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1078-0947
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Discrete & Continuous Dynamical Systems - A
March 2012 , Volume 32 , Issue 3
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2012, 32(3): 703-715
doi: 10.3934/dcds.2012.32.703
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Abstract:
We consider the Cauchy problem for (energy-subcritical) nonlinear Schrödinger equations with sub-quadratic external potentials and an additional angular momentum rotation term. This equation is a well-known model for superfluid quantum gases in rotating traps. We prove global existence (in the energy space) for defocusing nonlinearities without any restriction on the rotation frequency, generalizing earlier results given in [11, 12]. Moreover, we find that the rotation term has a considerable influence in proving finite time blow-up in the focusing case.
We consider the Cauchy problem for (energy-subcritical) nonlinear Schrödinger equations with sub-quadratic external potentials and an additional angular momentum rotation term. This equation is a well-known model for superfluid quantum gases in rotating traps. We prove global existence (in the energy space) for defocusing nonlinearities without any restriction on the rotation frequency, generalizing earlier results given in [11, 12]. Moreover, we find that the rotation term has a considerable influence in proving finite time blow-up in the focusing case.
2012, 32(3): 717-751
doi: 10.3934/dcds.2012.32.717
+[Abstract](2033)
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We conduct the multifractal analysis of the level sets of the asymptotic behavior of almost additive continuous potentials $(\phi_n)_{n=1}^\infty$ on a topologically mixing subshift of finite type $X$ endowed itself with a metric associated with such a potential. We work without additional regularity assumption other than continuity. Our approach differs from those used previously to deal with this question under stronger assumptions on the potentials. As a consequence, it provides a new description of the structure of the spectrum in terms of weak concavity. Also, the lower bound for the spectrum is obtained as a consequence of the study sets of points at which the asymptotic behavior of $\phi_n(x)$ is localized, i.e. depends on the point $x$ rather than being equal to a constant. Specifically, we compute the Hausdorff dimension of sets of the form $\{x\in X: \lim_{n\to\infty} \phi_n(x)/n=\xi(x)\}$, where $\xi$ is a given continuous function. This has interesting geometric applications to fixed points in the asymptotic average for dynamical systems in $\mathbb{R}^d$, as well as the fine local behavior of the harmonic measure on conformal planar Cantor sets.
We conduct the multifractal analysis of the level sets of the asymptotic behavior of almost additive continuous potentials $(\phi_n)_{n=1}^\infty$ on a topologically mixing subshift of finite type $X$ endowed itself with a metric associated with such a potential. We work without additional regularity assumption other than continuity. Our approach differs from those used previously to deal with this question under stronger assumptions on the potentials. As a consequence, it provides a new description of the structure of the spectrum in terms of weak concavity. Also, the lower bound for the spectrum is obtained as a consequence of the study sets of points at which the asymptotic behavior of $\phi_n(x)$ is localized, i.e. depends on the point $x$ rather than being equal to a constant. Specifically, we compute the Hausdorff dimension of sets of the form $\{x\in X: \lim_{n\to\infty} \phi_n(x)/n=\xi(x)\}$, where $\xi$ is a given continuous function. This has interesting geometric applications to fixed points in the asymptotic average for dynamical systems in $\mathbb{R}^d$, as well as the fine local behavior of the harmonic measure on conformal planar Cantor sets.
2012, 32(3): 753-793
doi: 10.3934/dcds.2012.32.753
+[Abstract](2612)
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Abstract:
In this article, we consider an a priori unstable Hamiltonian system with three degrees of freedom, for which we construct a drifting solution with an optimal time of instability. Such a result has been already proved by Berti, Bolle and Biasco using variational arguments, and by Treschev using his separatrix map theory. Our approach is new: it is based on a special type of symbolic dynamics corresponding to the random iteration of a family of twist maps of the annulus, and it gives the first concrete application of this idea introduced by Moeckel in an abstract setting and further studied by Marco. Our method should also be useful in obtaining the optimal time of instability in the more difficult context of a priori stable Hamiltonian systems.
In this article, we consider an a priori unstable Hamiltonian system with three degrees of freedom, for which we construct a drifting solution with an optimal time of instability. Such a result has been already proved by Berti, Bolle and Biasco using variational arguments, and by Treschev using his separatrix map theory. Our approach is new: it is based on a special type of symbolic dynamics corresponding to the random iteration of a family of twist maps of the annulus, and it gives the first concrete application of this idea introduced by Moeckel in an abstract setting and further studied by Marco. Our method should also be useful in obtaining the optimal time of instability in the more difficult context of a priori stable Hamiltonian systems.
