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Discrete & Continuous Dynamical Systems - A
October 2006 , Volume 14 , Issue 4
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2006, 14(4): 617-630
doi: 10.3934/dcds.2006.14.617
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Abstract:
For many classes of symplectic manifolds, the Hamiltonian flow of a function with sufficiently large variation must have a fast periodic orbit. This principle is the base of the notion of Hofer-Zehnder capacity and some other symplectic invariants and leads to numerous results concerning existence of periodic orbits of Hamiltonian flows. Along these lines, we show that given a negatively curved manifold $M$, a neigbourhood $U_{R}$ of $M$ in T*M, a sufficiently $C^{1}$-small magnetic field $\sigma$ and a non-trivial free homotopy class of loops $\alpha$, then the magnetic flow of certain Hamiltonians supported in $U_{R}$ with big enough minimum, has a one-periodic orbit in $\alpha$. As a consequence, we obtain estimates for the relative Hofer-Zehnder capacity and the Biran-Polterovich-Salamon capacity of a neighbourhood of $M$.
For many classes of symplectic manifolds, the Hamiltonian flow of a function with sufficiently large variation must have a fast periodic orbit. This principle is the base of the notion of Hofer-Zehnder capacity and some other symplectic invariants and leads to numerous results concerning existence of periodic orbits of Hamiltonian flows. Along these lines, we show that given a negatively curved manifold $M$, a neigbourhood $U_{R}$ of $M$ in T*M, a sufficiently $C^{1}$-small magnetic field $\sigma$ and a non-trivial free homotopy class of loops $\alpha$, then the magnetic flow of certain Hamiltonians supported in $U_{R}$ with big enough minimum, has a one-periodic orbit in $\alpha$. As a consequence, we obtain estimates for the relative Hofer-Zehnder capacity and the Biran-Polterovich-Salamon capacity of a neighbourhood of $M$.
2006, 14(4): 631-642
doi: 10.3934/dcds.2006.14.631
+[Abstract](1237)
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Abstract:
In this paper we consider twist mappings of the torus, $\overline{T}: T^2\rightarrow T^2,$ and their vertical rotation intervals $\rho _V( T)=[\rho _V^{-},\rho _V^{+}],$ which are closed intervals such that for any $\omega \in ]\rho _V^{-},\rho _V^{+}$[ there exists a compact $ \overline{T}$-invariant set $\overline{Q}_\omega $ with $\rho _V(\overline{x} )=\omega$ for any $\overline{x}\in \overline{Q}_\omega ,$ where $\rho _V( \overline{x})$ is the vertical rotation number of $\overline{x}.$ In case $ \omega $ is a rational number, $\overline{Q}_\omega $ is a periodic orbit (this study began in [1] and [2]). Here we analyze how $\rho _V^{-}$ and $ \rho _V^{+}$ behave as we perturb $\overline{T}$ when they assume rational values. In particular we prove that for analytic area-preserving mappings these functions are locally constant at rational values.
In this paper we consider twist mappings of the torus, $\overline{T}: T^2\rightarrow T^2,$ and their vertical rotation intervals $\rho _V( T)=[\rho _V^{-},\rho _V^{+}],$ which are closed intervals such that for any $\omega \in ]\rho _V^{-},\rho _V^{+}$[ there exists a compact $ \overline{T}$-invariant set $\overline{Q}_\omega $ with $\rho _V(\overline{x} )=\omega$ for any $\overline{x}\in \overline{Q}_\omega ,$ where $\rho _V( \overline{x})$ is the vertical rotation number of $\overline{x}.$ In case $ \omega $ is a rational number, $\overline{Q}_\omega $ is a periodic orbit (this study began in [1] and [2]). Here we analyze how $\rho _V^{-}$ and $ \rho _V^{+}$ behave as we perturb $\overline{T}$ when they assume rational values. In particular we prove that for analytic area-preserving mappings these functions are locally constant at rational values.
