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Discrete & Continuous Dynamical Systems - A
February 2005 , Volume 13 , Issue 2
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2005, 13(2): 239-269
doi: 10.3934/dcds.2005.13.239
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Abstract:
Morales, Pacifico and Pujals proved recently that every robustly transitive singular set for a 3-dimensional flow must be partially hyperbolic. In this paper we generalize the result to higher dimensions. By definition, an isolated invariant set $\Lambda$ of a $C^1$ vector field $S$ is called robustly transitive if there exist an isolating neighborhood $U$ of $\Lambda$ in $M$ and a $C^1$ neighborhood $\mathcal U$ of $S$ in $\mathcal X^1(M)$ such that for every $X\in\mathcal U$, the maximal invariant set of $X$ in $U$ is non-trivially transitive. Such a set $\Lambda$ is called singular if it contains a singularity. The set $\Lambda$ is called strongly homogeneous of index $i$, if there exist an isolating neighborhood $U$ of $\Lambda$ in $M$ and a $C^1$ neighborhood $\mathcal U$ of $S$ in $\mathcal X^1(M)$ such that all periodic orbits of all $X\in\mathcal U$ contained in $U$ have the same index $i$. We prove in this paper that any robustly transitive singular set that is strongly homogeneous must be partially hyperbolic, as long as the indices of singularities and periodic orbits fit in certain way. As corollaries we obtain that every robust singular attractor (or repeller) that is strongly homogeneous must be partially hyperbolic and, if dim$M\le 4$, every robustly transitive singular set that is strongly homogeneous must be partially hyperbolic. The main novelty of the proofs in this paper is an extension of the usual linear Poincaré flow "to singularities".
Morales, Pacifico and Pujals proved recently that every robustly transitive singular set for a 3-dimensional flow must be partially hyperbolic. In this paper we generalize the result to higher dimensions. By definition, an isolated invariant set $\Lambda$ of a $C^1$ vector field $S$ is called robustly transitive if there exist an isolating neighborhood $U$ of $\Lambda$ in $M$ and a $C^1$ neighborhood $\mathcal U$ of $S$ in $\mathcal X^1(M)$ such that for every $X\in\mathcal U$, the maximal invariant set of $X$ in $U$ is non-trivially transitive. Such a set $\Lambda$ is called singular if it contains a singularity. The set $\Lambda$ is called strongly homogeneous of index $i$, if there exist an isolating neighborhood $U$ of $\Lambda$ in $M$ and a $C^1$ neighborhood $\mathcal U$ of $S$ in $\mathcal X^1(M)$ such that all periodic orbits of all $X\in\mathcal U$ contained in $U$ have the same index $i$. We prove in this paper that any robustly transitive singular set that is strongly homogeneous must be partially hyperbolic, as long as the indices of singularities and periodic orbits fit in certain way. As corollaries we obtain that every robust singular attractor (or repeller) that is strongly homogeneous must be partially hyperbolic and, if dim$M\le 4$, every robustly transitive singular set that is strongly homogeneous must be partially hyperbolic. The main novelty of the proofs in this paper is an extension of the usual linear Poincaré flow "to singularities".
2005, 13(2): 271-275
doi: 10.3934/dcds.2005.13.271
+[Abstract](1323)
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Abstract:
In this note we extend the hypercyclicity criterion (HC) to $C_0$- semigroups on separable Banach spaces and characterize semigroups satisfying (HC). Using (HC) we show the hypercyclicity of the translation group on the weighted spaces $L^p_\rho(\mathbb R,\mathbb C)$ or $C_{0,\rho}(\mathbb R,\mathbb C)$ for admissible weight functions $\rho$.
In this note we extend the hypercyclicity criterion (HC) to $C_0$- semigroups on separable Banach spaces and characterize semigroups satisfying (HC). Using (HC) we show the hypercyclicity of the translation group on the weighted spaces $L^p_\rho(\mathbb R,\mathbb C)$ or $C_{0,\rho}(\mathbb R,\mathbb C)$ for admissible weight functions $\rho$.
2005, 13(2): 277-290
doi: 10.3934/dcds.2005.13.277
+[Abstract](1142)
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Abstract:
The local well-posedness for the nonlinear Dirac equation with special forms of quadratic nonlinearities in one space dimension is obtained by two approaches. One is to apply the Fourier restriction norm method of Bourgain [2, 3] by showing the bilinear estimates for the nonlinearities. Another is to study the explicit solutions for wave equations and derive another bilinear estimates similar with Bournaveas [4].
