complexity of I. We show that if $w(c)$ is more complicated than a finite collection of convergent sequences then I can have arbitrarily high limit complexity. We give a complete description of the limit complexity of I for any possible $\w(c)$.
Non topologically weakly mixing interval exchanges
Hadda Hmili
2010, 27(3): 1079-1091 doi: 10.3934/dcds.2010.27.1079 +[Abstract](96) +[PDF](200.1KB)
In this paper, we prove a criterion for the existence of continuous non constant eigenfunctions for interval exchange transformations which are non topologically weakly mixing. We first construct, for any $m>3$, uniquely ergodic interval exchange transformations of Q-rank $2$ with irrational eigenvalues associated to continuous eigenfunctions which are not topologically weakly mixing; this answers a question of Ferenczi and Zamboni [5]. Moreover we construct, for any even integer $m \geq 4$, interval exchange transformations of Q-rank $2$ with both irrational eigenvalues (associated to continuous eigenfunctions) and non trivial rational eigenvalues (associated to piecewise continuous eigenfunctions); these examples can be chosen to be either uniquely ergodic or non minimal.
Well-posedness and existence of standing waves for the fourth order nonlinear Schrödinger type equation
Jun-ichi Segata
2010, 27(3): 1093-1105 doi: 10.3934/dcds.2010.27.1093 +[Abstract](85) +[PDF](225.1KB)
We consider the fourth order nonlinear Schrödinger type equation (4NLS). The first purpose is to revisit the well-posedness theory of (4NLS). In [8], [9], [20] and [21], they proved the time-local well-posedness of (4NLS) in H *(R) with $s>1/2$ by using the Fourier restriction method. In this paper we give another proof of above result by using simpler approach than the Fourier restriction method. The second purpose is to construct the exact standing wave solution to (4NLS).
Quasi-invariant measures, escape rates and the effect of the hole
Wael Bahsoun and Christopher Bose
2010, 27(3): 1107-1121 doi: 10.3934/dcds.2010.27.1107 +[Abstract](91) +[PDF](259.1KB)
Let $T$ be a piecewise expanding interval map and $T_H$ be an abstract perturbation of $T$ into an interval map with a hole. Given a number , 0 < < l, we compute an upper-bound on the size of a hole needed for the existence of an absolutely continuous conditionally invariant measure (accim) with escape rate not greater than -ln(1-). The two main ingredients of our approach are Ulam's method and an abstract perturbation result of Keller and Liverani.
Classes of singular $pq-$Laplacian semipositone systems
Eun Kyoung Lee, R. Shivaji and Jinglong Ye
2010, 27(3): 1123-1132 doi: 10.3934/dcds.2010.27.1123 +[Abstract](121) +[PDF](152.0KB)
We consider the positive solutions to classes of $pq-$Laplacian semipositone systems with Dirichlet boundary conditions, in particular, we study strongly coupled reaction terms which tend to $-\infty$ at the origin and satisfy a combined sublinear condition at $\infty.$ By using the method of sub-super solutions we establish our results.
Hyperbolicity of $C^1$-stably expansive homoclinic classes
Keonhee Lee and Manseob Lee
2010, 27(3): 1133-1145 doi: 10.3934/dcds.2010.27.1133 +[Abstract](90) +[PDF](190.5KB)
Let $f$ be a diffeomorphism of a compact $C^\infty$ manifold, and let $p$ be a hyperbolic periodic point of $f$. In this paper we introduce the notion of $C^1$-stable expansivity for a closed $f$-invariant set, and prove that $(i)$ the chain recurrent set $\mathcal {R}(f)$ of $f$ is $C^1$-stably expansive if and only if $f$ satisfies both Axiom A and no-cycle condition, $(ii)$ the homoclinic class $H_f(p)$ of $f$ associated to $p$ is $C^1$-stably expansive if and only if $H_f(p)$ is hyperbolic, and $(iii)$ $C^1$-generically, the homoclinic class $H_f(p)$ is $C^1$-stably expansive if and only if $H_f(p)$ is $C^1$-persistently expansive.
Rotating modes in the Frenkel-Kontorova model with periodic interaction potential
Wen-Xin Qin
2010, 27(3): 1147-1158 doi: 10.3934/dcds.2010.27.1147 +[Abstract](79) +[PDF](269.8KB)
