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Discrete & Continuous Dynamical Systems - A
April 2011 , Volume 30 , Issue 1
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2011, 30(1): 1-16
doi: 10.3934/dcds.2011.30.1
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Abstract:
This paper is a first step in the study of the recurrence behaviour in random dynamical systems and randomly perturbed dynamical systems. In particular we define a concept of quenched and annealed return times for systems generated by the composition of random maps. We moreover prove that for super-polynomially mixing systems, the random recurrence rate is equal to the local dimension of the stationary measure.
This paper is a first step in the study of the recurrence behaviour in random dynamical systems and randomly perturbed dynamical systems. In particular we define a concept of quenched and annealed return times for systems generated by the composition of random maps. We moreover prove that for super-polynomially mixing systems, the random recurrence rate is equal to the local dimension of the stationary measure.
2011, 30(1): 17-37
doi: 10.3934/dcds.2011.30.17
+[Abstract](1941)
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Abstract:
We study Markov processes generated by iterated function systems (IFS). The constituent maps of the IFS are monotonic transformations of the interval. We first obtain an upper bound on the number of SRB (Sinai-Ruelle-Bowen) measures for the IFS. Then, when all the constituent maps have common fixed points at 0 and 1, theorems are given to analyze properties of the ergodic invariant measures $\delta_0$ and $\delta_1$. In particular, sufficient conditions for $\delta_0$ and/or $\delta_1$ to be, or not to be, SRB measures are given. We apply some of our results to asset market games.
We study Markov processes generated by iterated function systems (IFS). The constituent maps of the IFS are monotonic transformations of the interval. We first obtain an upper bound on the number of SRB (Sinai-Ruelle-Bowen) measures for the IFS. Then, when all the constituent maps have common fixed points at 0 and 1, theorems are given to analyze properties of the ergodic invariant measures $\delta_0$ and $\delta_1$. In particular, sufficient conditions for $\delta_0$ and/or $\delta_1$ to be, or not to be, SRB measures are given. We apply some of our results to asset market games.
2011, 30(1): 39-53
doi: 10.3934/dcds.2011.30.39
+[Abstract](1624)
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Abstract:
In this paper we consider the relation between the notions of exponential stability and admissibility, in the general context of nonuniform exponential behavior. In particular, we show that with respect to certain adapted norms related to the nonuniform behavior, if any $L^p$ space, with $p\in(1,\infty]$, is admissible for a given evolution process, then this process is a nonuniform exponential dichotomy. In addition, for each nonuniform exponential dichotomy we provide a collection of admissible Banach spaces, also defined in terms of the adapted norms.
In this paper we consider the relation between the notions of exponential stability and admissibility, in the general context of nonuniform exponential behavior. In particular, we show that with respect to certain adapted norms related to the nonuniform behavior, if any $L^p$ space, with $p\in(1,\infty]$, is admissible for a given evolution process, then this process is a nonuniform exponential dichotomy. In addition, for each nonuniform exponential dichotomy we provide a collection of admissible Banach spaces, also defined in terms of the adapted norms.
2011, 30(1): 55-76
doi: 10.3934/dcds.2011.30.55
+[Abstract](1548)
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Abstract:
We construct $C^k$ invariant center manifolds for differential equations $u'=A(t)u+f(t,u)$ obtained from sufficiently small perturbations of a nonuniform exponential trichotomy. We emphasize that our results are optimal, in the sense that the invariant manifolds are as regular as the vector field. In addition, we can also consider linear perturbations with the same method.
We construct $C^k$ invariant center manifolds for differential equations $u'=A(t)u+f(t,u)$ obtained from sufficiently small perturbations of a nonuniform exponential trichotomy. We emphasize that our results are optimal, in the sense that the invariant manifolds are as regular as the vector field. In addition, we can also consider linear perturbations with the same method.
2011, 30(1): 77-113
doi: 10.3934/dcds.2011.30.77
+[Abstract](1703)
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Abstract:
In this paper, we study the generic behavior of the solutions to a large class of evolution equations. The Schrödinger evolution is considered as an application.
In this paper, we study the generic behavior of the solutions to a large class of evolution equations. The Schrödinger evolution is considered as an application.
