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Discrete & Continuous Dynamical Systems - A

2008 , Volume 20 , Issue 4

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Uniqueness of solutions for second order Bellman-Isaacs equations with mixed boundary conditions
Monica Motta and Caterina Sartori
2008, 20(4): 739-765 doi: 10.3934/dcds.2008.20.739 +[Abstract](232) +[PDF](340.3KB)
Abstract:
We investigate the uniqueness of solutions for boundary value problems in bounded and unbounded domains involving nonlinear degenerate second order Bellman-Isaacs equations and mixed boundary conditions (Dirichlet, generalized Dirichlet and state constrained conditions). These boundary value problems arise from exit or stopping time stochastic differential games or optimal control problems with constraints, such as state and integral constraints.
Existence of periodic solutions of a system of damped wave equations in thin domains
Bouthaina Abdelhedi
2008, 20(4): 767-800 doi: 10.3934/dcds.2008.20.767 +[Abstract](224) +[PDF](354.5KB)
Abstract:
We prove the persistence of the existence of a time-periodic solution both in the non-autonomous and autonomous cases for a system of damped wave equations in a thin domain. The methods used are a combination of a refined version of the fixed point theorem and a functional characterization of periodic solutions.
A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds
Andrei Agrachev, Ugo Boscain and Mario Sigalotti
2008, 20(4): 801-822 doi: 10.3934/dcds.2008.20.801 +[Abstract](232) +[PDF](906.6KB)
Abstract:
We consider a generalization of Riemannian geometry that naturally arises in the framework of control theory. Let $X$ and $Y$ be two smooth vector fields on a two-dimensional manifold $M$. If $X$ and $Y$ are everywhere linearly independent, then they define a classical Riemannian metric on $M$ (the metric for which they are orthonormal) and they give to $M$ the structure of metric space. If $X$ and $Y$ become linearly dependent somewhere on $M$, then the corresponding Riemannian metric has singularities, but under generic conditions the metric structure is still well defined. Metric structures that can be defined locally in this way are called almost-Riemannian structures. They are special cases of \ar s, which are naturally defined in terms of submodules of the space of smooth vector fields on $M$. Almost-Riemannian structures show interesting phenomena, in particular those which concern the relation between curvature, presence of conjugate points, and topology of the manifold. The main result of the paper is a generalization to almost-Riemannian structures of the Gauss-Bonnet formula.
Topological invariants for singularities of real vector fields in dimension three
C. Alonso-González, M. I. Camacho and F. Cano
2008, 20(4): 823-847 doi: 10.3934/dcds.2008.20.823 +[Abstract](255) +[PDF](294.6KB)
Abstract:
We give topological invariants for a wide class of abso-lutely isolated singularities of three-dimensional real vector fields. Our invariants are complete once the desingularization morphism $\pi$ is fixed. They are obtained in terms of a finite set of configurations depending only on the eigenvalues of the singularities.
Physical measures at the boundary of hyperbolic maps
Vítor Araújo and Ali Tahzibi
2008, 20(4): 849-876 doi: 10.3934/dcds.2008.20.849 +[Abstract](229) +[PDF](394.0KB)
Abstract:
We consider diffeomorphisms of a compact manifold with a dominated splitting which is hyperbolic except for a "small" subset of points (Hausdorff dimension smaller than one, e.g. a denumerable subset) and prove the existence of physical measures and their stochastic stability. The physical measures are obtained as zero-noise limits which are shown to satisfy the Entropy Formula.
Optimal Lyapunov inequalities for disfocality and Neumann boundary conditions using $L^p$ norms
Antonio Cañada and Salvador Villegas
2008, 20(4): 877-888 doi: 10.3934/dcds.2008.20.877 +[Abstract](256) +[PDF](183.9KB)
Abstract:
Motivated by the applications to nonlinear resonant boundary value problems with Neumann boundary conditions, this paper is devoted to the study of $L^{p}$ Lyapunov-type inequalities ($1 \leq p \leq \infty$) with mixed boundary conditions. We carry out a complete treatment of the problem for any constant $p \geq 1.$ Our main result is derived from a detailed analysis of the relationship between the existence of nontrivial solutions of these two different boundary problems.
Smoothness of classical solutions to the Vlasov-Maxwell-Landau system near Maxwellians
Yemin Chen
2008, 20(4): 889-910 doi: 10.3934/dcds.2008.20.889 +[Abstract](242) +[PDF](264.9KB)
Abstract:
In this paper we prove that under the assumption that the electromagnetic field is smooth initially, even if the distribution function is not smooth initially, the classical solutions (both the distribution function and the electromagnetic field) to the Vlasov-Maxwell-Landau system become immediately smooth with respect to all variables.
On small oscillations of mechanical systems with time-dependent kinetic and potential energy
Nicola Guglielmi and László Hatvani
2008, 20(4): 911-926 doi: 10.3934/dcds.2008.20.911 +[Abstract](234) +[PDF](291.4KB)
Abstract:
Small oscillations of an undamped holonomic mechanical system with varying parameters are described by the equations

$\sum$nk=1$(a_{ik}(t)\ddot q_k+c_{ik}(t)q_k)=0, (i=1,2,\ldots,n).$(*)

A nontrivial solution $q_1^0,\ldots ,q_n^0$ is called small if

$\lim _{t\to \infty}q_k(t)=0 (k=1,2,\ldots n).