Employing a homotopy argument and the Leray-Schauder degree theory, we show the existence of rotating modes for the Frenkel-Kontorova model with periodic interaction potential. The solutions describing rotating modes are periodic and called rotating oscillating solutions, in which the phase of a fixed rotator increases by $2\pi$ per period, while its neighbors oscillate with small amplitudes around their equilibrium positions. We also discuss a fundamental difference between the Frenkel-Kontorova model with periodic interaction potential and that with convex interaction potential by demonstrating the nonexistence of the rotating modes for the latter case.
Boundary feedback stabilization of the two dimensional Navier-Stokes equations with finite dimensional controllers
Jean-Pierre Raymond and Laetitia Thevenet
2010, 27(3): 1159-1187 doi: 10.3934/dcds.2010.27.1159 +[Abstract](123) +[PDF](348.7KB)
We study the boundary stabilization of the two-dimensional Navier-Stokes equations about an unstable stationary solution by controls of finite dimension in feedback form. The main novelty is that the linear feedback control law is determined by solving an optimal control problem of finite dimension. More precisely, we show that, to stabilize locally the Navier-Stokes equations, it is sufficient to look for a boundary feedback control of finite dimension, able to stabilize the projection of the linearized equation onto the unstable subspace of the linearized Navier-Stokes operator. The feedback operator is obtained by solving an algebraic Riccati equation in a space of finite dimension, that is to say a matrix Riccati equation.
Quasistatic evolution for plasticity with softening: The spatially homogeneous case
Francesco Solombrino
2010, 27(3): 1189-1217 doi: 10.3934/dcds.2010.27.1189 +[Abstract](80) +[PDF](355.1KB)
The spatially uniform case of the problem of quasistatic evolution in small strain associative elastoplasticity with softening is studied. Through the introdution of a viscous approximation, the problem reduces to determine the limit behaviour of the solutions of a singularly perturbed system of ODE's in a finite dimensional Banach space. We see that the limit dynamics presents, for a generic choice of the initial data, the alternation of three possible regimes (elastic regime, slow dynamics, fast dynamics), which is determined by the sign of two scalar indicators, whose explicit expression is given.
Measures of intermediate entropies for skew product diffeomorphisms
Peng Sun
2010, 27(3): 1219-1231 doi: 10.3934/dcds.2010.27.1219 +[Abstract](138) +[PDF](180.3KB)
In this paper we study a skew product map $F$ preserving an ergodic measure $\mu$ of positive entropy. We show that if on the fibers the map are $C^{1+\alpha}$ diffeomorphisms with nonzero Lyapunov exponents, then $F$ has ergodic measures of arbitrary intermediate entropies. To construct these measures we find a set on which the return map is a skew product with horseshoes along fibers. We can control the average return time and show the maximal entropy of these measures can be arbitrarily close to $h_\mu(F)$.
On the spatial asymptotics of solutions of the Toda lattice
Gerald Teschl
2010, 27(3): 1233-1239 doi: 10.3934/dcds.2010.27.1233 +[Abstract](91) +[PDF](139.6KB)
We investigate the spatial asymptotics of decaying solutions of the Toda hierarchy and show that the asymptotic behaviour is preserved by the time evolution. In particular, we show that the leading asymptotic term is time independent. Moreover, we establish infinite propagation speed for the Toda lattice.
Homoclinic orbits for superlinear Hamiltonian systems without Ambrosetti-Rabinowitz growth condition
Jun Wang, Junxiang Xu and Fubao Zhang
2010, 27(3): 1241-1257 doi: 10.3934/dcds.2010.27.1241 +[Abstract](69) +[PDF](248.8KB)
In this paper we prove the existence of homoclinic orbits for the first order non-autonomous Hamiltonian system

$\dot{z}=\mathcal {J}H_{z}(t,z),$

where $H(t,z)$ depends periodically on $t$. We establish some existence results of the homoclinic orbits for weak superlinear cases. To this purpose, we apply a new linking theorem to provide bounded Palais-Samle sequences.

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