Attractor for the nonlinear Schr?dinger equation with a nonlocal nonlinear term

In this paper, we consider the long-time behavior of solutions of the dissipative 1D nonlinear Schrödinger (NLS) equation with nonlocal integral term and with periodic boundary conditions. We prove the existence of the global attractor \( \mathcal{A} \) for the nonlocal equation in the strong topology of H ¹(Ω). We also prove that the global attractor is regular, i.e., \( \mathcal{A} \subset {H^2}\left( \Omega \right) \), assuming that f(x) is of class C ². Furthermore, we estimate the number of the determining modes for this equation.     

Limit Cycles for Multidimensional Vector Fields. The Elliptic Case

We consider polynomial vector fields of the form and their polynomial perturbations of degree n. We present a sufficient condition that the perturbed system has an invariant surface close to the plane z = 0. We study limit cycles which appear on this surface. The linearized condition for limit cycles, bifurcating from the curves y ² – x ³ + 3x = h, leads to a certain 2- dimensional integral (which generalizes the elliptic integrals). We show that this integral has a representation R ₁(h)I ₁ + + R _e(h)I _e, where R _j (h) are rational functions with degrees of numerators and denominators bounded by O(n). In the case of constant and one-dimensional matrix A(x,y) we estimate the number of zeros of the integral by const n.     

Integration of Complex Differential Equations

Let X be a polynomial vector field in ²; then it defines an algebraic foliation on P(2). If admits a Liouvillian first integral on P(2), then it is transversely affine outside some algebraic invariant curve S P(2). If, moreover, for some irreducible component S₀ S, the singularities q Sing S are generic, then either is given by a closed rational 1-form or it is a rational pull-back from a Bernoulli foliation This result has several applications such as the study of foliations with algebraic limit sets on P(2)(2), the classification polynomial complete vector fields over ², and topological rigidity of foliations on P(2). We also address the problem of moderate integration for germs of complex ordinary differential equations.     

Rolling of manifolds without spinning

The control model of rolling of a Riemannian manifold (M; g) onto another one $ \left( {\hat{M},\hat{g}} \right) $ consists of a state space Q of relative orientations (isometric linear maps) between their tangent spaces equipped with a so-called rolling distribution $ {\mathcal D} $ _R, which models the natural constraints of no-spinning and no-slipping of the rolling motion. It turns out that the distribution $ {\mathcal D} $ _R can be built as a sub-distribution of a so-called no-spinning distribution $ {{\mathcal{D}}_{\overline{\nabla}}} $ on Q that models only the no-spinning constraint of the rolling motion. One is thus motivated to study the control problem associated to $ {{\mathcal{D}}_{\overline{\nabla}}} $ and, in particular, the geometry of $ {{\mathcal{D}}_{\overline{\nabla}}} $ -orbits. Moreover, the definition of $ {{\mathcal{D}}_{\overline{\nabla}}} $ (contrary to the definition of $ {\mathcal D} $ _R) makes sense in the general context of vector bundles equipped with linear connections. The purpose of this paper is to study the distribution $ {{\mathcal{D}}_{\overline{\nabla}}} $ determined by the product connection $ \nabla \times \hat{\nabla} $ on a tensor bundle $ {E^{*}}\otimes \hat{E}\to M\times \hat{M} $ induced by linear connections ?, $ \hat{\nabla} $ on vector bundles $ E\to M,\,\,\,\hat{E}\to \hat{M} $ . We describe completely the orbit structure of $ {{\mathcal{D}}_{\overline{\nabla}}} $ in terms of the holonomy groups of ?, $ \hat{\nabla} $ and characterize the integral manifolds of it. Moreover, we describe the general formulas for the Lie brackets of vector elds in $ {E^{*}}\otimes \hat{E} $ in terms of $ {{\mathcal{D}}_{\overline{\nabla}}} $ and the vertical tangent distribution of $ {E^{*}}\otimes \hat{E}\to M\times \hat{M} $ . In the particular case of tangent bundles $ TM\to M,\,\,\,T\hat{M}\to \hat{M} $ and Levi-Civita connections, we describe in more detail how $ {{\mathcal{D}}_{\overline{\nabla}}} $ is related to the above mentioned rolling model, where these Lie brackets formulas provide an important tool for the study of controllability of the related control system.     

Regions Where the Exponential Map at Regular Points of Sub-Riemannian Manifolds is a Local Diffeomorphism

For a k-step sub-Riemannian manifold which admits a bracket generating vector at a point, we describe a region near the point where the exponential map is a local diffeomorphism. This is proved by taking the Taylor series of the exponential map and calculating the first nonzero term, which has order , where n is the topological dimension and is the Hausdorff dimension of the metric space associated to the sub-Riemannian manifold.      