2006, 14(4): 643-672
doi: 10.3934/dcds.2006.14.643
+[Abstract](1268)
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Abstract:
In this paper, we investigate the connections between controllability properties of distributed systems and existence of non zero entire functions subject to restrictions on their growth and on their sets of zeros. Exploiting these connections, we first show that, for generic bounded open domains in dimension $n\geq 2$, the steady--state controllability for the heat equation with boundary controls dependent only on time, does not hold. In a second step, we study a model of a water tank whose dynamics is given by a wave equation on a two-dimensional bounded open domain. We provide a condition which prevents steady-state controllability of such a system, where the control acts on the boundary and is only dependent on time. Using that condition, we prove that the steady-state controllability does not hold for generic tank shapes.
In this paper, we investigate the connections between controllability properties of distributed systems and existence of non zero entire functions subject to restrictions on their growth and on their sets of zeros. Exploiting these connections, we first show that, for generic bounded open domains in dimension $n\geq 2$, the steady--state controllability for the heat equation with boundary controls dependent only on time, does not hold. In a second step, we study a model of a water tank whose dynamics is given by a wave equation on a two-dimensional bounded open domain. We provide a condition which prevents steady-state controllability of such a system, where the control acts on the boundary and is only dependent on time. Using that condition, we prove that the steady-state controllability does not hold for generic tank shapes.
2006, 14(4): 673-688
doi: 10.3934/dcds.2006.14.673
+[Abstract](1500)
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Abstract:
We give a new type of sufficient condition for the existence of measures with maximal entropy for an interval map $f$, using some non-uniform hyperbolicity to compensate for a lack of smoothness of $f$. More precisely, if the topological entropy of a $C^1$ interval map is greater than the sum of the local entropy and the entropy of the critical points, then there exists at least one measure with maximal entropy. As a corollary, we obtain that any $C^r$ interval map $f$ such that htop(f) > 2log || f'||∞ / r possesses measures with maximal entropy.
We give a new type of sufficient condition for the existence of measures with maximal entropy for an interval map $f$, using some non-uniform hyperbolicity to compensate for a lack of smoothness of $f$. More precisely, if the topological entropy of a $C^1$ interval map is greater than the sum of the local entropy and the entropy of the critical points, then there exists at least one measure with maximal entropy. As a corollary, we obtain that any $C^r$ interval map $f$ such that htop(f) > 2log || f'||∞ / r possesses measures with maximal entropy.
2006, 14(4): 689-706
doi: 10.3934/dcds.2006.14.689
+[Abstract](1098)
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Abstract:
Consider a Stefan-like problem with Gibbs-Thomson and kinetic effects as a model of crystal growth from vapor. The equilibrium shape is assumed to be a regular circular cylinder. Our main concern is a problem whether or not a surface of cylindrical crystals (called a facet) is stable under evolution in the sense that its normal velocity is constant over the facet. If a facet is unstable, then it breaks or bends. A typical result we establish is that all facets are stable if the evolving crystal is near the equilibrium. The stability criterion we use is a variational principle for selecting the correct Cahn-Hoffman vector. The analysis of the phase plane of an evolving cylinder (identified with points in the plane) near the unique equilibrium provides a bound for ratio of velocities of top and lateral facets of the cylinders.
Consider a Stefan-like problem with Gibbs-Thomson and kinetic effects as a model of crystal growth from vapor. The equilibrium shape is assumed to be a regular circular cylinder. Our main concern is a problem whether or not a surface of cylindrical crystals (called a facet) is stable under evolution in the sense that its normal velocity is constant over the facet. If a facet is unstable, then it breaks or bends. A typical result we establish is that all facets are stable if the evolving crystal is near the equilibrium. The stability criterion we use is a variational principle for selecting the correct Cahn-Hoffman vector. The analysis of the phase plane of an evolving cylinder (identified with points in the plane) near the unique equilibrium provides a bound for ratio of velocities of top and lateral facets of the cylinders.
2006, 14(4): 707-719
doi: 10.3934/dcds.2006.14.707
+[Abstract](1332)
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Abstract:
Let $\Omega$ be a bounded domain in $\mathbb R^N$$(N\geq 4)$ with smooth boundary $\partial \Omega$ and the origin $0 \in \overline{\Omega}$, $\mu<0$, 2*=2N/(N-2). We obtain existence results of positive and sign-changing solutions to Dirichlet problem $-\Delta u=\mu\frac{ u}{|x|^2}$+|u|2*-2u+$\lambda u \ \text{on}\ \Omega,\ u=0 \ \text{on}\ \partial\Omega$, which also gives a positive answer to the open problem proposed by A. Ferrero and F. Gazzola in [Existence of solutions for singular critical growth semilinear elliptic equations, J. Differential Equations, 177(2001), 494-522].
