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## Discrete & Continuous Dynamical Systems - A

2016 , Volume 36 , Issue 11

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2016, 36(11): 5837-5879
doi: 10.3934/dcds.2016057

*+*[Abstract](513)*+*[PDF](646.8KB)**Abstract:**

In this paper the study of asymptotic stability of standing waves for a model of Schrödinger equation with spatially concentrated nonlinearity in dimension three. The nonlinearity studied is a power nonlinearity concentrated at the point $x=0$ obtained considering a contact (or $\delta$) interaction with strength $\alpha$, which consists of a singular perturbation of the Laplacian described by a selfadjoint operator $H_{\alpha}$, and letting the strength $\alpha$ depend on the wavefunction in a prescribed way: $i\dot u= H_\alpha u$, $\alpha=\alpha(u)$. For power nonlinearities in the range $(\frac{1}{\sqrt 2},1)$ there exist orbitally stable standing waves $\Phi_\omega$, and the linearization around them admits two imaginary eigenvalues (neutral modes, absent in the range $(0,\frac{1}{\sqrt 2})$ previously treated by the same authors) which in principle could correspond to non decaying states, so preventing asymptotic relaxation towards an equilibrium orbit. We prove that, in the range $(\frac{1}{\sqrt 2},\sigma^*)$ for a certain $\sigma^* \in (\frac{1}{\sqrt{2}}, \frac{\sqrt{3} +1}{2 \sqrt{2}}]$, the dynamics near the orbit of a standing wave asymptotically relaxes in the following sense: consider an initial datum $u(0)$, suitably near the standing wave $\Phi_{\omega_0}, $ then the solution $u(t)$ can be asymptotically decomposed as $$u(t) = e^{i\omega_{\infty} t +i b_1 \log (1 +\epsilon k_{\infty} t) + i \gamma_\infty} \Phi_{\omega_{\infty}} +U_t*\psi_{\infty} +r_{\infty}, \quad \textrm{as} \;\; t \rightarrow +\infty,$$ where $\omega_{\infty}$, $k_{\infty}, \gamma_\infty > 0$, $b_1 \in \mathbb{R}$, and $\psi_{\infty}$ and $r_{\infty} \in L^2(\mathbb{R}^3)$ , $U(t)$ is the free Schrödinger group and $$\| r_{\infty} \|_{L^2} = O(t^{-1/4}) \quad \textrm{as} \;\; t \rightarrow +\infty\ .$$ We stress the fact that in the present case and contrarily to the main results in the field, the admitted nonlinearity is $L^2$-subcritical.

2016, 36(11): 5881-5910
doi: 10.3934/dcds.2016058

*+*[Abstract](609)*+*[PDF](500.9KB)**Abstract:**

In this paper we are going to study a class of Schrödinger-Poisson system $$ \left\{ \begin{array}{ll} - \Delta u + (\lambda a(x)+1)u+ \phi u = f(u) \mbox{ in } \,\,\, \mathbb{R}^{3},\\ -\Delta \phi=4\pi u^2 \mbox{ in } \,\,\, \mathbb{R}^{3}.\\ \end{array} \right. $$ Assuming that the nonnegative function $a(x)$ has a potential well $int (a^{-1}(\{0\}))$ consisting of $k$ disjoint bounded components $\Omega_1, \Omega_2, ....., \Omega_k$ and the nonlinearity $f(t)$ has a subcritical growth, we are able to establish the existence of positive multi-bump solutions by variational methods.

2016, 36(11): 5911-5927
doi: 10.3934/dcds.2016059

*+*[Abstract](498)*+*[PDF](399.6KB)**Abstract:**

In this article we consider the general problem of translating definitions and results from the category of discrete-time dynamical systems to the category of flows. We consider the dynamics of homeomorphisms and flows on compact metric spaces, in particular Peano continua. As a translating tool, we construct continuous, symmetric and monotonous fields of local cross sections for an arbitrary flow without singular points. Next, we use this structure in the study of expansive flows on Peano continua. We show that expansive flows have not stable points and that every point contains a non-trivial continuum in its stable set. As a corollary we obtain that no Peano continuum with an open set homeomorphic to the plane admits an expansive flow. In particular, compact surfaces do not admit expansive flows without singular points.

