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Discrete & Continuous Dynamical Systems - A
March 2003 , Volume 9 , Issue 2
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2003, 9(2): 233-262
doi: 10.3934/dcds.2003.9.233
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Abstract:
Properly degenerate nearly--integrable Hamiltonian systems with two degrees of freedom such that the "intermediate system" depend explicitly upon the angle--variable conjugated to the non--degenerate action--variable are considered and, in particular, model problems motivated by classical examples of Celestial Mechanics, are investigated. Under suitable "convexity" assumptions on the intermediate Hamiltonian, it is proved that, in every energy surface, the action variables stay forever close to their initial values. In "non convex" cases, stability holds up to a small set where, in principle, the degenerate action--variable might (in exponentially long times) drift away from its initial value by a quantity independent of the perturbation. Proofs are based on a "blow up" (complex) analysis near separatrices, KAM techniques and energy conservation arguments.
Properly degenerate nearly--integrable Hamiltonian systems with two degrees of freedom such that the "intermediate system" depend explicitly upon the angle--variable conjugated to the non--degenerate action--variable are considered and, in particular, model problems motivated by classical examples of Celestial Mechanics, are investigated. Under suitable "convexity" assumptions on the intermediate Hamiltonian, it is proved that, in every energy surface, the action variables stay forever close to their initial values. In "non convex" cases, stability holds up to a small set where, in principle, the degenerate action--variable might (in exponentially long times) drift away from its initial value by a quantity independent of the perturbation. Proofs are based on a "blow up" (complex) analysis near separatrices, KAM techniques and energy conservation arguments.
2003, 9(2): 263-280
doi: 10.3934/dcds.2003.9.263
+[Abstract](2102)
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Abstract:
We introduce pointwise dimensions and spectra associated with Poincaré recurrences. These quantities are then calculated for any ergodic measure of positive entropy on a weakly specified subshift. We show that they satisfy a relation comparable to Young's formula for the Hausdorff dimension of measures invariant under surface diffeomorphisms. A key-result in establishing these formula is to prove that the Poincaré recurrence for a 'typical' cylinder is asymptotically its length. Examples are provided which show that this is not true for some systems with zero entropy. Similar results are obtained for special flows and we get a formula relating spectra for measures of the base to the ones of the flow.
We introduce pointwise dimensions and spectra associated with Poincaré recurrences. These quantities are then calculated for any ergodic measure of positive entropy on a weakly specified subshift. We show that they satisfy a relation comparable to Young's formula for the Hausdorff dimension of measures invariant under surface diffeomorphisms. A key-result in establishing these formula is to prove that the Poincaré recurrence for a 'typical' cylinder is asymptotically its length. Examples are provided which show that this is not true for some systems with zero entropy. Similar results are obtained for special flows and we get a formula relating spectra for measures of the base to the ones of the flow.
2003, 9(2): 281-286
doi: 10.3934/dcds.2003.9.281
+[Abstract](2443)
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Abstract:
We prove the existence of a compact $(L^2-H^1)$ attractor for a reaction-diffusion equation in $R^N$. This improves a previous result of B. Wang concerning the existence of a compact $(L^2-L^2)$ attractor for the same equation.
We prove the existence of a compact $(L^2-H^1)$ attractor for a reaction-diffusion equation in $R^N$. This improves a previous result of B. Wang concerning the existence of a compact $(L^2-L^2)$ attractor for the same equation.
2003, 9(2): 287-308
doi: 10.3934/dcds.2003.9.287
+[Abstract](2367)
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Abstract:
We study weak and orbital shadowing properties of dynamical systems related to the following approach: we look for exact trajectories lying in small neighborhoods of approximate ones (or containing approximate ones in their small neighborhoods) or for exact trajectories such that the Hausdorff distances between their closures and closures of approximate trajectories are small.
These properties are characterized for linear diffeomorphisms. We also study some $C^1$-open sets of diffeomorphisms defined in terms of these properties. It is shown that the $C^1$-interior of the set of diffeomorphisms having the orbital shadowing property coincides with the set of structurally stable diffeomorphisms.