2011, 30(1): 115-135
doi: 10.3934/dcds.2011.30.115
+[Abstract](1788)
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Abstract:
This work aims to investigate the regularity and the stability of the solutions for a class of partial functional differential equations with infinite delay. Here we suppose that the undelayed part generates an analytic semigroup and the delayed part is continuous with respect to fractional powers of the generator. First, we give a new characterization for the infinitesimal generator of the solution semigroup, which allows us to give necessary and sufficient conditions for the regularity of solutions. Second, we investigate the stability of the semigroup solution. We proved that one of the fundamental and wildly used assumption, in the computing of eigenvalues and eigenvectors, is an immediate consequence of the already considered ones. Finally, we discuss the asymptotic behavior of solutions.
This work aims to investigate the regularity and the stability of the solutions for a class of partial functional differential equations with infinite delay. Here we suppose that the undelayed part generates an analytic semigroup and the delayed part is continuous with respect to fractional powers of the generator. First, we give a new characterization for the infinitesimal generator of the solution semigroup, which allows us to give necessary and sufficient conditions for the regularity of solutions. Second, we investigate the stability of the semigroup solution. We proved that one of the fundamental and wildly used assumption, in the computing of eigenvalues and eigenvectors, is an immediate consequence of the already considered ones. Finally, we discuss the asymptotic behavior of solutions.
2011, 30(1): 137-167
doi: 10.3934/dcds.2011.30.137
+[Abstract](1881)
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Abstract:
We prove the existence and uniqueness, for wave speeds sufficiently large, of monotone traveling wave solutions connecting stable to unstable spatial equilibria for a class of $N$-dimensional lattice differential equations with unidirectional coupling. This class of lattice equations includes some cellular neural networks, monotone systems, and semi-discretizations for hyperbolic conservation laws with a source term. We obtain a variational characterization of the critical wave speed above which monotone traveling wave solutions are guaranteed to exist. We also discuss non-monotone waves, and the coexistence of monotone and non-monotone waves.
We prove the existence and uniqueness, for wave speeds sufficiently large, of monotone traveling wave solutions connecting stable to unstable spatial equilibria for a class of $N$-dimensional lattice differential equations with unidirectional coupling. This class of lattice equations includes some cellular neural networks, monotone systems, and semi-discretizations for hyperbolic conservation laws with a source term. We obtain a variational characterization of the critical wave speed above which monotone traveling wave solutions are guaranteed to exist. We also discuss non-monotone waves, and the coexistence of monotone and non-monotone waves.
2011, 30(1): 169-186
doi: 10.3934/dcds.2011.30.169
+[Abstract](1873)
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Abstract:
Invariance entropy for continuous-time control systems measures how often open-loop control functions have to be updated in order to render a subset of the state space invariant. In the present paper, we derive upper and lower bounds for the invariance entropy of control systems on smooth manifolds, using differential-geometric tools. As an example, we compute these bounds explicitly for projected bilinear control systems on the unit sphere. Moreover, we derive a formula for the invariance entropy of a control set for one-dimensional control-affine systems with a single control vector field.
Invariance entropy for continuous-time control systems measures how often open-loop control functions have to be updated in order to render a subset of the state space invariant. In the present paper, we derive upper and lower bounds for the invariance entropy of control systems on smooth manifolds, using differential-geometric tools. As an example, we compute these bounds explicitly for projected bilinear control systems on the unit sphere. Moreover, we derive a formula for the invariance entropy of a control set for one-dimensional control-affine systems with a single control vector field.
2011, 30(1): 187-207
doi: 10.3934/dcds.2011.30.187
+[Abstract](1409)
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Abstract:
One traditionally considers positive measures in the moment problem. However, this restriction makes its theory and application limited. The main purpose of this paper is to generalize it to deal with complex measures. More precisely, the theory of the truncated moment problem is extended to include complex measures. This extended theory provides considerable flexibility in its applications. In fact, we also develop an approximation technique based on control of moments. The key idea is to use the heat equation as a link that connects the generalized moment problem and this approximation technique. The backward moment of a measure is introduced as the moment of a solution to the heat equation at a backward time and then used to approximate the given measure. This approximation gives a geometric convergence order as the number of moments under control increases. Numerical examples are given that show the properties of approximation technique.
