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1078-0947
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Discrete & Continuous Dynamical Systems - A
June 2008 , Volume 21 , Issue 2
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2008, 21(2): 367-392
doi: 10.3934/dcds.2008.21.367
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Abstract:
We present an entropy formula of Ledrappier-Young type for invariant measures (maybe non-SRB) of $ C^2 $ endomorphisms (maybe non-invertible and with singularities) on a compact manifold via their inverse limit spaces. This result may be considered as the most general form of entropy formula for a deterministic system with an invariant measure, and a preliminary step to Eckmann-Ruelle conjecture. As an important application, we have proved the exact dimensionality of ergodic measures invariant under expanding maps.
We present an entropy formula of Ledrappier-Young type for invariant measures (maybe non-SRB) of $ C^2 $ endomorphisms (maybe non-invertible and with singularities) on a compact manifold via their inverse limit spaces. This result may be considered as the most general form of entropy formula for a deterministic system with an invariant measure, and a preliminary step to Eckmann-Ruelle conjecture. As an important application, we have proved the exact dimensionality of ergodic measures invariant under expanding maps.
2008, 21(2): 393-401
doi: 10.3934/dcds.2008.21.393
+[Abstract](1201)
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Abstract:
It is well known that every Axiom A diffeomorphism defined in the 2-sphere $S^{2}$ has a sink or a source [19]. A natural question is if this property is still true for higher dimensional Axiom A diffeomorphisms and Axiom A vector fields. In this paper we give a negative answer to this question: we prove that for every closed manifold of dimension $n\geq 3$ there are a $C^1$ open set of Axiom A diffeomorphisms and a $C^1$ open set of Axiom A vector fields without sinks and sources. We also show that a sufficient condition for an Axiom A vector field in $S^3$ to exhibit a sink or a source is that every torus in $S^3$ transverse to $X$ is unknotted.
It is well known that every Axiom A diffeomorphism defined in the 2-sphere $S^{2}$ has a sink or a source [19]. A natural question is if this property is still true for higher dimensional Axiom A diffeomorphisms and Axiom A vector fields. In this paper we give a negative answer to this question: we prove that for every closed manifold of dimension $n\geq 3$ there are a $C^1$ open set of Axiom A diffeomorphisms and a $C^1$ open set of Axiom A vector fields without sinks and sources. We also show that a sufficient condition for an Axiom A vector field in $S^3$ to exhibit a sink or a source is that every torus in $S^3$ transverse to $X$ is unknotted.
2008, 21(2): 403-413
doi: 10.3934/dcds.2008.21.403
+[Abstract](1382)
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Abstract:
A Riemannian manifold is said to be uniformly secure if there is a finite number $s$ such that all geodesics connecting an arbitrary pair of points in the manifold can be blocked by $s$ point obstacles. We prove that the number of geodesics with length $\leq T$ between every pair of points in a uniformly secure manifold grows polynomially as $T \to \infty$. By results of Gromov and Mañé, the fundamental group of such a manifold is virtually nilpotent, and the topological entropy of its geodesic flow is zero. Furthermore, if a uniformly secure manifold has no conjugate points, then it is flat. This follows from the virtual nilpotency of its fundamental group either via the theorems of Croke-Schroeder and Burago-Ivanov, or by more recent work of Lebedeva.
We derive from this that a compact Riemannian manifold with no conjugate points whose geodesic flow has positive topological entropy is totally insecure: the geodesics between any pair of points cannot be blocked by a finite number of point obstacles.
A Riemannian manifold is said to be uniformly secure if there is a finite number $s$ such that all geodesics connecting an arbitrary pair of points in the manifold can be blocked by $s$ point obstacles. We prove that the number of geodesics with length $\leq T$ between every pair of points in a uniformly secure manifold grows polynomially as $T \to \infty$. By results of Gromov and Mañé, the fundamental group of such a manifold is virtually nilpotent, and the topological entropy of its geodesic flow is zero. Furthermore, if a uniformly secure manifold has no conjugate points, then it is flat. This follows from the virtual nilpotency of its fundamental group either via the theorems of Croke-Schroeder and Burago-Ivanov, or by more recent work of Lebedeva.