2012, 32(3): 795-826
doi: 10.3934/dcds.2012.32.795
+[Abstract](3121)
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Abstract:
In this paper, we consider the following problem $$ \left\{ \begin{array}{ll} -\Delta u+u=u^{2^{*}-1}+\lambda(f(x,u)+h(x))\ \ \hbox{in}\ \mathbb{R}^{N},\\ u\in H^{1}(\mathbb{R}^{N}),\ \ u>0 \ \hbox{in}\ \mathbb{R}^{N}, \end{array} \right. (\star) $$ where $\lambda>0$ is a parameter, $2^* =\frac {2N}{N-2}$ is the critical Sobolev exponent and $N>4$, $f(x,t)$ and $h(x)$ are some given functions. We prove that there exists $0<\lambda^{*}<+\infty$ such that $(\star)$ has exactly two positive solutions for $\lambda\in(0,\lambda^{*})$ by Barrier method and Mountain Pass Lemma and no positive solutions for $\lambda >\lambda^*$. Moreover, if $\lambda=\lambda^*$, $(\star)$ has a unique solution $(\lambda^{*}, u_{\lambda^{*}})$, which means that $(\lambda^{*}, u_{\lambda^{*}})$ is a turning point in $H^{1}(\mathbb{R}^{N})$ for problem $(\star)$.
In this paper, we consider the following problem $$ \left\{ \begin{array}{ll} -\Delta u+u=u^{2^{*}-1}+\lambda(f(x,u)+h(x))\ \ \hbox{in}\ \mathbb{R}^{N},\\ u\in H^{1}(\mathbb{R}^{N}),\ \ u>0 \ \hbox{in}\ \mathbb{R}^{N}, \end{array} \right. (\star) $$ where $\lambda>0$ is a parameter, $2^* =\frac {2N}{N-2}$ is the critical Sobolev exponent and $N>4$, $f(x,t)$ and $h(x)$ are some given functions. We prove that there exists $0<\lambda^{*}<+\infty$ such that $(\star)$ has exactly two positive solutions for $\lambda\in(0,\lambda^{*})$ by Barrier method and Mountain Pass Lemma and no positive solutions for $\lambda >\lambda^*$. Moreover, if $\lambda=\lambda^*$, $(\star)$ has a unique solution $(\lambda^{*}, u_{\lambda^{*}})$, which means that $(\lambda^{*}, u_{\lambda^{*}})$ is a turning point in $H^{1}(\mathbb{R}^{N})$ for problem $(\star)$.
2012, 32(3): 827-846
doi: 10.3934/dcds.2012.32.827
+[Abstract](3009)
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Abstract:
We study blow-up, global existence and standing waves for the nonlinear Schrödinger equations with two-dimensional magnetic field in a cold plasma. Under certain conditions on initial data and initial energy, we derive finite time blow-up phenomena of the solutions to the equations under study. Using compactness and Lagrange multiplier method, we establish the existence of standing waves. Finally, by introducing invariant manifolds and utilizing potential well argument as well as concavity method, we obtain the sharp threshold for global existence and blowup.
We study blow-up, global existence and standing waves for the nonlinear Schrödinger equations with two-dimensional magnetic field in a cold plasma. Under certain conditions on initial data and initial energy, we derive finite time blow-up phenomena of the solutions to the equations under study. Using compactness and Lagrange multiplier method, we establish the existence of standing waves. Finally, by introducing invariant manifolds and utilizing potential well argument as well as concavity method, we obtain the sharp threshold for global existence and blowup.
2012, 32(3): 847-865
doi: 10.3934/dcds.2012.32.847
+[Abstract](2293)
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Abstract:
Let $u$ be a global in time solution of the Cauchy-Dirichlet problem for a semilinear heat equation, $$ \left\{ \begin{array}{ll} \partial_t u=\Delta u+u^p,\quad & x\in\Omega,\,\, t>0,\\ u=0,\quad & x\in\partial\Omega,\,\,t>0,\\ u(x,0)=\phi(x)\ge 0,\quad & x\in\Omega, \end{array} \right. $$ where $\partial_t=\partial/\partial t$, $p>1+2/N$, $N\ge 3$, $\Omega$ is a smooth domain in ${\bf R}^N$, and $\phi\in L^\infty(\Omega)$. In this paper we give a sufficient condition for the solution $u$ to behave like $\|u(t)\|_{L^\infty({\bf R}^N)}=O(t^{-1/(p-1)})$ as $t\to\infty$, and give a classification of the large time behavior of the solution $u$.