The local well-posedness for the nonlinear Dirac equation with special forms of quadratic nonlinearities in one space dimension is obtained by two approaches. One is to apply the Fourier restriction norm method of Bourgain [2, 3] by showing the bilinear estimates for the nonlinearities. Another is to study the explicit solutions for wave equations and derive another bilinear estimates similar with Bournaveas [4].
2005, 13(2): 291-338
doi: 10.3934/dcds.2005.13.291
+[Abstract](1344)
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Abstract:
We deal with the Fatou functions $f_\lambda(z)=z+e^{-z}+\lambda$, Re$\lambda\ge 1$. We consider the set $J_r(f_\lambda)$ consisting of those points of the Julia set of $f_\lambda$ whose real parts do not escape to infinity under positive iterates of $f_\lambda$. Our ultimate result is that the function $\lambda\mapsto$HD$(J_r(f_\lambda))$ is real-analytic. In order to prove it we develop the thermodynamic formalism of potentials of the form $-t$log$|F_\lambda'|$, where $F_\lambda$ is the projection of $f_\lambda$ to the infinite cylinder. It includes appropriately defined topological pressure, Perron-Frobenius operators, geometric and invariant generalized conformal measures (Gibbs states). We show that our Perron-Frobenius operators are quasicompact, that they embed into a family of operators depending holomorphically on an appropriate parameter and we obtain several other properties of these operators. We prove an appropriate version of Bowen's formula that the Hausdorff dimension of the set $J_r(f_\lambda)$ is equal to the unique zero of the pressure function. Since the formula for the topological pressure is independent of the set $J_r(f_\lambda)$, Bowen's formula also indicates that $J_r(f_\lambda)$ is the right set to deal with. What concerns geometry of the set $J_r(f_\lambda)$ we also prove that the HD$(J_r(f_\lambda))$-dimensional Hausdorff measure of the set $J_r(F_\lambda)$ is positive and finite whereas its HD$(J_r(f_\lambda))$-dimensional packing measure is locally infinite. This last property allows us to conclude that HD$(J_r(f_\lambda))<2$. We also study in detail the properties of quasiconformal conjugations between the maps $f_\lambda$. As a byproduct of our main course of reasoning we prove stochastic properties of the dynamical system generated by $F_\lambda$ and the invariant Gibbs states $\mu_t$ such as the Central Limit Theorem and the exponential decay of correlations.
We deal with the Fatou functions $f_\lambda(z)=z+e^{-z}+\lambda$, Re$\lambda\ge 1$. We consider the set $J_r(f_\lambda)$ consisting of those points of the Julia set of $f_\lambda$ whose real parts do not escape to infinity under positive iterates of $f_\lambda$. Our ultimate result is that the function $\lambda\mapsto$HD$(J_r(f_\lambda))$ is real-analytic. In order to prove it we develop the thermodynamic formalism of potentials of the form $-t$log$|F_\lambda'|$, where $F_\lambda$ is the projection of $f_\lambda$ to the infinite cylinder. It includes appropriately defined topological pressure, Perron-Frobenius operators, geometric and invariant generalized conformal measures (Gibbs states). We show that our Perron-Frobenius operators are quasicompact, that they embed into a family of operators depending holomorphically on an appropriate parameter and we obtain several other properties of these operators. We prove an appropriate version of Bowen's formula that the Hausdorff dimension of the set $J_r(f_\lambda)$ is equal to the unique zero of the pressure function. Since the formula for the topological pressure is independent of the set $J_r(f_\lambda)$, Bowen's formula also indicates that $J_r(f_\lambda)$ is the right set to deal with. What concerns geometry of the set $J_r(f_\lambda)$ we also prove that the HD$(J_r(f_\lambda))$-dimensional Hausdorff measure of the set $J_r(F_\lambda)$ is positive and finite whereas its HD$(J_r(f_\lambda))$-dimensional packing measure is locally infinite. This last property allows us to conclude that HD$(J_r(f_\lambda))<2$. We also study in detail the properties of quasiconformal conjugations between the maps $f_\lambda$. As a byproduct of our main course of reasoning we prove stochastic properties of the dynamical system generated by $F_\lambda$ and the invariant Gibbs states $\mu_t$ such as the Central Limit Theorem and the exponential decay of correlations.
