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1078-0947
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Discrete & Continuous Dynamical Systems - A
November 2003 , Volume 9 , Issue 6
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2003, 9(6): 1361-1386
doi: 10.3934/dcds.2003.9.1361
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Abstract:
In this paper we investigate randomly perturbed orbits. If a dynamical system is hyperbolic one can keep random perturbations from accumulating into large deviations by making small corrections. We study the converse problem. This leads naturally to the notion of sustainable orbits.
In this paper we investigate randomly perturbed orbits. If a dynamical system is hyperbolic one can keep random perturbations from accumulating into large deviations by making small corrections. We study the converse problem. This leads naturally to the notion of sustainable orbits.
2003, 9(6): 1387-1400
doi: 10.3934/dcds.2003.9.1387
+[Abstract](1959)
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Abstract:
We prove that spherically symmetric solutions of the Cauchy problem for the linear wave equation with the inverse-square potential satisfy a modified dispersive inequality that bounds the $L^\infty$ norm of the solution in terms of certain Besov norms of the data, with a factor that decays in $t$ for positive potentials. When the potential is negative we show that the decay is split between $t$ and $r$, and the estimate blows up at $r=0$. We also provide a counterexample showing that the use of Besov norms in dispersive inequalities for the wave equation are in general unavoidable.
We prove that spherically symmetric solutions of the Cauchy problem for the linear wave equation with the inverse-square potential satisfy a modified dispersive inequality that bounds the $L^\infty$ norm of the solution in terms of certain Besov norms of the data, with a factor that decays in $t$ for positive potentials. When the potential is negative we show that the decay is split between $t$ and $r$, and the estimate blows up at $r=0$. We also provide a counterexample showing that the use of Besov norms in dispersive inequalities for the wave equation are in general unavoidable.
2003, 9(6): 1401-1409
doi: 10.3934/dcds.2003.9.1401
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We present analogies between Diophantine conditions appearing in the theory of Small Divisors and classical Transcendental Number Theory. Let K be a number field. Using Bertrand's postulate, we give a simple proof that $e$ is transcendental over Liouville fields K$(\theta)$ where $\theta $ is a Liouville number with explicit very good rational approximations. The result extends to any Liouville field K$(\Theta )$ generated by a family $\Theta$ of Liouville numbers satisfying a Diophantine condition (the transcendence degree can be uncountable). This Diophantine condition is similar to the one appearing in Moser's theorem of simultanneous linearization of commuting holomorphic germs.
We present analogies between Diophantine conditions appearing in the theory of Small Divisors and classical Transcendental Number Theory. Let K be a number field. Using Bertrand's postulate, we give a simple proof that $e$ is transcendental over Liouville fields K$(\theta)$ where $\theta $ is a Liouville number with explicit very good rational approximations. The result extends to any Liouville field K$(\Theta )$ generated by a family $\Theta$ of Liouville numbers satisfying a Diophantine condition (the transcendence degree can be uncountable). This Diophantine condition is similar to the one appearing in Moser's theorem of simultanneous linearization of commuting holomorphic germs.
2003, 9(6): 1411-1422
doi: 10.3934/dcds.2003.9.1411
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In this paper, some new dynamical systems which are determined by a semigroup $\Phi$ of maps in a closed interval $I$ are studied.The main peculiarity of these systems is that $\Phi$ is generated by two noncommuting maps. Introducing certain closed subsets $\mathcal T_1$ and $\mathcal T_2$ in $I$ makes it possible to determine some specific orbits corresponding to $\Phi$ and some specific attractors in $I$. These orbits play a crucial role in solving a wide variety problems in such diverse fields of analysis as functional and functional-integral equations, integral geometry, boundary problems for hyperbolic partial differential equations of higher $(>2)$ order. In the first part of this work we describe some conditions which ensure the existence of attractors in question of a special structure. In the second part several new problems in the above-mentioned fields of analysis are formulated, and we trace how the above dynamic approach works in solving this problems.