We study weak and orbital shadowing properties of dynamical systems related to the following approach: we look for exact trajectories lying in small neighborhoods of approximate ones (or containing approximate ones in their small neighborhoods) or for exact trajectories such that the Hausdorff distances between their closures and closures of approximate trajectories are small.
These properties are characterized for linear diffeomorphisms. We also study some $C^1$-open sets of diffeomorphisms defined in terms of these properties. It is shown that the $C^1$-interior of the set of diffeomorphisms having the orbital shadowing property coincides with the set of structurally stable diffeomorphisms.
2003, 9(2): 309-321
doi: 10.3934/dcds.2003.9.309
+[Abstract](2103)
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Abstract:
We prove that the well-known $3/2$ stability condition established for the Wright equation (WE) still holds if the nonlinearity $p(\exp(-x)-1)$ in WE is replaced by a decreasing or unimodal smooth function $f$ with $f'(0)<0$ satisfying the standard negative feedback and below boundedness conditions and having everywhere negative Schwarz derivative.
We prove that the well-known $3/2$ stability condition established for the Wright equation (WE) still holds if the nonlinearity $p(\exp(-x)-1)$ in WE is replaced by a decreasing or unimodal smooth function $f$ with $f'(0)<0$ satisfying the standard negative feedback and below boundedness conditions and having everywhere negative Schwarz derivative.
2003, 9(2): 323-338
doi: 10.3934/dcds.2003.9.323
+[Abstract](1852)
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Abstract:
We describe new examples of global attractors arising in planar piecewise rotations with two convex atoms. The dynamics inside these attractors is proved to be equivalent to that from models of digital filters.
We also discuss some subtleties on the definition of piecewise isometric attractors and their properties, motivated not only by our new examples but also by others existing in the literature. More precisely, we require a minimality condition so that uniqueness is guaranteed and also, we establish equivalent forms of attractivity under some regularity assumption. The notion of quasi-invariance is introduced as it proves to be a suitable concept in the context of planar piecewise rotations.
We describe new examples of global attractors arising in planar piecewise rotations with two convex atoms. The dynamics inside these attractors is proved to be equivalent to that from models of digital filters.
We also discuss some subtleties on the definition of piecewise isometric attractors and their properties, motivated not only by our new examples but also by others existing in the literature. More precisely, we require a minimality condition so that uniqueness is guaranteed and also, we establish equivalent forms of attractivity under some regularity assumption. The notion of quasi-invariance is introduced as it proves to be a suitable concept in the context of planar piecewise rotations.
2003, 9(2): 339-358
doi: 10.3934/dcds.2003.9.339
+[Abstract](2151)
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Abstract:
We consider the equation
We consider the equation
$\alpha x''(t)=-x'(t)+F(x(t),t)-$sign$x(t-h),\quad\alpha=$const$>0,\ $ $h=$const$>0,$
which is a model for a scalar system with a discontinuous negative delayed feedback, and study the dynamics of oscillations with emphasis on the existence, frequency and stability of periodic oscillations. Our main conclusion is that, in the autonomous case $F(x,t)\equiv F(x)$, for $|F(x)|<1$, there are periodic solutions with different frequencies of oscillations, though only slowly-oscillating solutions are (orbitally) stable. Under extra conditions we show the uniqueness of a periodic slowly-oscillating solution. We also give a criterion for the existence of bounded oscillations in the case of unbounded function $F(x,t)$. Our approach consists basically in reducing the problem to the study of dynamics of some discrete scalar system.
2003, 9(2): 359-378
doi: 10.3934/dcds.2003.9.359
+[Abstract](1718)
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Abstract:
We consider Birkhoff averages of an observable $\phi$ along orbits of a continuous map $f:X \rightarrow X$ with respect to a non-invariant measure $m$. In the simple case where the averages converge $m$-almost everywhere, one may discuss the distribution of values of the average in a natural way. We extend this analysis to the case where convergence does not hold $m$-almost everywhere. The case that the averages converge $m$-almost everywhere is shown to be related to the recently defined notion of "predictable" behavior, which is a condition on the existence of pointwise asymptotic measures (SRB measures). A heteroclinic attractor is an example of a system which is not predictable. We define a more general notion called "statistically predictable" behavior which is weaker than predictability, but is strong enough to allow meaningful statistical properties, i.e. distribution of Birkhoff averages, to be analyzed. Statistical predictability is shown to imply the existence of an asymptotic measure, but not vice versa. We investigate the relationship between the various notions of asymptotic measures and distributions of Birkhoff average. Analysis of the heteroclinic attractor is used to illustrate the applicability of the concepts.