We derive from this that a compact Riemannian manifold with no conjugate points whose geodesic flow has positive topological entropy is totally insecure: the geodesics between any pair of points cannot be blocked by a finite number of point obstacles.
2008, 21(2): 415-443
doi: 10.3934/dcds.2008.21.415
+[Abstract](1377)
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Abstract:
We first prove the existence and uniqueness of pullback and random attractors for abstract multi-valued non-autonomous and random dynamical systems. The standard assumption of compactness of these systems can be replaced by the assumption of asymptotic compactness. Then, we apply the abstract theory to handle a random reaction-diffusion equation with memory or delay terms which can be considered on the complete past defined by $\mathbb{R}^{-}$. In particular, we do not assume the uniqueness of solutions of these equations.
We first prove the existence and uniqueness of pullback and random attractors for abstract multi-valued non-autonomous and random dynamical systems. The standard assumption of compactness of these systems can be replaced by the assumption of asymptotic compactness. Then, we apply the abstract theory to handle a random reaction-diffusion equation with memory or delay terms which can be considered on the complete past defined by $\mathbb{R}^{-}$. In particular, we do not assume the uniqueness of solutions of these equations.
2008, 21(2): 445-475
doi: 10.3934/dcds.2008.21.445
+[Abstract](1270)
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Abstract:
We consider the question of existence of periodic solutions (called breather solutions or discrete solitons) for the Discrete Nonlinear Schrödinger Equation with saturable and power nonlinearity. Theoretical and numerical results are proved concerning the existence and nonexistence of periodic solutions by a variational approach and a fixed point argument. In the variational approach we are restricted to DNLS lattices with Dirichlet boundary conditions. It is proved that there exists parameters (frequency or nonlinearity parameters) for which the corresponding minimizers satisfy explicit upper and lower bounds on the power. The numerical studies performed indicate that these bounds behave as thresholds for the existence of periodic solutions. The fixed point method considers the case of infinite lattices. Through this method, the existence of a threshold is proved in the case of saturable nonlinearity and an explicit theoretical estimate which is independent on the dimension is given. The numerical studies, testing the efficiency of the bounds derived by both methods, demonstrate that these thresholds are quite sharp estimates of a threshold value on the power needed for the the existence of a breather solution. This it justified by the consideration of limiting cases with respect to the size of the nonlinearity parameters and nonlinearity exponents.
We consider the question of existence of periodic solutions (called breather solutions or discrete solitons) for the Discrete Nonlinear Schrödinger Equation with saturable and power nonlinearity. Theoretical and numerical results are proved concerning the existence and nonexistence of periodic solutions by a variational approach and a fixed point argument. In the variational approach we are restricted to DNLS lattices with Dirichlet boundary conditions. It is proved that there exists parameters (frequency or nonlinearity parameters) for which the corresponding minimizers satisfy explicit upper and lower bounds on the power. The numerical studies performed indicate that these bounds behave as thresholds for the existence of periodic solutions. The fixed point method considers the case of infinite lattices. Through this method, the existence of a threshold is proved in the case of saturable nonlinearity and an explicit theoretical estimate which is independent on the dimension is given. The numerical studies, testing the efficiency of the bounds derived by both methods, demonstrate that these thresholds are quite sharp estimates of a threshold value on the power needed for the the existence of a breather solution. This it justified by the consideration of limiting cases with respect to the size of the nonlinearity parameters and nonlinearity exponents.