Let $u$ be a global in time solution of the Cauchy-Dirichlet problem for a semilinear heat equation, $$ \left\{ \begin{array}{ll} \partial_t u=\Delta u+u^p,\quad & x\in\Omega,\,\, t>0,\\ u=0,\quad & x\in\partial\Omega,\,\,t>0,\\ u(x,0)=\phi(x)\ge 0,\quad & x\in\Omega, \end{array} \right. $$ where $\partial_t=\partial/\partial t$, $p>1+2/N$, $N\ge 3$, $\Omega$ is a smooth domain in ${\bf R}^N$, and $\phi\in L^\infty(\Omega)$. In this paper we give a sufficient condition for the solution $u$ to behave like $\|u(t)\|_{L^\infty({\bf R}^N)}=O(t^{-1/(p-1)})$ as $t\to\infty$, and give a classification of the large time behavior of the solution $u$.
2012, 32(3): 867-889
doi: 10.3934/dcds.2012.32.867
+[Abstract](2459)
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Abstract:
In this paper, we consider a $n$-species Lotka-Volterra mutualism system with stochastic perturbation. Sufficient criteria for persistence in mean and stationary distribution of the system are established. Besides, we show the large white noise will make the system nonpersistent. Finally, we illustrate the dynamic behavior of the system with $n=3$ and their approximations via a range of numerical experiments.
In this paper, we consider a $n$-species Lotka-Volterra mutualism system with stochastic perturbation. Sufficient criteria for persistence in mean and stationary distribution of the system are established. Besides, we show the large white noise will make the system nonpersistent. Finally, we illustrate the dynamic behavior of the system with $n=3$ and their approximations via a range of numerical experiments.
2012, 32(3): 891-900
doi: 10.3934/dcds.2012.32.891
+[Abstract](2407)
+[PDF](335.8KB)
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This paper provides a way for determining the Kolmogorov-Sinai entropy of time-discrete dynamical systems on the base of quantifying ordinal patterns obtained from a finite set of observables. As a consequence, it is shown that the Kolmogorov-Sinai entropy is bounded from above by a quantity which generalizes the concept of permutation entropy. In this framework, the determination of the Kolmogorov-Sinai entropy of a multidimensional system by use of only a single one-dimensional observable and Takens' embedding theorem is discussed.
This paper provides a way for determining the Kolmogorov-Sinai entropy of time-discrete dynamical systems on the base of quantifying ordinal patterns obtained from a finite set of observables. As a consequence, it is shown that the Kolmogorov-Sinai entropy is bounded from above by a quantity which generalizes the concept of permutation entropy. In this framework, the determination of the Kolmogorov-Sinai entropy of a multidimensional system by use of only a single one-dimensional observable and Takens' embedding theorem is discussed.
2012, 32(3): 901-933
doi: 10.3934/dcds.2012.32.901
+[Abstract](2367)
+[PDF](576.2KB)
Abstract:
We investigate positive equilibria and phase portraits of predator-prey systems with constant harvesting rates arising in ecology. These systems are generalizations of the well--known predator--prey systems with Beddington--DeAngelis functional responses. We seek the ranges of the five parameters involved for which the equilibria of the systems to be positive and obtain all the positive equilibria of the systems. We prove that these positive equilibria are saddles, topological saddles, nodes, saddle-nodes, foci, centers, or cusps by providing suitable ranges of the five parameters. These results show how the harvesting rate and another parameter used in the Beddington--DeAngelis functional responses affect the dynamical behaviors of these systems. In particular, if the harvesting rate is larger than $1/4$ or the parameter just mentioned is too large, then the mutual extinction occurs.
We investigate positive equilibria and phase portraits of predator-prey systems with constant harvesting rates arising in ecology. These systems are generalizations of the well--known predator--prey systems with Beddington--DeAngelis functional responses. We seek the ranges of the five parameters involved for which the equilibria of the systems to be positive and obtain all the positive equilibria of the systems. We prove that these positive equilibria are saddles, topological saddles, nodes, saddle-nodes, foci, centers, or cusps by providing suitable ranges of the five parameters. These results show how the harvesting rate and another parameter used in the Beddington--DeAngelis functional responses affect the dynamical behaviors of these systems. In particular, if the harvesting rate is larger than $1/4$ or the parameter just mentioned is too large, then the mutual extinction occurs.