2016, 36(11): 5929-5949
doi: 10.3934/dcds.2016060

*+*[Abstract](467)*+*[PDF](1769.6KB)**Abstract:**

For a stochastically perturbed equation $x_{n+1} =\max\{f(x_n)+l\chi_{n+1}, 0 \}$ with $f(x) < x$ on $(0,m)$, which corresponds to the Allee effect, we observe that for very small perturbation amplitude $l$, the eventual behavior is similar to a non-perturbed case: there is extinction for small initial values in $(0,m-\varepsilon)$ and persistence for $x_0 \in (m + \delta, H]$ for some $H$ satisfying $H > f(H)> m$. As the amplitude grows, an interval $(m-\varepsilon, m + \delta)$ of initial values arises and expands, such that with a certain probability, $x_n$ sustains in $[m, H]$, and possibly eventually gets into the interval $(0,m-\varepsilon)$, with a positive probability. Lower estimates for these probabilities are presented. If $H$ is large enough, as the amplitude of perturbations grows, the Allee effect disappears: a solution persists for any positive initial value.

2016, 36(11): 5951-5970
doi: 10.3934/dcds.2016061

*+*[Abstract](352)*+*[PDF](802.5KB)**Abstract:**

In this work we define a stochastic adding machine associated to the Fibonacci base and to a probabilities sequence $\overline{p}=(p_i)_{i\geq 1}$. We obtain a Markov chain whose states are the set of nonnegative integers. We study probabilistic properties of this chain, such as transience and recurrence. We also prove that the spectrum associated to this Markov chain is connected to the fibered Julia sets for a class of endomorphisms in $\mathbb{C}^2$.

2016, 36(11): 5971-5991
doi: 10.3934/dcds.2016062

*+*[Abstract](518)*+*[PDF](461.5KB)**Abstract:**

In this paper, we study the linear stability of the criss-cross orbit in the planar equal-mass three-body problem. In each period of the criss-cross orbit, the configurations of three masses are switching from a straight line to an isosceles triangle eight times. By analyzing its symmetry properties and variational characterization, we show that the criss-cross orbit is linearly stable via index theory.

2016, 36(11): 5993-6022
doi: 10.3934/dcds.2016063

*+*[Abstract](429)*+*[PDF](524.9KB)**Abstract:**

We study the well-posedness of the Euler-Poincaré-Arnold equations on the semi-direct products of the group of orientation-preserving diffeomorphisms of the circle with itself. To achieve this goal, according to the previous results obtained in [5], we had to extend the results obtained in [10] for the general case of inertia operators of pseudo-differential type.

2016, 36(11): 6023-6064
doi: 10.3934/dcds.2016064

*+*[Abstract](560)*+*[PDF](1337.1KB)**Abstract:**

In this paper we study, from a variational and geometrical point of view, second-order variational problems on Lie groupoids and the construction of variational integrators for optimal control problems. First, we develop variational techniques for second-order variational problems on Lie groupoids and their applications to the construction of variational integrators for optimal control problems of mechanical systems. Next, we show how Lagrangian submanifolds of a symplectic groupoid gives intrinsically the discrete dynamics for second-order systems, both unconstrained and constrained, and we study the geometric properties of the implicit flow which defines the dynamics in a Lagrangian submanifold. We also study the theory of reduction by symmetries and the corresponding Noether theorem.