Let $\Omega$ be a bounded domain in $\mathbb R^N$$(N\geq 4)$ with smooth boundary $\partial \Omega$ and the origin $0 \in \overline{\Omega}$, $\mu<0$, 2*=2N/(N-2). We obtain existence results of positive and sign-changing solutions to Dirichlet problem $-\Delta u=\mu\frac{ u}{|x|^2}$+|u|2*-2u+$\lambda u \ \text{on}\ \Omega,\ u=0 \ \text{on}\ \partial\Omega$, which also gives a positive answer to the open problem proposed by A. Ferrero and F. Gazzola in [Existence of solutions for singular critical growth semilinear elliptic equations, J. Differential Equations, 177(2001), 494-522].
2006, 14(4): 721-736
doi: 10.3934/dcds.2006.14.721
+[Abstract](1438)
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Abstract:
We study the delay equation $\dot{x}(t)=-\mu x(t)+f(x(t-1))$ with $\mu>0$ and a nonmonotone $C^1$-function $f$ obeying $x f(x)>0$ (positive feedback) outside a small neighbourhood of zero. By means of a computer-assisted method we prove the existence of asymptotically orbitally stable periodic solutions. The main idea behind our proof is the reduction of the infinite-dimensional dynamics to a finite-dimensional map. In particular, for two classes of nonlinearities $f$ we construct two types of solutions, the dynamics of which is reduced to a one- and a two-dimensional map, respectively.
We study the delay equation $\dot{x}(t)=-\mu x(t)+f(x(t-1))$ with $\mu>0$ and a nonmonotone $C^1$-function $f$ obeying $x f(x)>0$ (positive feedback) outside a small neighbourhood of zero. By means of a computer-assisted method we prove the existence of asymptotically orbitally stable periodic solutions. The main idea behind our proof is the reduction of the infinite-dimensional dynamics to a finite-dimensional map. In particular, for two classes of nonlinearities $f$ we construct two types of solutions, the dynamics of which is reduced to a one- and a two-dimensional map, respectively.
2006, 14(4): 737-751
doi: 10.3934/dcds.2006.14.737
+[Abstract](1311)
+[PDF](278.6KB)
Abstract:
We consider the generalized Gierer-Meinhardt system
We consider the generalized Gierer-Meinhardt system
$\frac{\partial u_{j}}{\partial t}=d_{j}$Δ $u_{j}- a_{j}u_{j}
+g_{j}(x,u) \ \ \text{in}\ \ \Omega\times[0,T) $ ,
$\frac{\partial u_{j}}{\partial\nu}=0\ \ \text{on}\ \
\partial\Omega\times[
0,T) $,
$u_{j}(x,0) =\varphi_{j}(x)\ \ \text{in}\ \ \Omega$
where $\Omega$ is a smooth bounded domain in $\mathbb{R}^{n}$ with $\nu$ its unit outer normal, $j=1,2$, $u=(u_1,u_2)$ and
$g_{1}(x,u) =\rho_{1}(x,u) \frac{u_{1}^{p}}{u_{2}^{q}}+\sigma_{1}(x) $ ,
$g_{2}(x,u) =\rho_{2}(x,u) \frac{u_{1}^{r}}{u_{2}^{s}}+\sigma_{2}(x) $.
Here $d_{j}, a_{j}$ are positive constants, $\rho_{1}\geq0$, $\rho
_{2}>0,\sigma_{j}\geq0$ are bounded smooth functions and $p,q,r,s$
are
nonnegative constants satisfying $0<\frac{p-1}{r}<\frac{q}{s+1}$.
We show that there is a unique global solution when p-1 < r, which
improves 1987 result of K. Masuda and K. Takahashi
[10]. Asymptotic bounds of global solutions are also
established which yield new a priori estimates of stationary
solutions.