Existence of Weak Solutions for Generalized Quasilinear Schrödinger Equations

This paper shows the existence of nontrivial weak solutions for the generalized quasilinear Schrödinger equations

$$ -div(g^{p}(u)|\nabla u|^{p-2}\nabla u)+g^{p-1}(u)g^{\prime}(u)|\nabla u|^{p}+ V(x)|u|^{p-2}u=h(u),\,\, x\in \mathbb{R}^{N}, $$

where N ≥ 3, \(g(s): \mathbb {R}\rightarrow \mathbb {R}^{+}\) is C ¹ nondecreasing function with respect to |s|, V is a positive potential bounded away from zero and h(u) is a nonlinear term of subcritical type. By introducing a variable replacement and using minimax methods, we show the existence of a nontrivial solution in \(C^{\alpha }_{loc}(\mathbb {R}^{N})\).

     

Multifractal Analysis of Ergodic Averages: A Generalization of Eggleston's Theorem

Let V be a finite set, S be an infinite countable commutative semigroup, { _s , s S} be the semigroup of translations in the function space X = V ^S , A = {A _n } be a sequence of finite sets in S, f be a continuous function on X with values in a separable real Banach space B, and let B. We introduce in X a scale metric generating the product topology. Under some assumptions on f and A, we evaluate the Hausdorff dimension of the set X _f,,Adefined by the following formula:

On High Dimensional Schrödinger Equation with Quasi-Periodic Potentials

In this paper, we consider the high dimensional Schrödinger equation \( -\frac {d^{2}y}{dt^{2}} + u(t)y= Ey, y\in \mathbb {R}^{n}, \) where u(t) is a real analytic quasi-periodic symmetric matrix, \(E= \text {diag}({\lambda _{1}^{2}}, \ldots , {\lambda _{n}^{2}})\) is a diagonal matrix with λ _j >0,j=1,…,n, being regarded as parameters, and prove that if the basic frequencies of u satisfy a Bruno-Rüssmann’s non-resonant condition, then for most of sufficiently large λ _j ,j=1,…,n, there exist n pairs of conjugate quasi-periodic solutions.     

Genericity for Non-Wandering Surface Flows

Consider the set \(\chi ^{0}_{\text {nw}}\) of non-wandering continuous flows on a closed surface M. Then we show that such a flow can be approximated by a non-wandering flow v such that the complement M?Per(v) of the set of periodic points is the union of finitely many centers and finitely many homoclinic saddle connections. Using the approximation, the following are equivalent for a continuous non-wandering flow v on a closed connected surface M: (1) the non-wandering flow v is topologically stable in \(\chi ^{0}_{\text {nw}}\); (2) the orbit space M/v is homeomorphic to a closed interval; (3) the closed connected surface M is not homeomorphic to a torus but consists of periodic orbits and at most two centers. Moreover, we show that a closed connected surface has a topologically stable continuous non-wandering flow in \(\chi ^{0}_{\text {nw}}\) if and only if the surface is homeomorphic to either the sphere \(\mathbb {S}^{2}\), the projective plane \(\mathbb {P}^{2}\), or the Klein bottle \(\mathbb {K}^{2}\).     

Multifractal Spectrum for Barycentric Averages

Let \(\left (X,\nu \right ) \) and Y be a measured space and a C A T(0) space, respectively. If \(\mathcal {M}_{2}(Y)\) is the set of measures on Y with finite second moment then a map \(bar:\mathcal {M}_{2}(Y)\rightarrow Y\) can be defined. Also, for any x∈X and for a map \(\varphi :X\rightarrow Y\), a sequence \(\left \{\mathcal {E}_{N,\varphi }(x)\right \} \) of empirical measures on Y can be introduced. The sequence \(\left \{ bar\left (\mathcal {E}_{N,\varphi }(x)\right ) \right \} \) replaces in C A T(0) spaces the usual ergodic averages for real valuated maps. It converges in Y (to a map \(\overline {\varphi }\left (x\right )\)) almost surely for any x∈X (Austin J Topol Anal. 2011;3: 145–152). In this work, we shall consider the following multifractal decomposition in X:

$$K_{y,\varphi}=\left\{ x:\lim\limits_{N\rightarrow\infty}bar\left(\mathcal{E}_{N,\varphi}(x)\right) =y\right\} , $$

and we will obtain a variational formula for this multifractal spectrum.

     

First Variation of the Hausdorff Measure of Non-horizontal Submanifolds in Sub-Riemannian Stratified Lie Groups

We determine necessary conditions for a non-horizontal submanifold of a sub-Riemannian stratified Lie group to be of minimal measure. We calculate the first variation of the measure for a non-horizontal submanifold and find that the minimality condition implies the tensor equation H + σ = 0, where H is analogous to the mean curvature and σ is the mean torsion. We also discuss new examples of minimal non-horizontal submanifolds in the Heisenberg group, in particular surfaces in \(\mathbb {H}^{2}\).     