2008, 21(2): 477-500
doi: 10.3934/dcds.2008.21.477
+[Abstract](1536)
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Abstract:
We introduce a renormalization group framework for the study of quasiperiodic skew flows on Lie groups of real or complex $n\times n$ matrices, for arbitrary Diophantine frequency vectors in $R^{d}$ and dimensions $d,n$. In cases where the Lie algebra component of the vector field is small, it is shown that there exists an analytic manifold of reducible skew systems, for each Diophantine frequency vector. More general near-linear flows are mapped to this case by increasing the dimension of the torus. This strategy is applied for the group of unimodular $2\times 2$ matrices, where the stable manifold is identified with the set of skew systems having a fixed fibered rotation number. Our results apply to vector fields of class Cγ, with $\gamma$ depending on the number of independent frequencies, and on the Diophantine exponent.
We introduce a renormalization group framework for the study of quasiperiodic skew flows on Lie groups of real or complex $n\times n$ matrices, for arbitrary Diophantine frequency vectors in $R^{d}$ and dimensions $d,n$. In cases where the Lie algebra component of the vector field is small, it is shown that there exists an analytic manifold of reducible skew systems, for each Diophantine frequency vector. More general near-linear flows are mapped to this case by increasing the dimension of the torus. This strategy is applied for the group of unimodular $2\times 2$ matrices, where the stable manifold is identified with the set of skew systems having a fixed fibered rotation number. Our results apply to vector fields of class Cγ, with $\gamma$ depending on the number of independent frequencies, and on the Diophantine exponent.
2008, 21(2): 501-512
doi: 10.3934/dcds.2008.21.501
+[Abstract](1384)
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Abstract:
Extending our results of [17], we confirm that Entropy Conjecture holds for every continuous self-map of a compact $K(\pi,1)$ manifold with the fundamental group $\pi$ torsion free and virtually nilpotent, in particular for every continuous map of an infra-nilmanifold. In fact we prove a stronger version, a lower estimate of the topological entropy of a map by the logarithm of the spectral radius of exterior power of an associated "linearization matrix" with integer entries.
From this, referring to known estimates of Mahler measure of polynomials, we deduce some absolute lower bounds for the entropy.
Extending our results of [17], we confirm that Entropy Conjecture holds for every continuous self-map of a compact $K(\pi,1)$ manifold with the fundamental group $\pi$ torsion free and virtually nilpotent, in particular for every continuous map of an infra-nilmanifold. In fact we prove a stronger version, a lower estimate of the topological entropy of a map by the logarithm of the spectral radius of exterior power of an associated "linearization matrix" with integer entries.
From this, referring to known estimates of Mahler measure of polynomials, we deduce some absolute lower bounds for the entropy.
2008, 21(2): 513-535
doi: 10.3934/dcds.2008.21.513
+[Abstract](1420)
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Abstract:
We establish uniqueness of viscosity solutions for some boundary value problems arising from stochastic optimal control problems with unbounded, possibly singular, controls. They involve a nonlinear degenerate second order Bellman-Isaacs equation and mixed boundary conditions (Dirichlet, generalized Dirichlet and state constrained conditions).
We establish uniqueness of viscosity solutions for some boundary value problems arising from stochastic optimal control problems with unbounded, possibly singular, controls. They involve a nonlinear degenerate second order Bellman-Isaacs equation and mixed boundary conditions (Dirichlet, generalized Dirichlet and state constrained conditions).
2008, 21(2): 537-549
doi: 10.3934/dcds.2008.21.537
+[Abstract](1279)
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Abstract:
Given a finite union $P$ of rational simplexes, we assign to $P$ numerical invariants $\lambda_{0}, \lambda_{1},\ldots,\lambda_{\dim P};$ each $\lambda_{i}$ is the suitably normalized volume of the $i$-dimensional part of $P$. We then prove that every finitely generated projective lattice-ordered abelian group $G$ with order-unit $u$ has a faithful invariant positive linear functional $s: G\to \mathbb R$. For each $g\in G$, $s(g)$ is the integral of $g$ over the maximal spectrum of $G$, the latter being canonically identified with a rational polyhedron $P$. Volume elements are measured by the $\lambda_{i}$'s. The proof uses the polyhedral versions of the Włodarczyk-Morelli theorem on decompositions of birational toric maps in blow-ups and blow-downs, and of the De Concini-Procesi theorem on elimination of points of indeterminacy.