It is known that in the scalar case ($n=1$, $a_{11}(t)\equiv 1$, $c_{11}(t)=:c(t)$) there exists a small solution if $c$ is increasing and it tends to infinity as $t\to \infty$.
    Sufficient conditions for the existence of a small solution of the general system (*) are given in the case when coefficients $a_{ik}$, $c_{ik}$ are step functions. The method of proofs is based upon a transformation reducing the ODE (*) to a discrete dynamical system. The results are illustrated by the examples of the coupled harmonic oscillator and the double pendulum.

Blowup rate estimates for the heat equation with a nonlinear gradient source term
Jong-Shenq Guo and Bei Hu
2008, 20(4): 927-937 doi: 10.3934/dcds.2008.20.927 +[Abstract](288) +[PDF](187.3KB)
Abstract:
The gradient blowup rate of the equation $u_t = \Delta u + |\nabla u|^p$, where $p>2$, is studied. It is shown that the blowup rate will never match that of the self-similar variables. In the one space dimensional case when assumptions are made on the initial data so that the solution is monotonically increasing in time, the exact blowup rate is found.
Dynamics of local map of a discrete Brusselator model: eventually trapping regions and strange attractors
Hunseok Kang
2008, 20(4): 939-959 doi: 10.3934/dcds.2008.20.939 +[Abstract](236) +[PDF](860.9KB)
Abstract:
The reaction-diffusion equation for the Brusselator model produces a coupled map lattice (CML) by discretization. The two-dimensional nonlinear local map of this lattice has rich and interesting dynamics. In [7] we studied the dynamics of the local map, focusing on trajectories escaping to infinity, and the Julia set. In this paper we build a correspondence between CML and its local map via traveling waves, and then using this correspondence we study asymptotic properties of this CML. We show the existence of a bounded region in which every trajectory in the Julia set is eventually trapped. We also find a region where every bounded trajectory visits. Finally, we present some strange attractors that are numerically observed in the Julia set.
Lyapunov exponents and the dimension of the attractor for 2d shear-thinning incompressible flow
P. Kaplický and Dalibor Pražák
2008, 20(4): 961-974 doi: 10.3934/dcds.2008.20.961 +[Abstract](241) +[PDF](220.1KB)
Abstract:
The equations describing planar motion of a homogeneous, incompressible generalized Newtonian fluid are considered. The stress tensor is given constitutively as $\T=\nu(1+\mu|\Du|^2)^{\frac{p-2}2}\Du$, where $\Du$ is the symmetric part of the velocity gradient. The equations are complemented by periodic boundary conditions.
    For the solution semigroup the Lyapunov exponents are computed using a slightly generalized form of the Lieb-Thirring inequality and consequently the fractal dimension of the global attractor is estimated for all $p\in(4/3,2]$.
On a certain kind of polynomials of degree 4 with disconnected Julia set
Koh Katagata
2008, 20(4): 975-987 doi: 10.3934/dcds.2008.20.975 +[Abstract](256) +[PDF](312.1KB)
Abstract:
For a polynomial of degree at least two, the Julia set and the filled-in Julia set are either connected or have uncountably many components. In the case that the Julia set of a polynomial of degree 4 is neither connected nor totally disconnected, there exists a homeomorphism between the set of all components of the filled-in Julia set and some subset of the corresponding symbol space. Furthermore the polynomial is topologically conjugate to the shift map via the homeomorphism. Moreover there exists a homeomorphism between the Julia sets of the polynomial and that of a certain polynomial semigroup.
Asymptotics of the Arnold tongues in problems at infinity
Victor Kozyakin, Alexander M. Krasnosel’skii and Dmitrii Rachinskii
2008, 20(4): 989-1011 doi: 10.3934/dcds.2008.20.989 +[Abstract](329) +[PDF](328.0KB)
Abstract:
We consider discrete time systems $x_{k+1}=U(x_{k};\lambda)$, $x\in\R^{N}$, with a complex parameter $\lambda$, and study their trajectories of large amplitudes. The expansion of the map $U(\cdot;\lambda)$ at infinity contains a principal linear term, a bounded positively homogeneous nonlinearity, and a smaller vanishing part. We study Arnold tongues: the sets of parameter values for which the large-amplitude periodic trajectories exist. The Arnold tongues in problems at infinity generically are thick triangles [4]; here we obtain asymptotic estimates for the length of the Arnold tongues and for the length of their triangular part. These estimates allow us to study subfurcation at infinity. In the related problems on small-amplitude periodic orbits near an equilibrium, similarly defined Arnold tongues have the form of narrow beaks. For standard pictures associated with the Neimark-Sacker bifurcation of smooth discrete time systems at an equilibrium, the Arnold tongues have asymptotically zero width except for the strong resonance points. The different shape of the tongues in the problem at infinity is due to the non-polynomial form of the principal homogeneous nonlinear term of the map $U(\cdot;\lambda)$: this form implies non-degeneracy of the nonlinear terms in the expansion of the map iterations and non-degeneracy of the corresponding resonance functions.