2016, 36(11): 6065-6099
doi: 10.3934/dcds.2016065

*+*[Abstract](406)*+*[PDF](1355.8KB)**Abstract:**

The configuration manifold $M$ of a mechanical system consisting of two unconstrained rigid bodies in $\mathbb{R}^n$ is a manifold with boundary (typically with singularities.) A full description of the system requires boundary conditions specifying how orbits should be continued after collisions, that is, the assignment of a

*collision map*at each tangent space on the boundary of $M$ giving the post-collision state of the system for each pre-collision state. We give a complete description of the space of linear collision maps satisfying energy and (linear and angular) momentum conservation, time reversibility, and the natural requirement that impulse forces only act at the point of contact of the colliding bodies. These assumptions are stated geometrically in terms of a family of vector subbundles of the tangent bundle to $\partial M$: the

*diagonal, non-slipping,*and

*impulse*subbundles. Collision maps are shown to be the isometric involutions that restrict to the identity on the non-slipping subspace. We then make a few observations of a dynamical nature about

*non-standard billiard systems*, among which is a sufficient condition for the billiard map on the space of boundary states to preserve the canonical measure on constant energy hypersurfaces.

2016, 36(11): 6101-6116
doi: 10.3934/dcds.2016066

*+*[Abstract](406)*+*[PDF](416.0KB)**Abstract:**

In the present study we prove rigorously that in the long-wave limit, the unidirectional solutions of a class of nonlocal wave equations to which the improved Boussinesq equation belongs are well approximated by the solutions of the Camassa-Holm equation over a long time scale. This general class of nonlocal wave equations model bidirectional wave propagation in a nonlocally and nonlinearly elastic medium whose constitutive equation is given by a convolution integral. To justify the Camassa-Holm approximation we show that approximation errors remain small over a long time interval. To be more precise, we obtain error estimates in terms of two independent, small, positive parameters $\epsilon$ and $\delta$ measuring the effect of nonlinearity and dispersion, respectively. We further show that similar conclusions are also valid for the lower order approximations: the Benjamin-Bona-Mahony approximation and the Korteweg-de Vries approximation.

2016, 36(11): 6117-6132
doi: 10.3934/dcds.2016067

*+*[Abstract](606)*+*[PDF](440.5KB)**Abstract:**

This paper is concerned with the classical Timoshenko system for vibrations of thin rods. It has been studied by many authors and most of known results are concerned with decay rates of the energy, controllability and numerical approximations. There are just a few references on the long-time dynamics of such systems. Motivated by this scenario we establish the existence of global and exponential attractors for a class of semilinear Timoshenko systems with linear frictional damping acting on the whole system and without assuming the well-known equal wave speeds condition.

2016, 36(11): 6133-6166
doi: 10.3934/dcds.2016068

*+*[Abstract](516)*+*[PDF](571.6KB)**Abstract:**

We consider a parametric nonlinear Robin problem driven by the $p -$Laplacian plus an indefinite potential and a Carathéodory reaction which is $(p-1) -$ superlinear without satisfying the Ambrosetti - Rabinowitz condition. We prove a bifurcation-type result describing the dependence of the set of positive solutions on the parameter. We also prove the existence of nodal solutions. Our proofs use tools from critical point theory, Morse theory and suitable truncation techniques.

2016, 36(11): 6167-6185
doi: 10.3934/dcds.2016069

*+*[Abstract](404)*+*[PDF](479.1KB)**Abstract:**

We develop several aspects of the infinite-dimensional Weak KAM theory using a random variables' approach. We prove that the infinite-dimensional cell problem admits a viscosity solution that is a fixed point of the Lax-Oleinik semigroup. Furthermore, we show the existence of invariant minimizing measures and calibrated curves defined on $\mathbb{R}$.

2016, 36(11): 6187-6199
doi: 10.3934/dcds.2016070

*+*[Abstract](530)*+*[PDF](368.5KB)**Abstract:**

Let $f$ be a smooth self-map of the $m$-dimensional sphere $S^m$. Under the assumption that $f$ preserves latitudinal foliations with the fibres $S^1$, we estimate from below the number of fixed points of the iterates of $f$. The paper generalizes the results obtained by Pugh and Shub and by Misiurewicz.

2016, 36(11): 6201-6256
doi: 10.3934/dcds.2016071

*+*[Abstract](516)*+*[PDF](1262.9KB)**Abstract:**

This is a survey of results on long time behavior and attractors for Hamiltonian nonlinear partial differential equations, considering the global attraction to stationary states, stationary orbits, and solitons, the adiabatic effective dynamics of the solitons, and the asymptotic stability of the solitary manifolds. The corresponding numerical results and relations to quantum postulates are considered.