One traditionally considers positive measures in the moment problem. However, this restriction makes its theory and application limited. The main purpose of this paper is to generalize it to deal with complex measures. More precisely, the theory of the truncated moment problem is extended to include complex measures. This extended theory provides considerable flexibility in its applications. In fact, we also develop an approximation technique based on control of moments. The key idea is to use the heat equation as a link that connects the generalized moment problem and this approximation technique. The backward moment of a measure is introduced as the moment of a solution to the heat equation at a backward time and then used to approximate the given measure. This approximation gives a geometric convergence order as the number of moments under control increases. Numerical examples are given that show the properties of approximation technique.
2011, 30(1): 209-218
doi: 10.3934/dcds.2011.30.209
+[Abstract](1432)
+[PDF](181.5KB)
Abstract:
There is a natural continuous realization of the one-sided Bernoulli shift on the $p$-adic integers as the map that shifts the coefficients of the $p$-adic expansion to the left. We study this map's Mahler power series expansion. We prove strong results on $p$-adic valuations of the coefficients in this expansion, and show that certain natural maps (including many polynomials) are in a sense small perturbations of the shift. As a result, these polynomials share the shift map's important dynamical properties. This provides a novel approach to an earlier result of the authors.
There is a natural continuous realization of the one-sided Bernoulli shift on the $p$-adic integers as the map that shifts the coefficients of the $p$-adic expansion to the left. We study this map's Mahler power series expansion. We prove strong results on $p$-adic valuations of the coefficients in this expansion, and show that certain natural maps (including many polynomials) are in a sense small perturbations of the shift. As a result, these polynomials share the shift map's important dynamical properties. This provides a novel approach to an earlier result of the authors.
2011, 30(1): 219-225
doi: 10.3934/dcds.2011.30.219
+[Abstract](1611)
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Abstract:
We consider the KdV equation on the Sobolev space of periodic distributions. We obtain estimates of the solution of the KdV in terms of action variables.
We consider the KdV equation on the Sobolev space of periodic distributions. We obtain estimates of the solution of the KdV in terms of action variables.
2011, 30(1): 227-241
doi: 10.3934/dcds.2011.30.227
+[Abstract](1652)
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Abstract:
In this paper, we consider the relation between $p > 1$ and critical dimension of the extremal solution of the semilinear equation
In this paper, we consider the relation between $p > 1$ and critical dimension of the extremal solution of the semilinear equation
$\beta \Delta^{2}u-\tau \Delta u=\frac{\lambda}{(1-u)^{p}} \mbox{in} B$,
$0 < u \leq 1 \mbox{in} B$,
$u=\Delta u=0 \mbox{on} \partial B$,
where $B$ is the unit ball in $R^{n}$, $\lambda>0$ is a parameter, $\tau>0, \beta>0,p>1$ are fixed constants. By Hardy-Rellich inequality, we find that when $p$ is large enough, the critical dimension is 13.
2011, 30(1): 243-252
doi: 10.3934/dcds.2011.30.243
+[Abstract](1336)
+[PDF](366.9KB)
Abstract:
In this paper, we consider a one-parameter family $F_{\lambda }$ of continuous maps on $\mathbb{R}^{m}$ or $\mathbb{R}^{m}\times \mathbb{R}^{k}$ with the singular map $F_{0}$ having one of the forms (i) $F_{0}(x)=f(x),$ (ii) $F_{0}(x,y)=(f(x),g(x))$, where $g:\mathbb{R}^{m}\rightarrow \mathbb{R} ^{k}$ is continuous, and (iii) $F_{0}(x,y)=(f(x),g(x,y))$, where $g:\mathbb{R}^{m}\times \mathbb{R}^{k}\rightarrow \mathbb{R}^{k}$ is continuous and locally trapping along the second variable $y$. We show that if $f:\mathbb{R}^{m}\rightarrow \mathbb{R}^{m}$ is a $C^{1}$ diffeomorphism having a topologically crossing homoclinic point, then $F_{\lambda }$ has positive topological entropy for all $\lambda $ close enough to $0$.
