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Discrete & Continuous Dynamical Systems - A
October 2008 , Volume 20 , Issue 4
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2008, 20(4): 739-765
doi: 10.3934/dcds.2008.20.739
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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.
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.
2008, 20(4): 767-800
doi: 10.3934/dcds.2008.20.767
+[Abstract](1887)
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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.
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.
2008, 20(4): 801-822
doi: 10.3934/dcds.2008.20.801
+[Abstract](2179)
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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.
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.
2008, 20(4): 823-847
doi: 10.3934/dcds.2008.20.823
+[Abstract](2150)
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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.
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.
2008, 20(4): 849-876
doi: 10.3934/dcds.2008.20.849
+[Abstract](2020)
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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.
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.
2008, 20(4): 877-888
doi: 10.3934/dcds.2008.20.877
+[Abstract](1853)
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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.
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.
2008, 20(4): 889-910
doi: 10.3934/dcds.2008.20.889
+[Abstract](1878)
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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.
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.
2008, 20(4): 911-926
doi: 10.3934/dcds.2008.20.911
+[Abstract](2094)
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Abstract:
Small oscillations of an undamped holonomic mechanical system with varying parameters are described by the equations
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.
2008, 20(4): 927-937
doi: 10.3934/dcds.2008.20.927
+[Abstract](2500)
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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.
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.
2008, 20(4): 939-959
doi: 10.3934/dcds.2008.20.939
+[Abstract](1870)
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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.
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.
2008, 20(4): 961-974
doi: 10.3934/dcds.2008.20.961
+[Abstract](2136)
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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 $p\in(4/3,2]$.
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 $p\in(4/3,2]$.
2008, 20(4): 975-987
doi: 10.3934/dcds.2008.20.975
+[Abstract](2087)
+[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.
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.
2008, 20(4): 989-1011
doi: 10.3934/dcds.2008.20.989
+[Abstract](2337)
+[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.
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.
2008, 20(4): 1013-1028
doi: 10.3934/dcds.2008.20.1013
+[Abstract](2232)
+[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.
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.
2008, 20(4): 1029-1038
doi: 10.3934/dcds.2008.20.1029
+[Abstract](1922)
+[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.
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.
2008, 20(4): 1039-1056
doi: 10.3934/dcds.2008.20.1039
+[Abstract](2193)
+[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.
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.
2008, 20(4): 1057-1093
doi: 10.3934/dcds.2008.20.1057
+[Abstract](2033)
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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.
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.
2008, 20(4): 1095-1109
doi: 10.3934/dcds.2008.20.1095
+[Abstract](1846)
+[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.
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.
2008, 20(4): 1111-1122
doi: 10.3934/dcds.2008.20.1111
+[Abstract](2388)
+[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.
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.
2008, 20(4): 1123-1139
doi: 10.3934/dcds.2008.20.1123
+[Abstract](2056)
+[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.
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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