2006, 14(4): 753-782
doi: 10.3934/dcds.2006.14.753
+[Abstract](1400)
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Abstract:
In this article, we present a new approach to averaging in non-Hamiltonian systems with periodic forcing. The results here do not depend on the existence of a small parameter. In fact, we show that our averaging method fits into an appropriate nonlinear equivalence problem, and that this problem can be solved formally by using the Lie transform framework to linearize it. According to this approach, we derive formal coordinate transformations associated with both first-order and higher-order averaging, which result in more manageable formulae than the classical ones.
Using these transformations, it is possible to correct the solution of an averaged system by recovering the oscillatory components of the original non-averaged system. In this framework, the inverse transformations are also defined explicitly by formal series; they allow the estimation of appropriate initial data for each higher-order averaged system, respecting the equivalence relation.
Finally, we show how these methods can be used for identifying and computing periodic solutions for a very large class of nonlinear systems with time-periodic forcing. We test the validity of our approach by analyzing both the first-order and the second-order averaged system for a problem in atmospheric chemistry.
In this article, we present a new approach to averaging in non-Hamiltonian systems with periodic forcing. The results here do not depend on the existence of a small parameter. In fact, we show that our averaging method fits into an appropriate nonlinear equivalence problem, and that this problem can be solved formally by using the Lie transform framework to linearize it. According to this approach, we derive formal coordinate transformations associated with both first-order and higher-order averaging, which result in more manageable formulae than the classical ones.
Using these transformations, it is possible to correct the solution of an averaged system by recovering the oscillatory components of the original non-averaged system. In this framework, the inverse transformations are also defined explicitly by formal series; they allow the estimation of appropriate initial data for each higher-order averaged system, respecting the equivalence relation.
Finally, we show how these methods can be used for identifying and computing periodic solutions for a very large class of nonlinear systems with time-periodic forcing. We test the validity of our approach by analyzing both the first-order and the second-order averaged system for a problem in atmospheric chemistry.
2006, 14(4): 783-800
doi: 10.3934/dcds.2006.14.783
+[Abstract](1178)
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Abstract:
In this paper, we consider the boundedness of all the solutions and the existence of quasi-periodic solutions for Duffing equations
In this paper, we consider the boundedness of all the solutions and the existence of quasi-periodic solutions for Duffing equations
$\frac{d^2x}{dt^2}+\arctan x=p(t),$
where $p(t+1)=p(t)$ is a smooth function.
2006, 14(4): 801-820
doi: 10.3934/dcds.2006.14.801
+[Abstract](1414)
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Abstract:
In this paper we consider the Cauchy problem for the abstract nonlinear evolution equation in a Hilbert space $\H$
In this paper we consider the Cauchy problem for the abstract nonlinear evolution equation in a Hilbert space $\H$
$\A(u'(t))+ \B(u(t))-\lambda u(t)$ ∋ $f \mbox{in } \H \mbox{ for a.e. }t\in (0,+\infty)$
$u(0)=u_{0},$
where $\A$ is a maximal (possibly multivalued) monotone operator from the Hilbert space $\H$ to itself, while $\B$ is the subdifferential of a proper, convex and lower semicontinuous function φ:$\H\rightarrow (-\infty,+\infty]$ with compact sublevels in $\H$ satisfying a suitable compatibility condition. Finally, $\lambda$ is a positive constant. The existence of solutions is proved by using an approximation-a priori estimates-passage to the limit procedure. The main result of this paper is that the set of all the solutions generates a Generalized Semiflow in the sense of John M. Ball [8] in the phase space given by the domain of the potential φ. This process is shown to be point dissipative and asymptotically compact; moreover the global attractor, which attracts all the trajectories of the system with respect to a metric strictly linked to the constraint imposed on the unknown, is constructed. Applications to some problems involving PDEs are given.