In this paper, we consider a one-parameter family $F_{\lambda }$ of continuous maps on $\mathbb{R}^{m}$ or $\mathbb{R}^{m}\times \mathbb{R}^{k}$ with the singular map $F_{0}$ having one of the forms (i) $F_{0}(x)=f(x),$ (ii) $F_{0}(x,y)=(f(x),g(x))$, where $g:\mathbb{R}^{m}\rightarrow \mathbb{R} ^{k}$ is continuous, and (iii) $F_{0}(x,y)=(f(x),g(x,y))$, where $g:\mathbb{R}^{m}\times \mathbb{R}^{k}\rightarrow \mathbb{R}^{k}$ is continuous and locally trapping along the second variable $y$. We show that if $f:\mathbb{R}^{m}\rightarrow \mathbb{R}^{m}$ is a $C^{1}$ diffeomorphism having a topologically crossing homoclinic point, then $F_{\lambda }$ has positive topological entropy for all $\lambda $ close enough to $0$.
2011, 30(1): 253-259
doi: 10.3934/dcds.2011.30.253
+[Abstract](1387)
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Abstract:
We prove that the initial value problem (IVP) for the BBM equation is ill-posed for data in $H^s(\R)$, $s<0$ in the sense that the flow-map $u_0\mapsto u(t)$ that associates to initial data $u_0$ the solution $u$ cannot be continuous at the origin from $H^s(\R)$ to even $\mathcal{D}'(\R)$ at any fixed $t>0$ small enough. This result is sharp.
We prove that the initial value problem (IVP) for the BBM equation is ill-posed for data in $H^s(\R)$, $s<0$ in the sense that the flow-map $u_0\mapsto u(t)$ that associates to initial data $u_0$ the solution $u$ cannot be continuous at the origin from $H^s(\R)$ to even $\mathcal{D}'(\R)$ at any fixed $t>0$ small enough. This result is sharp.
2011, 30(1): 261-298
doi: 10.3934/dcds.2011.30.261
+[Abstract](1757)
+[PDF](537.8KB)
Abstract:
We introduce and explore random conformal graph directed Mar-kov systems governed by measure-preserving ergodic dynamical systems. We first develop the symbolic thermodynamic formalism for random finitely primitive subshifts of finite type with a countable alphabet (by establishing tightness in a narrow topology). We then construct fibrewise conformal and invariant measures along with fibrewise topological pressure. This enables us to define the expected topological pressure $\mathcal EP(t)$ and to prove a variant of Bowen's formula which identifies the Hausdorff dimension of almost every limit set fiber with $\inf\{t:\mathcal EP(t)\leq0\}$, and is the unique zero of the expected pressure if the alphabet is finite or the system is regular. We introduce the class of essentially random systems and we show that in the realm of systems with finite alphabet their limit set fibers are never homeomorphic in a bi-Lipschitz fashion to the limit sets of deterministic systems; they thus make up a drastically new world. We also provide a large variety of examples, with exact computations of Hausdorff dimensions, and we study in detail the small random perturbations of an arbitrary elliptic function.
We introduce and explore random conformal graph directed Mar-kov systems governed by measure-preserving ergodic dynamical systems. We first develop the symbolic thermodynamic formalism for random finitely primitive subshifts of finite type with a countable alphabet (by establishing tightness in a narrow topology). We then construct fibrewise conformal and invariant measures along with fibrewise topological pressure. This enables us to define the expected topological pressure $\mathcal EP(t)$ and to prove a variant of Bowen's formula which identifies the Hausdorff dimension of almost every limit set fiber with $\inf\{t:\mathcal EP(t)\leq0\}$, and is the unique zero of the expected pressure if the alphabet is finite or the system is regular. We introduce the class of essentially random systems and we show that in the realm of systems with finite alphabet their limit set fibers are never homeomorphic in a bi-Lipschitz fashion to the limit sets of deterministic systems; they thus make up a drastically new world. We also provide a large variety of examples, with exact computations of Hausdorff dimensions, and we study in detail the small random perturbations of an arbitrary elliptic function.
2011, 30(1): 299-312
doi: 10.3934/dcds.2011.30.299
+[Abstract](1536)
+[PDF](1051.9KB)
Abstract:
We consider the collinear three-body problem where the particles 1, 2, 3 with masses $m_{1}, m_{2}, m_{3}$ are on a common line in this order. We restrict the mass parameters to those that satisfy an "allowable" condition. We associate the binary collision of particle 1 and 2 with symbol "1", one of 2 and 3 with "3", and the triple collision with "2", and then represent the patterns of collisions of orbits in the time evolution using the symbol sequences. Inducing a tiling on an appropriate cross section, we prove that for any symbol sequence with some condition, there exists an orbit realizing the sequence. Furthermore we show the existence of infinitely many periodic solutions. Our novel result is the non-integrability of the collinear three-body problem with the allowable masses.