This theory differs significantly from the theory of attractors of dissipative systems where the attraction to stationary states is due to an energy dissipation caused by a friction. For the Hamilton equations the friction and energy dissipation are absent, and the attraction is caused by radiation which brings the energy irrevocably to infinity.

2016, 36(11): 6257-6284
doi: 10.3934/dcds.2016072

*+*[Abstract](544)*+*[PDF](563.2KB)**Abstract:**

Let $X$ be a Hadamard manifold, and $\Gamma\subset Is(X)$ a non-elementary discrete subgroup of isometries of $X$ which contains a rank one isometry. We relate the ergodic theory of the geodesic flow of the quotient orbifold $M=X/\Gamma$ to the behavior of the Poincaré series of $\Gamma$. Precisely, the aim of this paper is to extend the so-called theorem of Hopf-Tsuji-Sullivan -- well-known for manifolds of pinched negative curvature -- to the framework of rank one orbifolds. Moreover, we derive some important properties for $\Gamma$-invariant conformal densities supported on the geometric limit set of $\Gamma$.

2016, 36(11): 6285-6306
doi: 10.3934/dcds.2016073

*+*[Abstract](539)*+*[PDF](383.4KB)**Abstract:**

Two exponential stabilization results are proved for a vibrating structure subject to an interfacial slip. More precisely, the structure consists of two identical beams of Timoshenko type and clamped together but allowing for a longitudinal movement between the layers. We will stabilize the system through a transverse friction and also through a viscoelastic damping.

2016, 36(11): 6307-6330
doi: 10.3934/dcds.2016074

*+*[Abstract](482)*+*[PDF](491.4KB)**Abstract:**

In this paper we investigate the problem of prescribing the $Q$-curvature, on the sphere of any dimension with prescribed conical singularities. We also give the asymptotic behaviour of the solutions that we find and we prove their uniqueness in the negative curvature case. We focus mainly on the odd dimensional case, more specifically the three dimensional sphere.

2016, 36(11): 6331-6377
doi: 10.3934/dcds.2016075

*+*[Abstract](484)*+*[PDF](662.3KB)**Abstract:**

We consider classes of diffeomorphisms of Euclidean space with partial asymptotic expansions at infinity; the remainder term lies in a weighted Sobolev space whose properties at infinity fit with the desired application. We show that two such classes of asymptotic diffeomorphisms form topological groups under composition. As such, they can be used in the study of fluid dynamics according to the approach of V. Arnold [1].

2016, 36(11): 6379-6411
doi: 10.3934/dcds.2016076

*+*[Abstract](424)*+*[PDF](773.3KB)**Abstract:**

We analyze the continuum limit of a thresholding algorithm for motion by mean curvature of one dimensional interfaces in various space-time discrete regimes. The algorithm can be viewed as a time-splitting scheme for the Allen-Cahn equation which is a typical model for the motion of materials phase boundaries. Our results extend the existing statements which are applicable mostly in semi-discrete (continuous in space and discrete in time) settings. The motivations of this work are twofolds: to investigate the interaction between multiple small parameters in nonlinear singularly perturbed problems, and to understand the anisotropy in curvature for interfaces in spatially discrete environments. In the current work, the small parameters are the spatial and temporal discretization step sizes: $\triangle x = h$ and $\triangle t = \tau$. We have identified the limiting description of the interfacial velocity in the (i) sub-critical ($h \ll \tau$), (ii) critical ($h = O(\tau)$), and (iii) super-critical ($h \gg \tau$) regimes. The first case gives the classical isotropic motion by mean curvature, while the second produces intricate pinning and de-pinning phenomena, and anisotropy in the velocity function of the interface. The last case produces no motion (complete pinning).