2012, 32(3): 935-959
doi: 10.3934/dcds.2012.32.935
+[Abstract](2092)
+[PDF](569.8KB)
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We consider self-affine tilings in the Euclidean space and the associated tiling dynamical systems, namely, the translation action on the orbit closure of the given tiling. We investigate the spectral properties of the system. It turns out that the presence of the discrete component depends on the algebraic properties of the eigenvalues of the expansion matrix $\phi$ for the tiling. Assuming that $\phi$ is diagonalizable over $\mathbb{C}$ and all its eigenvalues are algebraic conjugates of the same multiplicity, we show that the dynamical system has a relatively dense discrete spectrum if and only if it is not weakly mixing, and if and only if the spectrum of $\phi$ is a "Pisot family." Moreover, this is equivalent to the Meyer property of the associated discrete set of "control points" for the tiling.
We consider self-affine tilings in the Euclidean space and the associated tiling dynamical systems, namely, the translation action on the orbit closure of the given tiling. We investigate the spectral properties of the system. It turns out that the presence of the discrete component depends on the algebraic properties of the eigenvalues of the expansion matrix $\phi$ for the tiling. Assuming that $\phi$ is diagonalizable over $\mathbb{C}$ and all its eigenvalues are algebraic conjugates of the same multiplicity, we show that the dynamical system has a relatively dense discrete spectrum if and only if it is not weakly mixing, and if and only if the spectrum of $\phi$ is a "Pisot family." Moreover, this is equivalent to the Meyer property of the associated discrete set of "control points" for the tiling.
2012, 32(3): 961-975
doi: 10.3934/dcds.2012.32.961
+[Abstract](2047)
+[PDF](408.6KB)
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We give approximations for the Gibbs states of arbitrary Hölder potentials $\phi$, with the help of weighted sums of atomic measures on preimage sets, in the case of smooth non-invertible maps hyperbolic on folded basic sets $\Lambda$. The endomorphism may have also stable directions on $\Lambda$ and is non-expanding in general. Folding of the phase space means that we do not have a foliation structure for the local unstable manifolds (instead they depend on the whole past and may intersect each other both inside and outside $\Lambda$). We consider here simultaneously all $n$-preimages in $\Lambda$ of a point, instead of the usual way of taking only the consecutive preimages from some given prehistory. We thus obtain the weighted distribution of consecutive preimage sets, with respect to various equilibrium measures on the saddle-type folded set $\Lambda$. In particular we obtain the distribution of preimage sets on $\Lambda$, with respect to the measure of maximal entropy. Our result is not a direct application of Birkhoff Ergodic Theorem on the inverse limit $\hat \Lambda$, since the set of prehistories of a point is uncountable in general, and the speed of convergence may vary for different prehistories in $\hat \Lambda$. For hyperbolic toral endomorphisms, we obtain the distribution of the consecutive preimage sets towards an inverse SRB measure, for Lebesgue-almost all points.
We give approximations for the Gibbs states of arbitrary Hölder potentials $\phi$, with the help of weighted sums of atomic measures on preimage sets, in the case of smooth non-invertible maps hyperbolic on folded basic sets $\Lambda$. The endomorphism may have also stable directions on $\Lambda$ and is non-expanding in general. Folding of the phase space means that we do not have a foliation structure for the local unstable manifolds (instead they depend on the whole past and may intersect each other both inside and outside $\Lambda$). We consider here simultaneously all $n$-preimages in $\Lambda$ of a point, instead of the usual way of taking only the consecutive preimages from some given prehistory. We thus obtain the weighted distribution of consecutive preimage sets, with respect to various equilibrium measures on the saddle-type folded set $\Lambda$. In particular we obtain the distribution of preimage sets on $\Lambda$, with respect to the measure of maximal entropy. Our result is not a direct application of Birkhoff Ergodic Theorem on the inverse limit $\hat \Lambda$, since the set of prehistories of a point is uncountable in general, and the speed of convergence may vary for different prehistories in $\hat \Lambda$. For hyperbolic toral endomorphisms, we obtain the distribution of the consecutive preimage sets towards an inverse SRB measure, for Lebesgue-almost all points.
2012, 32(3): 977-989
doi: 10.3934/dcds.2012.32.977
+[Abstract](1867)
+[PDF](369.5KB)
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We show that special perturbations of a particular holomorphic map on $\mathbf{P}^k$ give us examples of maps that possess chaotic non-algebraic attractors. Furthermore, we study the dynamics of the maps on the attractors. In particular, we construct invariant hyperbolic measures supported on the attractors with nice dynamical properties.