We consider the collinear three-body problem where the particles 1, 2, 3 with masses $m_{1}, m_{2}, m_{3}$ are on a common line in this order. We restrict the mass parameters to those that satisfy an "allowable" condition. We associate the binary collision of particle 1 and 2 with symbol "1", one of 2 and 3 with "3", and the triple collision with "2", and then represent the patterns of collisions of orbits in the time evolution using the symbol sequences. Inducing a tiling on an appropriate cross section, we prove that for any symbol sequence with some condition, there exists an orbit realizing the sequence. Furthermore we show the existence of infinitely many periodic solutions. Our novel result is the non-integrability of the collinear three-body problem with the allowable masses.
2011, 30(1): 313-363
doi: 10.3934/dcds.2011.30.313
+[Abstract](1517)
+[PDF](729.1KB)
Abstract:
We consider the dynamics of semi-hyperbolic semigroups generated by finitely many rational maps on the Riemann sphere. Assuming that the nice open set condition holds it is proved that there exists a geometric measure on the Julia set with exponent $h$ equal to the Hausdorff dimension of the Julia set. Both $h$-dimensional Hausdorff and packing measures are finite and positive on the Julia set and are mutually equivalent with Radon-Nikodym derivatives uniformly separated from zero and infinity. All three fractal dimensions, Hausdorff, packing and box counting are equal. It is also proved that for the canonically associated skew-product map there exists a unique $h$-conformal measure. Furthermore, it is shown that this conformal measure admits a unique Borel probability absolutely continuous invariant (under the skew-product map) measure. In fact these two measures are equivalent, and the invariant measure is metrically exact, hence ergodic.
We consider the dynamics of semi-hyperbolic semigroups generated by finitely many rational maps on the Riemann sphere. Assuming that the nice open set condition holds it is proved that there exists a geometric measure on the Julia set with exponent $h$ equal to the Hausdorff dimension of the Julia set. Both $h$-dimensional Hausdorff and packing measures are finite and positive on the Julia set and are mutually equivalent with Radon-Nikodym derivatives uniformly separated from zero and infinity. All three fractal dimensions, Hausdorff, packing and box counting are equal. It is also proved that for the canonically associated skew-product map there exists a unique $h$-conformal measure. Furthermore, it is shown that this conformal measure admits a unique Borel probability absolutely continuous invariant (under the skew-product map) measure. In fact these two measures are equivalent, and the invariant measure is metrically exact, hence ergodic.
2011, 30(1): 365-374
doi: 10.3934/dcds.2011.30.365
+[Abstract](1538)
+[PDF](350.2KB)
Abstract:
It is shown that optimal network plans can be obtained as a limit of point allocations. These problems are obtained by minimizing the mass transportation on the set of atomic measures of a prescribed number of atoms.
It is shown that optimal network plans can be obtained as a limit of point allocations. These problems are obtained by minimizing the mass transportation on the set of atomic measures of a prescribed number of atoms.
2011, 30(1): 375-377
doi: 10.3934/dcds.2011.30.375
+[Abstract](1275)
+[PDF](268.9KB)
Abstract:
The earlier paper [2] contains a lower bound of the solution in terms of its $L^1$ norm, which is incorrect. In this note we explain the mistake and present a correction to it under the restriction that the permeability constant $m$ satisfies $1< m <2$. As a consequence, the quantitative estimates on the convergence rate (Main Theorem (c) and Theorem 3.6 in [2] ) only hold for $1<\m<2$. For $\m\geq 2$ a partial convergence rate is obtained.
The earlier paper [2] contains a lower bound of the solution in terms of its $L^1$ norm, which is incorrect. In this note we explain the mistake and present a correction to it under the restriction that the permeability constant $m$ satisfies $1< m <2$. As a consequence, the quantitative estimates on the convergence rate (Main Theorem (c) and Theorem 3.6 in [2] ) only hold for $1<\m<2$. For $\m\geq 2$ a partial convergence rate is obtained.
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