Given a finite union $P$ of rational simplexes, we assign to $P$ numerical invariants $\lambda_{0}, \lambda_{1},\ldots,\lambda_{\dim P};$ each $\lambda_{i}$ is the suitably normalized volume of the $i$-dimensional part of $P$. We then prove that every finitely generated projective lattice-ordered abelian group $G$ with order-unit $u$ has a faithful invariant positive linear functional $s: G\to \mathbb R$. For each $g\in G$, $s(g)$ is the integral of $g$ over the maximal spectrum of $G$, the latter being canonically identified with a rational polyhedron $P$. Volume elements are measured by the $\lambda_{i}$'s. The proof uses the polyhedral versions of the Włodarczyk-Morelli theorem on decompositions of birational toric maps in blow-ups and blow-downs, and of the De Concini-Procesi theorem on elimination of points of indeterminacy.
2008, 21(2): 551-569
doi: 10.3934/dcds.2008.21.551
+[Abstract](1249)
+[PDF](271.3KB)
Abstract:
We consider generic properties of Lagrangians. Our main result is a Kupka-Smale Theorem for the Lagrangian setting. We show that for convex and superlinear Lagrangians defined on a compact surface and $k\in \mathbb{R}$, then generically, in Mañé's sense, the energy level $k$ is regular and all periodic orbits in this level are nondegenerate at all orders (the linearized Poincaré map, restricted to this energy level, does not have roots of unity as eigenvalues). Moreover, all heteroclinic intersections in this level are transversal. The results that we present are true in dimension $n \geq 2$, with the exception of Theorem 4.5, which we are only able to prove in dimension 2.
We consider generic properties of Lagrangians. Our main result is a Kupka-Smale Theorem for the Lagrangian setting. We show that for convex and superlinear Lagrangians defined on a compact surface and $k\in \mathbb{R}$, then generically, in Mañé's sense, the energy level $k$ is regular and all periodic orbits in this level are nondegenerate at all orders (the linearized Poincaré map, restricted to this energy level, does not have roots of unity as eigenvalues). Moreover, all heteroclinic intersections in this level are transversal. The results that we present are true in dimension $n \geq 2$, with the exception of Theorem 4.5, which we are only able to prove in dimension 2.
2008, 21(2): 571-593
doi: 10.3934/dcds.2008.21.571
+[Abstract](1459)
+[PDF](273.1KB)
Abstract:
Almost sure asymptotic stability of stochastic difference and differential equations with non-anticipating memory terms is studied in $\R^1$. Sufficient criteria are obtained with help of Lyapunov-Krasovskiĭ-type functionals, martingale decomposition and semi-martingale convergence theorems. The results allow numerical methods for stochastic differential equations with memory to be studied in terms of their ability to reproduce almost sure stability.
Almost sure asymptotic stability of stochastic difference and differential equations with non-anticipating memory terms is studied in $\R^1$. Sufficient criteria are obtained with help of Lyapunov-Krasovskiĭ-type functionals, martingale decomposition and semi-martingale convergence theorems. The results allow numerical methods for stochastic differential equations with memory to be studied in terms of their ability to reproduce almost sure stability.
2008, 21(2): 595-600
doi: 10.3934/dcds.2008.21.595
+[Abstract](1175)
+[PDF](325.8KB)
Abstract:
Let $\lambda \in ( 0,1)$ and $p\in( 0,1)$. Consider the following random sum
Let $\lambda \in ( 0,1)$ and $p\in( 0,1)$. Consider the following random sum
$Y_{\lambda}^p$:=$\sum_{n=0}^{\infty}\pm \lambda ^n$
where the "$+$" and "$-$" signs are chosen independently with
probability $p$ and $1-p$. Let $\nu_\lambda^p$ be the distribution
of the random sum $\nu_\lambda^p(E)$:=$Prob(Y_{\lambda}^p \in
E)$ for every set $E$. The conjecture is that for every $p \in
(0,1)$ the measure $\nu_\lambda^p$ is absolutely continuous w.r.t.