We consider Birkhoff averages of an observable $\phi$ along orbits of a continuous map $f:X \rightarrow X$ with respect to a non-invariant measure $m$. In the simple case where the averages converge $m$-almost everywhere, one may discuss the distribution of values of the average in a natural way. We extend this analysis to the case where convergence does not hold $m$-almost everywhere. The case that the averages converge $m$-almost everywhere is shown to be related to the recently defined notion of "predictable" behavior, which is a condition on the existence of pointwise asymptotic measures (SRB measures). A heteroclinic attractor is an example of a system which is not predictable. We define a more general notion called "statistically predictable" behavior which is weaker than predictability, but is strong enough to allow meaningful statistical properties, i.e. distribution of Birkhoff averages, to be analyzed. Statistical predictability is shown to imply the existence of an asymptotic measure, but not vice versa. We investigate the relationship between the various notions of asymptotic measures and distributions of Birkhoff average. Analysis of the heteroclinic attractor is used to illustrate the applicability of the concepts.
2003, 9(2): 379-382
doi: 10.3934/dcds.2003.9.379
+[Abstract](1502)
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Abstract:
Let $f:\hat \mathbf C\rightarrow \hat \mathbf C$ be a rational map of degree $n\geq 3$ and with exactly two critical points. Assume that the Julia set $J(f)$ is a proper subcontinuum of $\hat \mathbf C$ and there is no completely invariant Fatou component under the iterates $f^{2}$. It is shown that if there is no buried points in $J(f)$, then the Julia set $J(f)$ is a Lakes of Wada continuum, and hence is either an indecomposable continuum or the union of two indecomposable continua.
Let $f:\hat \mathbf C\rightarrow \hat \mathbf C$ be a rational map of degree $n\geq 3$ and with exactly two critical points. Assume that the Julia set $J(f)$ is a proper subcontinuum of $\hat \mathbf C$ and there is no completely invariant Fatou component under the iterates $f^{2}$. It is shown that if there is no buried points in $J(f)$, then the Julia set $J(f)$ is a Lakes of Wada continuum, and hence is either an indecomposable continuum or the union of two indecomposable continua.
2003, 9(2): 383-397
doi: 10.3934/dcds.2003.9.383
+[Abstract](1831)
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Abstract:
Connections between admissibility and uniform exponential dichotomy of discrete evolution families are studied. Discrete and continuous characterizations for uniform exponential dichotomy of evolution families are given. A new version for a theorem due to Van Minh, Räbiger and Schnaubelt, for the discrete case is obtained.
Connections between admissibility and uniform exponential dichotomy of discrete evolution families are studied. Discrete and continuous characterizations for uniform exponential dichotomy of evolution families are given. A new version for a theorem due to Van Minh, Räbiger and Schnaubelt, for the discrete case is obtained.
2003, 9(2): 399-412
doi: 10.3934/dcds.2003.9.399
+[Abstract](2075)
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Abstract:
We prove the existence of kernel sections for the process generated by a damped non-autonomous wave equation when there is nonlinear damping and the nonlinearity has a critically growing exponent. We show uniform boundedness of the Hausdorff dimension of the kernel sections. Finally, we point out that in the case of autonomous systems with linear damping, the obtained upper bound of the Hausdorff dimension decreases as the damping grows for suitable large damping.
We prove the existence of kernel sections for the process generated by a damped non-autonomous wave equation when there is nonlinear damping and the nonlinearity has a critically growing exponent. We show uniform boundedness of the Hausdorff dimension of the kernel sections. Finally, we point out that in the case of autonomous systems with linear damping, the obtained upper bound of the Hausdorff dimension decreases as the damping grows for suitable large damping.