In this paper, some new dynamical systems which are determined by a semigroup $\Phi$ of maps in a closed interval $I$ are studied.The main peculiarity of these systems is that $\Phi$ is generated by two noncommuting maps. Introducing certain closed subsets $\mathcal T_1$ and $\mathcal T_2$ in $I$ makes it possible to determine some specific orbits corresponding to $\Phi$ and some specific attractors in $I$. These orbits play a crucial role in solving a wide variety problems in such diverse fields of analysis as functional and functional-integral equations, integral geometry, boundary problems for hyperbolic partial differential equations of higher $(>2)$ order. In the first part of this work we describe some conditions which ensure the existence of attractors in question of a special structure. In the second part several new problems in the above-mentioned fields of analysis are formulated, and we trace how the above dynamic approach works in solving this problems.
2003, 9(6): 1423-1446
doi: 10.3934/dcds.2003.9.1423
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In this paper we study the role of uniform Bernoulli measure in the dynamics of cellular automata of algebraic origin.
First we show a representation result for classes of permutative cellular automata: those with associative type local rule are the product of a group cellular automaton with a translation map, and if they satisfy a scaling condition, they are the product of an affine cellular automaton (the alphabet is an Abelian group) with a translation map.
For cellular automata of this type with an Abelian factor group, and starting from a translation invariant probability measure with complete connections and summable decay, it is shown that the Cesàro mean of the iteration of this measure by the cellular automaton converges to the product of the uniform Bernoulli measure with a shift invariant measure.
Finally, the following characterization is shown for affine cellular automaton whose alphabet is a group of prime order: the uniform Bernoulli measure is the unique invariant probability measure which has positive entropy for the automaton, and is either ergodic for the shift or ergodic for the $\mathbb Z^2$-action induced by the shift and the automaton, together with a condition on the rational eigenvalues of the automaton.
In this paper we study the role of uniform Bernoulli measure in the dynamics of cellular automata of algebraic origin.
First we show a representation result for classes of permutative cellular automata: those with associative type local rule are the product of a group cellular automaton with a translation map, and if they satisfy a scaling condition, they are the product of an affine cellular automaton (the alphabet is an Abelian group) with a translation map.
For cellular automata of this type with an Abelian factor group, and starting from a translation invariant probability measure with complete connections and summable decay, it is shown that the Cesàro mean of the iteration of this measure by the cellular automaton converges to the product of the uniform Bernoulli measure with a shift invariant measure.
Finally, the following characterization is shown for affine cellular automaton whose alphabet is a group of prime order: the uniform Bernoulli measure is the unique invariant probability measure which has positive entropy for the automaton, and is either ergodic for the shift or ergodic for the $\mathbb Z^2$-action induced by the shift and the automaton, together with a condition on the rational eigenvalues of the automaton.
2003, 9(6): 1447-1464
doi: 10.3934/dcds.2003.9.1447
+[Abstract](1542)
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We prove that a non uniformly expanding one-dimensional system defined by an interval map with an ergodic non atomic Borel probability $\mu$ with positive Lyapunov exponent can be reduced to a Markov tower with good fractal geometrical properties. As a consequence we approximate $\mu$ by ergodic measures supported on hyperbolic Cantor sets of arbitrarily large dimension.
We prove that a non uniformly expanding one-dimensional system defined by an interval map with an ergodic non atomic Borel probability $\mu$ with positive Lyapunov exponent can be reduced to a Markov tower with good fractal geometrical properties. As a consequence we approximate $\mu$ by ergodic measures supported on hyperbolic Cantor sets of arbitrarily large dimension.
2003, 9(6): 1465-1492
doi: 10.3934/dcds.2003.9.1465
+[Abstract](1480)
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This paper is concerned with the classical Nicholson-Bailey model [15] defined by $f_\lambda(x,y)=(y(1-e^{-x}), \lambda y e^{-x})$. We show that for $\lambda=1$ a heteroclinic foliation exists and for $\lambda>1$ global strict oscillations take place. The important phenomenon of delay of stability loss is established for a general class of discrete dynamical systems, and it is applied to the study of nonexistence of periodic orbits for the Nicholson-Bailey model.
This paper is concerned with the classical Nicholson-Bailey model [15] defined by $f_\lambda(x,y)=(y(1-e^{-x}), \lambda y e^{-x})$. We show that for $\lambda=1$ a heteroclinic foliation exists and for $\lambda>1$ global strict oscillations take place. The important phenomenon of delay of stability loss is established for a general class of discrete dynamical systems, and it is applied to the study of nonexistence of periodic orbits for the Nicholson-Bailey model.