We show that special perturbations of a particular holomorphic map on $\mathbf{P}^k$ give us examples of maps that possess chaotic non-algebraic attractors. Furthermore, we study the dynamics of the maps on the attractors. In particular, we construct invariant hyperbolic measures supported on the attractors with nice dynamical properties.
2012, 32(3): 991-1009
doi: 10.3934/dcds.2012.32.991
+[Abstract](2250)
+[PDF](388.9KB)
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We investigate the asymptotic behavior of solutions of a class of non-autonomous Newton-Boussinesq equation in two-dimensional bounded domain. The existence of pullback global attractors is proved in $L^2(\Omega)\times L^2(\Omega)$ and $H^1(\Omega)\times H^1(\Omega)$, respectively.
We investigate the asymptotic behavior of solutions of a class of non-autonomous Newton-Boussinesq equation in two-dimensional bounded domain. The existence of pullback global attractors is proved in $L^2(\Omega)\times L^2(\Omega)$ and $H^1(\Omega)\times H^1(\Omega)$, respectively.
2012, 32(3): 1011-1046
doi: 10.3934/dcds.2012.32.1011
+[Abstract](2704)
+[PDF](617.3KB)
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This paper studies traveling front solutions of convex polyhedral shapes in bistable reaction-diffusion equations including the Allen-Cahn equations or the Nagumo equations. By taking the limits of such solutions as the lateral faces go to infinity, we construct a three-dimensional traveling front solution for any given $g\in C^{\infty}(S^{1})$ with $\min_{0\leq \theta\leq 2\pi}g(\theta)=0$.
This paper studies traveling front solutions of convex polyhedral shapes in bistable reaction-diffusion equations including the Allen-Cahn equations or the Nagumo equations. By taking the limits of such solutions as the lateral faces go to infinity, we construct a three-dimensional traveling front solution for any given $g\in C^{\infty}(S^{1})$ with $\min_{0\leq \theta\leq 2\pi}g(\theta)=0$.
2012, 32(3): 1047-1053
doi: 10.3934/dcds.2012.32.1047
+[Abstract](1796)
+[PDF](307.0KB)
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The straight-line flow on almost every staircase and on almost every square tiled staircase is recurrent. For almost every square tiled staircase the set of periodic orbits is dense in the phase space.
The straight-line flow on almost every staircase and on almost every square tiled staircase is recurrent. For almost every square tiled staircase the set of periodic orbits is dense in the phase space.
2012, 32(3): 1055-1063
doi: 10.3934/dcds.2012.32.1055
+[Abstract](2042)
+[PDF](363.2KB)
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We describe Veech groups of flat surfaces arising from irrational angled polygonal billiards or irreducible stable abelian differentials. For irrational polygonal billiards, we prove that these groups are non-discrete subgroups of $\rm SO(2,\mathbf{R})$ and we calculate their rank.
We describe Veech groups of flat surfaces arising from irrational angled polygonal billiards or irreducible stable abelian differentials. For irrational polygonal billiards, we prove that these groups are non-discrete subgroups of $\rm SO(2,\mathbf{R})$ and we calculate their rank.
2012, 32(3): 1065-1094
doi: 10.3934/dcds.2012.32.1065
+[Abstract](2213)
+[PDF](488.3KB)
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The main aim of this paper is to establish the LaSalle-type theorem to locate limit sets for neutral stochastic functional differential equations with infinite delay, from which some criteria on attraction, boundedness and the almost sure stability with general decay rate and robustness are obtained. To make our theory more applicable, by the $M$-matrix theory, this paper also examines some conditions under which attraction and stability are guaranteed. These conditions also show that attraction and stability are robust with respect to stochastic perturbations. By specializing the general decay rate as the exponential decay rate and the polynomial decay rate, this paper examines two neutral stochastic integral-differential equations and shows that they are exponentially attractive and polynomially stable, respectively.
The main aim of this paper is to establish the LaSalle-type theorem to locate limit sets for neutral stochastic functional differential equations with infinite delay, from which some criteria on attraction, boundedness and the almost sure stability with general decay rate and robustness are obtained. To make our theory more applicable, by the $M$-matrix theory, this paper also examines some conditions under which attraction and stability are guaranteed. These conditions also show that attraction and stability are robust with respect to stochastic perturbations. By specializing the general decay rate as the exponential decay rate and the polynomial decay rate, this paper examines two neutral stochastic integral-differential equations and shows that they are exponentially attractive and polynomially stable, respectively.
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