2016, 36(11): 6413-6451
doi: 10.3934/dcds.2016077

*+*[Abstract](399)*+*[PDF](639.0KB)**Abstract:**

For a $C^2$ Axiom A flow $\phi_t: M \longrightarrow M$ on a Riemannian manifold $M$ and a basic set $\Lambda$ for $\phi_t$ we consider the Ruelle transfer operator $L_{f - s \tau + z g}$, where $f$ and $g$ are real-valued Hölder functions on $\Lambda$, $\tau$ is the roof function and $s, z \in \mathbb{C}$ are complex parameters. Under some assumptions about $\phi_t$ we establish estimates for the iterations of this Ruelle operator in the spirit of the estimates for operators with one complex parameter (see [4], [21], [22]). Two cases are covered: (i) for arbitrary Hölder $f,g$ when $|Im z| \leq B |Im s|^\mu$ for some constants $B > 0$, $0 < \mu < 1$ ($\mu = 1$ for Lipschitz $f,g$), (ii) for Lipschitz $f,g$ when $|Im s| \leq B_1 |Im z|$ for some constant $B_1 > 0$ . Applying these estimates, we obtain a non zero analytic extension of the zeta function $\zeta(s, z)$ for $P_f - \epsilon < Re (s) < P_f$ and $|z|$ small enough with a simple pole at $s = s(z)$. Two other applications are considered as well: the first concerns the Hannay-Ozorio de Almeida sum formula, while the second deals with the asymptotic of the counting function $\pi_F(T)$ for weighted primitive periods of the flow $\phi_t.$

2016, 36(11): 6453-6473
doi: 10.3934/dcds.2016078

*+*[Abstract](458)*+*[PDF](474.0KB)**Abstract:**

We study the following Kirchhoff type problem: \begin{equation*} \left\{ \begin{array}{ccc} -\left(a+b\int_{\Omega}|\nabla u|^2dx \right) \Delta u=f(x,u), &\mbox{in} \ \ \Omega, \\ u=0, &\text{on} \ \partial \Omega. \end{array} \right. \end{equation*} Note that $F(x,t)=\int_0^1 f(x,s)ds$ is the primitive function of $f$. In the first result, we prove the existence of solutions by applying the $G-$Linking Theorem when the quotient $\frac{4F(x,t)}{bt^4}$ stays between $\mu_k$ and $\mu_{k+1}$ allowing for resonance with $\mu_{k+1}$ at infinity. In the second result, for the case that the quotient $\frac{4F(x,t)}{bt^4}$ stays between $\mu_1$ and $\mu'_{2}$ allowing for resonance with $\mu'_{2}$ at infinity, we find a nontrivial solution by using the classical Linking Theorem and argument of the characterization of $\mu'_2$. Meanwhile, similar results are obtained for degenerate problem.

2016, 36(11): 6475-6485
doi: 10.3934/dcds.2016079

*+*[Abstract](442)*+*[PDF](378.7KB)**Abstract:**

We extend a result regarding the Random Backward Iteration algorithm for drawing Julia sets (known to work for certain rational semigroups containing a non-Möbius element) to a class of Möbius semigroups which includes certain settings not yet been dealt with in the literature, namely, when the Julia set is not a thick attractor in the sense given in [8].

2016, 36(11): 6487-6522
doi: 10.3934/dcds.2016080

*+*[Abstract](567)*+*[PDF](584.5KB)**Abstract:**

We consider the evolutionary Hamilton-Jacobi equation depending on the unknown function with the continuous initial condition on a connected closed manifold. Under certain assumptions on $H(x,u,p)$ with respect to $u$ and $p$, we provide an

*implicit variational principle*. By introducing an implicitly defined

*solution semigroup*and an

*admissible value set*$\mathcal{C}_H$, we extend weak KAM theory to certain more general cases, in which $H$ depends on the unknown function $u$ explicitly. As an application, we show that for $0\notin \mathcal{C}_H$, as $t\rightarrow +\infty$, the viscosity solution of \begin{equation*} \begin{cases} \partial_tu(x,t)+H(x,u(x,t),\partial_xu(x,t))=0,\\ u(x,0)=\varphi(x), \end{cases} \end{equation*} diverges, otherwise for $0\in \mathcal{C}_H$, it converges to a weak KAM solution of the stationary Hamilton-Jacobi equation \begin{equation*} H(x,u(x),\partial_xu(x))=0. \end{equation*}

2016, 36(11): 6523-6532
doi: 10.3934/dcds.2016081

*+*[Abstract](421)*+*[PDF](360.8KB)**Abstract:**

We give a new proof of Bian-Guan's constant rank theorem for nonlinear elliptic equations. Our approach is to use a linear expression of the eigenvalues of the Hessian instead of quotients of elementary symmetric functions.