The pointwise estimates of solutions for dissipative wave equation in multi-dimensions
Yongqin Liu and Weike Wang
2008, 20(4): 1013-1028 doi: 10.3934/dcds.2008.20.1013 +[Abstract](358) +[PDF](204.6KB)
Abstract:
In this paper we focus on the pointwise estimates of the solution to the Cauchy problem for the dissipative wave equation in multi-dimensions. By using the method of Green function combined with the Fourier analysis, we obtain the pointwise estimates of the solution, which yields the $L^p(1\leq p\leq\infty)$ decay estimates of the solution.
Semi-hyperbolicity and hyperbolicity
Marcin Mazur, Jacek Tabor and Piotr Kościelniak
2008, 20(4): 1029-1038 doi: 10.3934/dcds.2008.20.1029 +[Abstract](261) +[PDF](147.8KB)
Abstract:
We prove that for $\mathcal{C}^1$-diffeomorfisms semi-hyperbolicity of an invariant set implies its hyperbolicity. Moreover, we provide some exact estimations of hyperbolicity constants by semi-hyperbolicity ones, which can be useful in strict numerical computations.
Chain recurrence in multidimensional time discrete dynamical systems
Piotr Oprocha
2008, 20(4): 1039-1056 doi: 10.3934/dcds.2008.20.1039 +[Abstract](263) +[PDF](245.9KB)
Abstract:
The author investigates the behavior of multidimensional time discrete dynamical systems. Problems of expansivity, P.O.T.P. and chain recurrence are considered in particular. The main result of this article is a general version of Spectral Decomposition Theorem.
Boundary stabilization for the wave equation in a bounded cylindrical domain
Kim Dang Phung
2008, 20(4): 1057-1093 doi: 10.3934/dcds.2008.20.1057 +[Abstract](274) +[PDF](334.5KB)
Abstract:
We provide a polynomial decay rate for the energy of the wave equation with a dissipative boundary condition in a cylindrical trapped domain. A new kind of interpolation estimate for the wave equation with mixed Dirichlet-Neumann boundary condition is established from a construction based on a Fourier integral operator involving a good choice of weight functions.
Right-permutative cellular automata on topological Markov chains
Marcelo Sobottka
2008, 20(4): 1095-1109 doi: 10.3934/dcds.2008.20.1095 +[Abstract](249) +[PDF](236.5KB)
Abstract:
In this paper we consider cellular automata $(G,\Phi)$ with algebraic local rules and such that $G$ is a topological Markov chain which has a structure compatible to this local rule. We characterize such cellular automata and study the convergence of the Cesàro mean distribution of the iterates of any probability measure with complete connections and summable decay.
On circle rotations and the shrinking target properties
Jimmy Tseng
2008, 20(4): 1111-1122 doi: 10.3934/dcds.2008.20.1111 +[Abstract](277) +[PDF](191.9KB)
Abstract:
We generalize the monotone shrinking target property (MSTP) to the $s$-exponent monotone shrinking target property ($s$MSTP) and give a necessary and sufficient condition for a circle rotation to have $s$MSTP.
    Using another variant of MSTP, we obtain a new, very short, proof of a known result, which concerns the behavior of irrational rotations and implies a logarithm law similar to D. Sullivan's logarithm law for geodesics.
Stability of traveling waves with critical speeds for $P$-degree Fisher-type equations
Yaping Wu and Xiuxia Xing
2008, 20(4): 1123-1139 doi: 10.3934/dcds.2008.20.1123 +[Abstract](285) +[PDF](267.3KB)
Abstract:
This paper is concerned with the stability of the traveling front solutions with critical speeds for a class of $p$-degree Fisher-type equations. By detailed spectral analysis and sub-supper solution method, we first show that the traveling front solutions with critical speeds are globally exponentially stable in some exponentially weighted spaces. Furthermore by Evans's function method, appropriate space decomposition and detailed semigroup decaying estimates, we can prove that the waves with critical speeds are locally asymptotically stable in some polynomially weighted spaces, which verifies some asymptotic phenomena obtained by numerical simulations.

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