2003, 9(6): 1493-1518
doi: 10.3934/dcds.2003.9.1493
+[Abstract](1726)
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The main results give hypotheses ensuring that a non-autonomous first order Hamiltonian system has a global branch of homoclinic solutions bifurcating from an eigenvalue of odd multiplicity of the linearization. The system is required to be asymptotically periodic (as time goes to plus and minus infinity) and these limit problems should have no homoclinic solutions. Furthermore, the asymptotic limits of the linearization should have no characteristic multipliers on the unit circle. The proof uses the topological degree for proper Fredholm maps of index zero.
The main results give hypotheses ensuring that a non-autonomous first order Hamiltonian system has a global branch of homoclinic solutions bifurcating from an eigenvalue of odd multiplicity of the linearization. The system is required to be asymptotically periodic (as time goes to plus and minus infinity) and these limit problems should have no homoclinic solutions. Furthermore, the asymptotic limits of the linearization should have no characteristic multipliers on the unit circle. The proof uses the topological degree for proper Fredholm maps of index zero.
2003, 9(6): 1519-1532
doi: 10.3934/dcds.2003.9.1519
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This paper deals with a reaction-diffusion system with nonlocal sources. Under appropriate hypotheses, we obtain that the solution either exists globally or blows up in finite time by making use of super and sub solution techniques. In the situation when the solution blows up in finite time, we show that the blow-up set is the whole domain, which is quite different from the results with local sources. Furthermore, we obtain the blow-up rate of the solution.
This paper deals with a reaction-diffusion system with nonlocal sources. Under appropriate hypotheses, we obtain that the solution either exists globally or blows up in finite time by making use of super and sub solution techniques. In the situation when the solution blows up in finite time, we show that the blow-up set is the whole domain, which is quite different from the results with local sources. Furthermore, we obtain the blow-up rate of the solution.
2003, 9(6): 1533-1548
doi: 10.3934/dcds.2003.9.1533
+[Abstract](1561)
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In this paper a class of linear maps on the 2-torus and some planar piecewise isometries are discussed. For these discontinuous maps, by introducing codings underlying the map operations, symbolic descriptions of the dynamics and admissibility conditions for itineraries are given, and explicit expressions in terms of the codings for periodic points are presented.
In this paper a class of linear maps on the 2-torus and some planar piecewise isometries are discussed. For these discontinuous maps, by introducing codings underlying the map operations, symbolic descriptions of the dynamics and admissibility conditions for itineraries are given, and explicit expressions in terms of the codings for periodic points are presented.
2003, 9(6): 1549-1570
doi: 10.3934/dcds.2003.9.1549
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We consider time optimal control problems governed by semilinear parabolic equations with Dirichlet boundary controls in the presence of a target state constraint. To establish optimality conditions for the terminal time $T$, we define a new Hamiltonian functional. Due to regularity results for the state and the adjoint state variables, this Hamiltonian belongs to $L_{l o c}^r(0,T)$ for some $r>1$. By proving that it satisfies a differential equation corresponding to an optimality condition for $T$, we deduce that it belongs to $W^{1,1}(0,T)$. This result answers to the question: how to define Hamiltonian functionals for infinite dimensional problems with variable endpoints (see [10], p. 282 and p. 595).
We consider time optimal control problems governed by semilinear parabolic equations with Dirichlet boundary controls in the presence of a target state constraint. To establish optimality conditions for the terminal time $T$, we define a new Hamiltonian functional. Due to regularity results for the state and the adjoint state variables, this Hamiltonian belongs to $L_{l o c}^r(0,T)$ for some $r>1$. By proving that it satisfies a differential equation corresponding to an optimality condition for $T$, we deduce that it belongs to $W^{1,1}(0,T)$. This result answers to the question: how to define Hamiltonian functionals for infinite dimensional problems with variable endpoints (see [10], p. 282 and p. 595).
2003, 9(6): 1571-1586
doi: 10.3934/dcds.2003.9.1571
+[Abstract](1371)
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We study the scattering theory for the coupled Wave-Schrödinger equation with the Yukawa type interaction, which is certain quadratic interaction, in three space dimensions. This equation belongs to the borderline between the short range case and the long range one. We construct modified wave operators for that equation for small scattered states with no restriction on the support of the Fourier transform of them.