2006, 14(4): 821-835
doi: 10.3934/dcds.2006.14.821
+[Abstract](1400)
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Abstract:
In 1992, S. Kolyada asked the question whether for triangular maps of the square zero topological entropy is equivalent to the fact that every recurrent point is uniformly recurrent. One of the implications was answered negatively by G.-L. Forti, L. Paganoni and J. Smítal in 1995. They showed that a zero entropy triangular map may have a recurrent point which is not uniformly recurrent. In this paper we show that neither the converse implication is true by constructing a triangular map of the square with positive topological entropy and with every chain recurrent point uniformly recurrent. In fact we first construct an appropriate minimal positive entropy system whose phase space is a nonhomogeneous subset of the square and then we extend it to a triangular map with required properties in the square.
In 1992, S. Kolyada asked the question whether for triangular maps of the square zero topological entropy is equivalent to the fact that every recurrent point is uniformly recurrent. One of the implications was answered negatively by G.-L. Forti, L. Paganoni and J. Smítal in 1995. They showed that a zero entropy triangular map may have a recurrent point which is not uniformly recurrent. In this paper we show that neither the converse implication is true by constructing a triangular map of the square with positive topological entropy and with every chain recurrent point uniformly recurrent. In fact we first construct an appropriate minimal positive entropy system whose phase space is a nonhomogeneous subset of the square and then we extend it to a triangular map with required properties in the square.
2006, 14(4): 837-843
doi: 10.3934/dcds.2006.14.837
+[Abstract](1741)
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Abstract:
Any forward-in-time self-similar (localized-in-space) suitable weak solution to the 3D Navier-Stokes equations is shown to be infinitely smooth in both space and time variables. As an application, a proof of infinite space and time regularity of a class of a priori singular small self-similar solutions in the critical weak Lebesgue space $L^{3,\infty}$ is given.
Any forward-in-time self-similar (localized-in-space) suitable weak solution to the 3D Navier-Stokes equations is shown to be infinitely smooth in both space and time variables. As an application, a proof of infinite space and time regularity of a class of a priori singular small self-similar solutions in the critical weak Lebesgue space $L^{3,\infty}$ is given.
2006, 14(4): 845-855
doi: 10.3934/dcds.2006.14.845
+[Abstract](1403)
+[PDF](233.4KB)
Abstract:
In 1954, F. Mautner gave a simple representation theoretic argument that for compact surfaces of constant negative curvature, invariance of a function along the geodesic flow implies invariance along the horocycle flows (these are facts which imply ergodicity of the geodesic flow itself), [M]. Many generalizations of this Mautner phenomenon exist in representation theory, [St1]. Here, we establish a new generalization, Theorem 2.1, whose novelty is mostly its method of proof, namely the Anosov-Hopf ergodicity argument from dynamical systems. Using some structural properties of Lie groups, we also show that stable ergodicity is equivalent to the unique ergodicity of the strong stable manifold foliations in the context of affine diffeomorphisms.
In 1954, F. Mautner gave a simple representation theoretic argument that for compact surfaces of constant negative curvature, invariance of a function along the geodesic flow implies invariance along the horocycle flows (these are facts which imply ergodicity of the geodesic flow itself), [M]. Many generalizations of this Mautner phenomenon exist in representation theory, [St1]. Here, we establish a new generalization, Theorem 2.1, whose novelty is mostly its method of proof, namely the Anosov-Hopf ergodicity argument from dynamical systems. Using some structural properties of Lie groups, we also show that stable ergodicity is equivalent to the unique ergodicity of the strong stable manifold foliations in the context of affine diffeomorphisms.
2006, 14(4): 857-859
doi: 10.3934/dcds.2006.14.857
+[Abstract](1253)
+[PDF](137.8KB)
Abstract:
It has been recently pointed out to us that there is an incorrect statement in the proof of our main theorem in the paper On the free boundary regularity result of Alt and Caffarelli which appeared in Discrete and Continuous Dynamical Systems, Volume 10, 2004, page 397-422. Fortunately, the incorrect statement does not affect the validity of our proof. We would like to set the record straight, with the following errata.
It has been recently pointed out to us that there is an incorrect statement in the proof of our main theorem in the paper On the free boundary regularity result of Alt and Caffarelli which appeared in Discrete and Continuous Dynamical Systems, Volume 10, 2004, page 397-422. Fortunately, the incorrect statement does not affect the validity of our proof. We would like to set the record straight, with the following errata.
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