2005, 13(2): 339-359
doi: 10.3934/dcds.2005.13.339
+[Abstract](1285)
+[PDF](453.3KB)
Abstract:
This work investigates the structure of a class of traveling wave solutions of delayed cellular neural networks distributed in the one-dimensional integer lattice $\mathbb Z^1$. The dynamics of a given cell is characterized by instantaneous self-feedback and neighborhood interaction with its two left neighbors in which one is instantaneous and the other is distributively delayed due to, for example, finite switching speed and finite velocity of signal transmission. Applying the method of step with the aid of positive roots of the corresponding characteristic function of the profile equation, we can directly figure out the solution in explicit form. We then partition the parameter space $(\alpha, \beta)$-plane into four regions such that the qualitative properties of traveling waves can be completely determined for each region. In addition to the existence of monotonic traveling wave solutions, we also find that, for certain parameters, there exist non-monotonic traveling wave solutions such as camel-like waves with many critical points.
This work investigates the structure of a class of traveling wave solutions of delayed cellular neural networks distributed in the one-dimensional integer lattice $\mathbb Z^1$. The dynamics of a given cell is characterized by instantaneous self-feedback and neighborhood interaction with its two left neighbors in which one is instantaneous and the other is distributively delayed due to, for example, finite switching speed and finite velocity of signal transmission. Applying the method of step with the aid of positive roots of the corresponding characteristic function of the profile equation, we can directly figure out the solution in explicit form. We then partition the parameter space $(\alpha, \beta)$-plane into four regions such that the qualitative properties of traveling waves can be completely determined for each region. In addition to the existence of monotonic traveling wave solutions, we also find that, for certain parameters, there exist non-monotonic traveling wave solutions such as camel-like waves with many critical points.
2005, 13(2): 361-383
doi: 10.3934/dcds.2005.13.361
+[Abstract](1574)
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Abstract:
A multidimensional piston problem for the Euler equations for compressible isentropic flow is analyzed. Thepiston initially locates at the origin and experiences compressiveand expansive motions with spherical symmetry. The initialsingularity at the origin is one of the difficulties for thisspherically symmetric piston problem. A local shock front solutionfor the compressive motion is constructed based on thelinearization at an approximate solution and the Newton iteration. A global entropy solution for the piston problem is constructed byusing a shock capturing approach and the method of compensatedcompactness.
A multidimensional piston problem for the Euler equations for compressible isentropic flow is analyzed. Thepiston initially locates at the origin and experiences compressiveand expansive motions with spherical symmetry. The initialsingularity at the origin is one of the difficulties for thisspherically symmetric piston problem. A local shock front solutionfor the compressive motion is constructed based on thelinearization at an approximate solution and the Newton iteration. A global entropy solution for the piston problem is constructed byusing a shock capturing approach and the method of compensatedcompactness.
2005, 13(2): 385-398
doi: 10.3934/dcds.2005.13.385
+[Abstract](1582)
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Abstract:
We prove that no finite time blow up can occur for nonlinear Schrödinger equations with quadratic potentials, provided that the potential has a sufficiently strong repulsive component. This is not obvious in general, since the energy associated to the linear equation is not positive. The proof relies essentially on two arguments: global in time Strichartz estimates, and a refined analysis of the linear equation, which makes it possible to control the nonlinear effects.
We prove that no finite time blow up can occur for nonlinear Schrödinger equations with quadratic potentials, provided that the potential has a sufficiently strong repulsive component. This is not obvious in general, since the energy associated to the linear equation is not positive. The proof relies essentially on two arguments: global in time Strichartz estimates, and a refined analysis of the linear equation, which makes it possible to control the nonlinear effects.
2005, 13(2): 399-411
doi: 10.3934/dcds.2005.13.399
+[Abstract](1447)
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Abstract:
We study the time of $n$th return of orbits to some given (union of) rectangle(s) of a Markov partition for an Axiom A diffeomorphism. Namely, we prove the existence of a scaled generating function for these returns with respect to any Gibbs measure. As a by-product, we derive precise large deviation estimates and a central limit theorem for these return times. We emphasize that we look at the limiting behavior in term of number of visits (the size of the visited set is kept fixed). Our approach relies on the spectral properties of a one-parameter family of induced transfer operators on unstable leaves crossing the visited set.
We study the time of $n$th return of orbits to some given (union of) rectangle(s) of a Markov partition for an Axiom A diffeomorphism. Namely, we prove the existence of a scaled generating function for these returns with respect to any Gibbs measure. As a by-product, we derive precise large deviation estimates and a central limit theorem for these return times. We emphasize that we look at the limiting behavior in term of number of visits (the size of the visited set is kept fixed). Our approach relies on the spectral properties of a one-parameter family of induced transfer operators on unstable leaves crossing the visited set.