Lebesgue measure and with the density in $L^2(R)$ for almost every
$\lambda\in (p^p\cdot(1-p)^{(1-p)},1).$
B. Solomyak and Y. Peres [3, [Corollary 1.4] proved that
for every $p \in (\frac{1}{3},\frac{2}{3})$ the distribution
$\nu_\lambda^p$ is absolutely continuous with $L^2(R)$ density for
almost every $\lambda \in (p^2+(1-p)^2,1).$ In this paper we
extend the parameter interval where a weakened version of the
conjecture still holds. Namely, we prove Corollary 3 that
for every $p \in (0,\frac{1}{3}]$ the measure
$\nu_\lambda^p$ is absolutely continuous with $L^2(R)$ density for
almost every $\lambda\in(F(p),1)$, where
$F(p)=(1-2p)^{2-\log41/\log 9}$, see Figure 3.
2008, 21(2): 601-623
doi: 10.3934/dcds.2008.21.601
+[Abstract](1615)
+[PDF](273.2KB)
Abstract:
We study the asymptotic exponential stability of traveling front solutions for a general monotone reaction-diffusion bistable system with some diffusion coefficients being zero. The main tools to obtain our results are comparison principle, suitably constructed super-sub solutions, and squeezing methods. No spectrum analysis of the linear operator associated with traveling front solutions under study is needed. Therefore, our results not only recover and/or complement earlier stability results in the literature, but also provide a simple method to show the asymptotic exponential stability of traveling front solutions for a general monotone reaction-diffusion bistable system with positive diffusion coefficients.
We study the asymptotic exponential stability of traveling front solutions for a general monotone reaction-diffusion bistable system with some diffusion coefficients being zero. The main tools to obtain our results are comparison principle, suitably constructed super-sub solutions, and squeezing methods. No spectrum analysis of the linear operator associated with traveling front solutions under study is needed. Therefore, our results not only recover and/or complement earlier stability results in the literature, but also provide a simple method to show the asymptotic exponential stability of traveling front solutions for a general monotone reaction-diffusion bistable system with positive diffusion coefficients.
2008, 21(2): 625-641
doi: 10.3934/dcds.2008.21.625
+[Abstract](1322)
+[PDF](227.9KB)
Abstract:
We consider non quasiconvex functionals of the form
We consider non quasiconvex functionals of the form
$\F(u) = \int_\O [f(x,Du(x))+h(x,u(x))]dx$
defined on Sobolev functions subject to Dirichlet boundary conditions. We give an existence result for minimum points, based on regularity assumptions on the minimizers of the relaxed functional, applying the method of extremization of the integral.
2008, 21(2): 643-663
doi: 10.3934/dcds.2008.21.643
+[Abstract](1281)
+[PDF](277.2KB)
Abstract:
In this paper, we consider the long time behavior of solutions for dissipative lattice dynamical systems with delays. We first prove a sufficient and necessary condition for the existence of a compact uniform attractor for the family of processes corresponding to the lattice dynamical systems with delays. Then we apply this result to prove the existence of a compact uniform attractor for the process associated to the retarded lattice Zakharov equations. As a consequence, some results for the non-delay lattice dynamical systems are deduced as particular cases.
In this paper, we consider the long time behavior of solutions for dissipative lattice dynamical systems with delays. We first prove a sufficient and necessary condition for the existence of a compact uniform attractor for the family of processes corresponding to the lattice dynamical systems with delays. Then we apply this result to prove the existence of a compact uniform attractor for the process associated to the retarded lattice Zakharov equations. As a consequence, some results for the non-delay lattice dynamical systems are deduced as particular cases.
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