2003, 9(2): 413-426
doi: 10.3934/dcds.2003.9.413
+[Abstract](2029)
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Abstract:
We consider the planar pendulum with support point oscillating in the vertical direction; the upside-down position of the pendulum corresponds to an equilibrium point for the projection of the motion on the pendulum phase space. By using the Lindstedt series method recently developed in literature starting from the pioneering work by Eliasson, we show that such an equilibrium point is stable for a full measure subset of the stability region of the linearized system inside the two-dimensional space of parameters, by proving the persistence of invariant KAM tori for the two-dimensional Hamiltonian system describing the model.
We consider the planar pendulum with support point oscillating in the vertical direction; the upside-down position of the pendulum corresponds to an equilibrium point for the projection of the motion on the pendulum phase space. By using the Lindstedt series method recently developed in literature starting from the pioneering work by Eliasson, we show that such an equilibrium point is stable for a full measure subset of the stability region of the linearized system inside the two-dimensional space of parameters, by proving the persistence of invariant KAM tori for the two-dimensional Hamiltonian system describing the model.
2003, 9(2): 427-442
doi: 10.3934/dcds.2003.9.427
+[Abstract](2378)
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Abstract:
We prove that Strichartz-type $L^p$ estimates hold for solutions of the linear wave equation with the inverse square potential, under the additional assumption that the Cauchy data are spherically symmetric. The estimates are then applied to prove global well-posedness in the critical norm for a nonlinear wave equation.
We prove that Strichartz-type $L^p$ estimates hold for solutions of the linear wave equation with the inverse square potential, under the additional assumption that the Cauchy data are spherically symmetric. The estimates are then applied to prove global well-posedness in the critical norm for a nonlinear wave equation.
2003, 9(2): 443-450
doi: 10.3934/dcds.2003.9.443
+[Abstract](1775)
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Abstract:
For points $x$ belonging to a basic set $\Lambda$ of an Axiom A holomorphic endomorphism of $\mathbb P^2$, one can construct the local stable manifold $W_{\varepsilon_0}^s(x)$ and the local unstable manifolds $W_{\varepsilon_0}^u(\hat x)$. A priori, $W_{\varepsilon_0}^u(\hat x)$ should depend on the entire prehistory $\hat x$ of $x$. However, all known examples have all their local unstable manifolds depending only on the base point $x$. Therefore a natural problem is to give actual examples where, for different prehistories of points in the basic sets of holomorphic Axiom A maps, we obtain different unstable manifolds. We solve this problem by considering the map $(z^4+\varepsilon w^2, w^4)$ and then also show that, by perturbing $(z^2+c, w^2)$, one can get also maps $f_\varepsilon$ which are injective on $\Lambda_\varepsilon$, their corresponding basic sets, hence the cardinality of the set $(f_\varepsilon|_{\Lambda_\varepsilon})^{-1}(x), x \in \Lambda_\varepsilon$, is not stable under perturbation.
For points $x$ belonging to a basic set $\Lambda$ of an Axiom A holomorphic endomorphism of $\mathbb P^2$, one can construct the local stable manifold $W_{\varepsilon_0}^s(x)$ and the local unstable manifolds $W_{\varepsilon_0}^u(\hat x)$. A priori, $W_{\varepsilon_0}^u(\hat x)$ should depend on the entire prehistory $\hat x$ of $x$. However, all known examples have all their local unstable manifolds depending only on the base point $x$. Therefore a natural problem is to give actual examples where, for different prehistories of points in the basic sets of holomorphic Axiom A maps, we obtain different unstable manifolds. We solve this problem by considering the map $(z^4+\varepsilon w^2, w^4)$ and then also show that, by perturbing $(z^2+c, w^2)$, one can get also maps $f_\varepsilon$ which are injective on $\Lambda_\varepsilon$, their corresponding basic sets, hence the cardinality of the set $(f_\varepsilon|_{\Lambda_\varepsilon})^{-1}(x), x \in \Lambda_\varepsilon$, is not stable under perturbation.