2005, 13(2): 413-428
doi: 10.3934/dcds.2005.13.413
+[Abstract](1434)
+[PDF](295.6KB)
Abstract:
We study standing wave solutions of the form $e^{i(\omega t+m\theta)}\phi(r)$ to nonlinear Schrödinger equation
We study standing wave solutions of the form $e^{i(\omega t+m\theta)}\phi(r)$ to nonlinear Schrödinger equation
$iu_t+\Delta u+|u|^{p-1}u=0\quad$ for $x\in \mathbb R^2$
and $t>0$, where $(r,\theta)$ are polar coordinates and $m\in\mathbb N$. Using the Evans function, we prove linear instability of standing wave solutions with nodes in the case where $p>3$.
2005, 13(2): 429-450
doi: 10.3934/dcds.2005.13.429
+[Abstract](1405)
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Abstract:
We consider a tridimensional phase-field model for a solidification/melting non-stationary process, which incorporates the physics of binary alloys, thermal properties and fluid motion of non-solidified material. The model is a free-boundary value problem consisting of a highly non-linear parabolic system including a phase-field equation, a heat equation, a concentration equation and a variant of the Navier-Stokes equations modified by a penalization term of Carman-Kozeny type to model the flow in mushy regions and a Boussinesq type term to take into account the effects of the differences in temperature and concentration in the flow. A proof of existence of generalized solutions for the system is given. For this, the problem is firstly approximated and a sequence of approximate solutions is obtained by Leray-Schauder's fixed point theorem. A solution of the original problem is then found by using compactness arguments.
We consider a tridimensional phase-field model for a solidification/melting non-stationary process, which incorporates the physics of binary alloys, thermal properties and fluid motion of non-solidified material. The model is a free-boundary value problem consisting of a highly non-linear parabolic system including a phase-field equation, a heat equation, a concentration equation and a variant of the Navier-Stokes equations modified by a penalization term of Carman-Kozeny type to model the flow in mushy regions and a Boussinesq type term to take into account the effects of the differences in temperature and concentration in the flow. A proof of existence of generalized solutions for the system is given. For this, the problem is firstly approximated and a sequence of approximate solutions is obtained by Leray-Schauder's fixed point theorem. A solution of the original problem is then found by using compactness arguments.
2005, 13(2): 451-468
doi: 10.3934/dcds.2005.13.451
+[Abstract](1492)
+[PDF](294.6KB)
Abstract:
Let $T:[0,1]\to [0,1]$ be a piecewise differentiable piecewise monotone map, and let $r>1$. It is well known that if $|T'|\le r$ (respectively $|T'|\ge r$) then $h_{t o p}(T)\le$ log $r$ (respectively $h_{t o p}(T)\ge$ log $r$). We show that if additionally $|T'| < r $ (respectively $ |T'| > r $) on some subinterval and $T$ is topologically transitive then the inequalities for the entropy are strict. We also give examples that the assumption of piecewise monotonicity is essential, even if $T$ is continuous. In one class of examples the dynamical dimension of the whole interval can be made arbitrarily small.
Let $T:[0,1]\to [0,1]$ be a piecewise differentiable piecewise monotone map, and let $r>1$. It is well known that if $|T'|\le r$ (respectively $|T'|\ge r$) then $h_{t o p}(T)\le$ log $r$ (respectively $h_{t o p}(T)\ge$ log $r$). We show that if additionally $|T'| < r $ (respectively $ |T'| > r $) on some subinterval and $T$ is topologically transitive then the inequalities for the entropy are strict. We also give examples that the assumption of piecewise monotonicity is essential, even if $T$ is continuous. In one class of examples the dynamical dimension of the whole interval can be made arbitrarily small.
2005, 13(2): 469-490
doi: 10.3934/dcds.2005.13.469
+[Abstract](1720)
+[PDF](257.9KB)
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We generalize the concept of Lyapunov exponent to transformations that are not necessarily differentiable. For fairly large classes of repellers and of hyperbolic sets of differentiable maps, the new exponents are shown to coincide with the classical ones. We also discuss the relation of the new Lyapunov exponents with the dimension theory of dynamical systems for invariant sets of continuous transformations.
We generalize the concept of Lyapunov exponent to transformations that are not necessarily differentiable. For fairly large classes of repellers and of hyperbolic sets of differentiable maps, the new exponents are shown to coincide with the classical ones. We also discuss the relation of the new Lyapunov exponents with the dimension theory of dynamical systems for invariant sets of continuous transformations.
2005, 13(2): 491-502
doi: 10.3934/dcds.2005.13.491
+[Abstract](1241)
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Necessary and sufficient conditions are given for master--slave synchronization of any pair of unidirectionally coupled one--dimensional affine cellular automata of rank one. In each case the synchronization condition is expressed in terms of the coupling and the arithmetic properties of the automaton local rule. The asymptotic behavior of finite length affine automata of rank one, subjected to Dirichlet boundary conditions, is shown to be equivalent to the synchronization problem.