Global Gradient Estimates for Nonlinear Elliptic Equations with Vanishing Neumann Data in a Convex Domain

In this paper, we obtain the following global L ^q estimates

$$\left|\mathbf{f}\right|^{p } \in L^{q}({\Omega}) \Rightarrow \left|\nabla u\right|^{p } \in L^{q}({\Omega}) \quad \text{for any} ~~q\ge 1 $$

in a convex domain Ω of weak solutions for nonlinear elliptic equations of p-Laplacian type with vanishing Neumann data

$$\begin{array}{@{}rcl@{}} \text{div} \left( \left( A \nabla u \cdot \nabla u\right)^{\frac{p -2}{2}} A \nabla u \right) & =& \text{div} \left( | \mathbf{f}|^{p-2} \mathbf{f} \right) \quad\text{in} ~~{\Omega},\\ \left( A \nabla u \cdot \nabla u\right)^{\frac{p -2}{2}} A \nabla u \cdot \mathbf{\nu} &=& | \mathbf{f}|^{p -2} \mathbf{f}\cdot \mathbf{\nu} \quad \quad \text{on}~~ \partial{\Omega}, \end{array} $$

where ν is the outwardpointing unit normal to ?Ω. Our argument is based on the works of Banerjee and Lewis (Nonlinear Anal 100:78–85, 2014), Kinnunen and Zhou (Comm Partial Differential Equations 24(11&12):2043–2068, 1999, Differential and Integral Equations 14(4):475–492, 2001), and Byun, Wang, and Zhou (Comm Pure Appl Math 57(10):1283–1310, 2004, J Funct Anal 20(3):617–637, 2007). In the proof of the above result, we only focus on the boundary case while the interior case can be obtained as a corollary.

     

Nonoscillation and Exponential Stability of Delay Differential Equations with Oscillating Coefficients

We prove that under some additional conditions, the nonoscillation of the scalar delay differential equation

Foliation Admitting Recurrent Leaves of Infinite Depth on Compact Two-Manifolds

Let be a foliation on a two-manifold M. Denote the topology closure of each leaf L of by . A sequence of proper inclusions , where each L _i is a recurrent leaf of , is called a nest of length k. The maximal length of various nests is known as the depth of the foliations . It is well known that if is orientable and M is compact, the depth of is at most one. In this paper, we show that on any orientable, compact two-manifold, there exist nonorientable foliations of infinite depth. This work negatively answers the Aranson conjecture [1]. This work was partially supported by FAPESP-Proj. Tematico No. 03/03107-9.     

Real-Formal Orbital Rigidity for Germs of Real Analytic Vector Fields on the Real Plane

For \(n \geqslant 2\), we consider \(\mathcal {V}^{\mathbb {R}}_{n}\) the class of germs of real analytic vector fields on \(\left (\mathbb {R}^{2}, \widehat {0}\right )\) with zero (n?1)-jet and nonzero n-jet. We prove, for generic germs of \(\mathcal {V}^{\mathbb {R}}_{n}\), that the real-formal orbital equivalence implies the real-analytic orbital equivalence, that is, the real-formal orbital rigidity takes place. This is the real analytic version of Voronin’s formal orbital rigidity theorem.     

Monotonic homotopy for trajectories of young systems

Let $ \mathbb{Y} $ be a Young system. Assume that the accessible set $ \mathcal{A} $ ( $ \mathbb{Y} $ ; x) of $ \mathbb{Y} $ starting from x is locally and semi-locally simply connected by trajectories of $ \mathbb{Y} $ . We prove that the covering space Γ( $ \mathbb{Y} $ ; x) of p-monotonically homotopic trajectories is identified to the universal covering space of $ \mathcal{A} $ ( $ \mathbb{Y} $ ; x).     

Second-Order Viability Problems with Perturbation in Hilbert Spaces

In this paper, we study the existence of viable solutions to the differential inclusion

$ \ddot{x}(t) \in f\left( {t,x(t),\dot{x}(t)} \right) + F\left( {x(t),\dot{x}(t)} \right), $

where f is a Carathéodory single-valued map and F is an upper semi-continuous multifunction with compact values contained in the Clarke subdifferential ? _c V of an uniformly regular function V.

     

Sphere Bundles Transverse to Holomorphic Vector Fields

We prove that for every pair of nonzero complex numbers λ ₁ and λ ₂ with \(\frac {\lambda _{1}}{\lambda _{2}}\not \in \mathbb {R}\) there is an embedding \(S^{2}\times S^{1}\rightarrow \mathbb {C}^{2}\) transverse to the linear holomorphic vector field \(Z(x,y)=\lambda _{1}x\frac {\partial }{\partial x}+\lambda _{2} y\frac {\partial }{\partial y}\) . This extends a previous result by Ito (1989).     

Output-Based Stabilization of Timoshenko Beam with the Boundary Control and Input Distributed Delay

In this paper, we consider the output-feedback exponential stabilization of Timoshenko beam with the boundary control and input distributed delay. Suppose that the outputs of controllers are of the forms \(\alpha _{1}u_{1}(t)+\beta _{1}u_{1}(t-\tau )+{\int }_{-\tau }^{0}g_{1}(\eta )u_{1} (t+\eta )d\eta \) and \(\alpha _{2}u_{2}(t)+\beta _{2}u_{2}(t-\tau ) +{\int }_{-\tau }^{0}g_{2}(\eta )u_{2}(t+\eta )d\eta \) respectively, where u ₁(t) and u ₂(t) are the inputs of controllers. Using the tricks of the Luenberger observer and partial state predictor, we translate the system with delay into a system without delay. And then, we design the feedback controls to stabilize the system without delay. Finally, we prove that under the choice of such controls, the original system also is stabilized exponentially.