2016, 36(11): 6533-6538
doi: 10.3934/dcds.2016082

*+*[Abstract](394)*+*[PDF](287.8KB)**Abstract:**

In this paper we give a complicated distributional chaos, that is, a mixing dynamical system with an invariant, extremal and transitive distributionally scrambled set.

2016, 36(11): 6539-6555
doi: 10.3934/dcds.2016083

*+*[Abstract](424)*+*[PDF](428.4KB)**Abstract:**

In this paper we provide characterization of integrablity of a system of vector fields via inverse Jacobian multipliers (matrix) and normalizers of smooth (or holomorphic) vector fields. These results improve and extend some well known ones, including the classical holomorphic Frobenius integrability theorem. Here we obtain the exact expression of an integrable system of vector fields acting on a smooth function via their known common first integrals. Moreover we characterize the relations between the integrability and the existence of normalizers for a system of vector fields. In the case of integrability of a system of vector fields we not only prove the existence of normalizers but also provide their exact expressions.

2016, 36(11): 6557-6580
doi: 10.3934/dcds.2016084

*+*[Abstract](367)*+*[PDF](515.5KB)**Abstract:**

The paper investigates the well-posedness and longtime dynamics of the semilinear wave equation with gentle dissipation： $u_{tt}-\triangle u+\gamma(-\triangle)^{\alpha} u_{t}+f(u)=g(x)$, with $\alpha\in(0,1/2)$. The main results are concerned with the relationships among the growth exponent $p$ of nonlinearity $f(u)$ and the well-posedness and longtime behavior of solutions of the equation. We show that (i) the well-posedness and longtime dynamics of the equation are of characters of parabolic equations as $1 \leq p < p^* \equiv \frac{N + 4\alpha}{(N-2)^+}$; (ii) the subclass $\mathbb{G}$ of limit solutions has a weak global attractor as $p^* \leq p < p^{**}\equiv \frac{N+2}{N-2}\ (N \geq 3)$.

2016, 36(11): 6581-6597
doi: 10.3934/dcds.2016085

*+*[Abstract](515)*+*[PDF](438.7KB)**Abstract:**

This article is devoted to the study of the irregular sets of Birkhoff averages in some nonuniformly hyperbolic systems via Pesin theory. Particularly, we give a conditional variational principle for the topological entropy of the irregular sets. Our result can be applied (i) to the diffeomorphisms on surfaces, (ii) to the nonuniformly hyperbolic diffeomorphisms described by Katok.

2016, 36(11): 6599-6622
doi: 10.3934/dcds.2016086

*+*[Abstract](436)*+*[PDF](501.0KB)**Abstract:**

In this work we consider a class of degenerate analytic maps of the form \begin{eqnarray*} \left\{ \begin{array}{l} \bar{x} =x+y^{m}+\epsilon f_1(x,y,\theta,\epsilon)+h_1(x,y,\theta,\epsilon),\\ \bar{y}=y+x^{n}+\epsilon f_2(x,y,\theta,\epsilon)+h_2(x,y,\theta,\epsilon),\\ \bar{\theta}=\theta+\omega, \end{array} \right. \end{eqnarray*} where $mn>1,n\geq m,$ $h_1 \ \mbox{and} \ h_2$ are of order $n+1$ in $z,$ and $\omega=(\omega_1,\omega_2,\ldots,\omega_{d})\in \Bbb{R}^{d}$ is a vector of rationally independent frequencies. It is shown that, under a generic non-degeneracy condition on $f$, if $\omega$ is Diophantine and $\epsilon>0$ is small enough, the map has at least one weakly hyperbolic invariant torus.

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