We study the scattering theory for the coupled Wave-Schrödinger equation with the Yukawa type interaction, which is certain quadratic interaction, in three space dimensions. This equation belongs to the borderline between the short range case and the long range one. We construct modified wave operators for that equation for small scattered states with no restriction on the support of the Fourier transform of them.
2003, 9(6): 1587-1606
doi: 10.3934/dcds.2003.9.1587
+[Abstract](1454)
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We study a nonlinear system of partial differential equations that is a viscous approximation for a multidimensional unsteady Euler potential flow governed by the conservation of mass and the Bernoulli law. The system consists of a transport equation for the density and the viscous nonhomogeneous Hamilton-Jacobi equation for the velocity potential. We establish the existence and regularity of global solutions for the nonlinear system with arbitrarily large periodic initial data. We also prove that the density in our global solutions has a positive lower bound, that is, our solutions always stay away from the vacuum, as long as the initial density has a positive lower bound.
We study a nonlinear system of partial differential equations that is a viscous approximation for a multidimensional unsteady Euler potential flow governed by the conservation of mass and the Bernoulli law. The system consists of a transport equation for the density and the viscous nonhomogeneous Hamilton-Jacobi equation for the velocity potential. We establish the existence and regularity of global solutions for the nonlinear system with arbitrarily large periodic initial data. We also prove that the density in our global solutions has a positive lower bound, that is, our solutions always stay away from the vacuum, as long as the initial density has a positive lower bound.
2003, 9(6): 1607-1624
doi: 10.3934/dcds.2003.9.1607
+[Abstract](1582)
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The generalized complexity of an orbit of a dynamical system is defined by the asymptotic behavior of the information that is necessary to describe $n$ steps of the orbit as $n$ increases. This local complexity indicator is also invariant up to topological conjugation and is suited for the study of $0$-entropy dynamical systems. First, we state a criterion to find systems with "non trivial" orbit complexity. Then, we consider also a global indicator of the complexity of the system. This global indicator generalizes the topological entropy, having non trivial values for systems were the number of essentially different orbits increases less than exponentially. Then we prove that if the system is constructive ( if the map can be defined up to any given accuracy by some algorithm) the orbit complexity is everywhere less or equal than the generalized topological entropy. Conversely there are compact non constructive examples where the inequality is reversed, suggesting that the notion of constructive map comes out naturally in this kind of complexity questions.
The generalized complexity of an orbit of a dynamical system is defined by the asymptotic behavior of the information that is necessary to describe $n$ steps of the orbit as $n$ increases. This local complexity indicator is also invariant up to topological conjugation and is suited for the study of $0$-entropy dynamical systems. First, we state a criterion to find systems with "non trivial" orbit complexity. Then, we consider also a global indicator of the complexity of the system. This global indicator generalizes the topological entropy, having non trivial values for systems were the number of essentially different orbits increases less than exponentially. Then we prove that if the system is constructive ( if the map can be defined up to any given accuracy by some algorithm) the orbit complexity is everywhere less or equal than the generalized topological entropy. Conversely there are compact non constructive examples where the inequality is reversed, suggesting that the notion of constructive map comes out naturally in this kind of complexity questions.
2003, 9(6): 1625-1639
doi: 10.3934/dcds.2003.9.1625
+[Abstract](2753)
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We consider a nonlinear Timoshenko system as an initial-boundary value problem in a one-dimensional bounded domain. The system has a dissipative mechanism through frictional damping being present only in the equation for the rotation angle. We first give an alternative proof for a sufficient and necessary condition for exponential stability for the linear case. Polynomial stability is proved in general. The global existence of small, smooth solutions and the exponential stability is investigated for the nonlinear case.
We consider a nonlinear Timoshenko system as an initial-boundary value problem in a one-dimensional bounded domain. The system has a dissipative mechanism through frictional damping being present only in the equation for the rotation angle. We first give an alternative proof for a sufficient and necessary condition for exponential stability for the linear case. Polynomial stability is proved in general. The global existence of small, smooth solutions and the exponential stability is investigated for the nonlinear case.
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