2003, 9(2): 451-470
doi: 10.3934/dcds.2003.9.451
+[Abstract](2264)
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Abstract:
We consider three degrees of freedom initially hyperbolic Hamiltonian systems $H_\mu$, where $0<\mu <$$<1$ is the perturbing parameter. We prove that, under some technical assumptions, the Arnold diffusion time can be of order $(1/\mu)$log$(1/\mu)$, as conjectured by P. Lochak.
Our method is based on the construction of a dual chain of hyperbolic periodic orbits surrounding a given transition chain of partially hyperbolic tori, whose parameters (angles, periods) can be related to parameters (diophantine condition, angles) of the original chain of tori. Using Easton's method of windows, we give a general formula for the time of drift along such a chain of hyperbolic periodic orbits. We then deduce the result for chain of partially hyperbolic tori.
We consider three degrees of freedom initially hyperbolic Hamiltonian systems $H_\mu$, where $0<\mu <$$<1$ is the perturbing parameter. We prove that, under some technical assumptions, the Arnold diffusion time can be of order $(1/\mu)$log$(1/\mu)$, as conjectured by P. Lochak.
Our method is based on the construction of a dual chain of hyperbolic periodic orbits surrounding a given transition chain of partially hyperbolic tori, whose parameters (angles, periods) can be related to parameters (diophantine condition, angles) of the original chain of tori. Using Easton's method of windows, we give a general formula for the time of drift along such a chain of hyperbolic periodic orbits. We then deduce the result for chain of partially hyperbolic tori.
2003, 9(2): 471-482
doi: 10.3934/dcds.2003.9.471
+[Abstract](1912)
+[PDF](222.1KB)
Abstract:
In this paper, we treat the coupled system of wave equations whose nonlinearities are $|u|^{p_j}|v|^{q_j}$ and propagation speeds may be different from each other. We study the lower bounds of $p_j$ and $q_j$ to assure the global existence of a class of small amplitude solutions which includes self-similar solutions. The exponent of self-similar solutions plays crucial role to find the lower bounds. Moreover, we prove that the discrepancy of propagation speeds allow us to bring them down. Conversely, if such conditions for the global existence do not hold, then no self-similar solution exists even for small initial data.
In this paper, we treat the coupled system of wave equations whose nonlinearities are $|u|^{p_j}|v|^{q_j}$ and propagation speeds may be different from each other. We study the lower bounds of $p_j$ and $q_j$ to assure the global existence of a class of small amplitude solutions which includes self-similar solutions. The exponent of self-similar solutions plays crucial role to find the lower bounds. Moreover, we prove that the discrepancy of propagation speeds allow us to bring them down. Conversely, if such conditions for the global existence do not hold, then no self-similar solution exists even for small initial data.
2003, 9(2): 483-496
doi: 10.3934/dcds.2003.9.483
+[Abstract](2055)
+[PDF](173.5KB)
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In this paper, we study the relationship between flow-invariant sets for an vector field $-f'(x)$ in a Banach space, and the critical points of the functional $f(x)$. The Mountain-Pass Lemma, for functionals defined on a Banach space, is generalized to a more general setting where the domain of the functional $f$ can be any flow-invariant set for $-f'(x)$. Furthermore, the intuitive approach taken in the proofs provides a new technique in proving multiple critical points.
In this paper, we study the relationship between flow-invariant sets for an vector field $-f'(x)$ in a Banach space, and the critical points of the functional $f(x)$. The Mountain-Pass Lemma, for functionals defined on a Banach space, is generalized to a more general setting where the domain of the functional $f$ can be any flow-invariant set for $-f'(x)$. Furthermore, the intuitive approach taken in the proofs provides a new technique in proving multiple critical points.
2003, 9(2): 497-503
doi: 10.3934/dcds.2003.9.497
+[Abstract](1500)
+[PDF](147.6KB)
Abstract:
The aim of this paper is to give a topological characterization of the sets that can be $\omega$-limit of continuous dynamical systems in the Klein bottle. This problem has been recently solved for the case of the projective plane answering a problem proposed by Anosov.
The aim of this paper is to give a topological characterization of the sets that can be $\omega$-limit of continuous dynamical systems in the Klein bottle. This problem has been recently solved for the case of the projective plane answering a problem proposed by Anosov.
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