Necessary and sufficient conditions are given for master--slave synchronization of any pair of unidirectionally coupled one--dimensional affine cellular automata of rank one. In each case the synchronization condition is expressed in terms of the coupling and the arithmetic properties of the automaton local rule. The asymptotic behavior of finite length affine automata of rank one, subjected to Dirichlet boundary conditions, is shown to be equivalent to the synchronization problem.
2005, 13(2): 503-514
doi: 10.3934/dcds.2005.13.503
+[Abstract](1653)
+[PDF](271.0KB)
Abstract:
In the homogenization of second order elliptic equations with periodic coefficients, it is well known that the rate of convergence of the zero order corrector $u_n -u^{h o m}$ in the $L^2$ norm is $1/n$, the same as the scale of periodicity (see Jikov et al [6]). It is possible to have the same rate of convergence in the case of almost periodic coefficients under some stringent structural conditions on the coefficients (see Kozlov [7]). The goal of this note is to construct almost periodic media where the rate of convergence is lower than $1/n$. To that aim, in the one dimensional setting, we introduce a family of random almost periodic coefficients for which we compute, using Fourier series analysis, the mean rate of convergence $r_n$ (mean with respect to the random parameter). This allows us to present examples where we find $r_n$>>$1/n^r$ for every $r>0$, showing a big contrast with the random case considered by Bourgeat and Piatnitski [2] where $r_n$~$1/\sqrt{n}$.
In the homogenization of second order elliptic equations with periodic coefficients, it is well known that the rate of convergence of the zero order corrector $u_n -u^{h o m}$ in the $L^2$ norm is $1/n$, the same as the scale of periodicity (see Jikov et al [6]). It is possible to have the same rate of convergence in the case of almost periodic coefficients under some stringent structural conditions on the coefficients (see Kozlov [7]). The goal of this note is to construct almost periodic media where the rate of convergence is lower than $1/n$. To that aim, in the one dimensional setting, we introduce a family of random almost periodic coefficients for which we compute, using Fourier series analysis, the mean rate of convergence $r_n$ (mean with respect to the random parameter). This allows us to present examples where we find $r_n$>>$1/n^r$ for every $r>0$, showing a big contrast with the random case considered by Bourgeat and Piatnitski [2] where $r_n$~$1/\sqrt{n}$.
2005, 13(2): 515-532
doi: 10.3934/dcds.2005.13.515
+[Abstract](1171)
+[PDF](270.5KB)
Abstract:
We consider a map called a double rotation, which is composed of two rotations on a circle. Specifically, a double rotation is a map on the interval $[0,1)$ that maps $x\in[0,c)$ to $\{x+\alpha\}$, and $x\in[c,1)$ to $\{x+\beta\}$. Although double rotations are discontinuous and noninvertible in general, we show that almost every double rotation can be reduced to a simple rotation, and the set of the parameter values such that the double rotation is irreducible to a rotation has a fractal structure. We also examine a characteristic number of a double rotation, which is called a discharge number. The graph of the discharge number as a function of $c$ reflects the fractal structure, and is very complicated.
We consider a map called a double rotation, which is composed of two rotations on a circle. Specifically, a double rotation is a map on the interval $[0,1)$ that maps $x\in[0,c)$ to $\{x+\alpha\}$, and $x\in[c,1)$ to $\{x+\beta\}$. Although double rotations are discontinuous and noninvertible in general, we show that almost every double rotation can be reduced to a simple rotation, and the set of the parameter values such that the double rotation is irreducible to a rotation has a fractal structure. We also examine a characteristic number of a double rotation, which is called a discharge number. The graph of the discharge number as a function of $c$ reflects the fractal structure, and is very complicated.
2005, 13(2): 533-540
doi: 10.3934/dcds.2005.13.533
+[Abstract](1648)
+[PDF](176.8KB)
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In this paper, various shadowing properties are considered for expansive homeomorphisms. More precisely, we show that the continuous shadowing property, the Lipschitz shadowing property, the limit shadowing property and the strong shadowing property are all equivalent to the (usual) shadowing property for expansive homeomorphisms on compact metric spaces.
In this paper, various shadowing properties are considered for expansive homeomorphisms. More precisely, we show that the continuous shadowing property, the Lipschitz shadowing property, the limit shadowing property and the strong shadowing property are all equivalent to the (usual) shadowing property for expansive homeomorphisms on compact metric spaces.
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