p-Laplacian方程的Aubry-Mather集

本文通过引进合适的作用-角变量变换并结合新的估计方法,对p-Laplacian方程的Poincare映射应用推广的Aubry-Mather定理,证明了一类p-Laplacian方程的Aubry-Mather集的存在性.     

一类可逆系统的Aubry-Mather集

本文利用推广的Aubry-Mather定理,获得了一类二阶可逆微分方程Aubry-Mather集存在的充分性条件.     

次线性不对称Duffing 方程的Aubry-Mather 集

本文通过引进合适的作用— 角变量变换并结合新的估计方法, 对次线性不对称Duffing 方程的Poincaré 映射应用推广的Aubry-Mather 定理, 获得了一类次线性不对称Duffing 方程的Aubry-Mather 集存在的充分性条件.     

MATHER SETS FOR SUBLINEAR DUFFING EQUATIONS

ＭＡＴＨＥＲＳＥＴＳＦＯＲＳＵＢＬＩＮＥＡＲＤＵＦＦＩＮＧＥＱＵＡＴＩＯＮＳ￥ＱＩＡＮＤＩＮＧＢＩＡＮ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＳｕｚｈｏｕＵｎｉｖｅｒｓｉｔｙ，２１５００６，Ｃｈｉｎａ．）Ａｂｓｔｒａｃｔ：Ｔｈｅｅｘｉｓｔｅ...     

Connecting orbits among Denjoy minimal sets for monotone twist map

In a Birkhoff region of instability for an exact area-preserving twist map, we construct some orbits connecting distinct Denjoy minimal sets. These sets correspond to the local, instead of global minimum of the Lagrangian action. In the earlier work, Mather constructed connecting orbits among Aubry-Mather sets and the global minimizer of the Lagrangian action.     

A General Mechanism of Diffusion in Hamiltonian Systems: Qualitative Results

We present a general mechanism to establish the existence of diffusing orbits in a large class of nearly integrable Hamiltonian systems. Our approach is based on following the “outer dynamics” along homoclinic orbits to a normally hyperbolic invariant manifold. The information on the outer dynamics is encoded by a geometrically defined “scattering map.” We show that for every finite sequence of successive iterations of the scattering map, there exists a true orbit that follows that sequence, provided that the inner dynamics is recurrent. We apply this result to prove the existence of diffusing orbits that cross large gaps in a priori unstable models of arbitrary degrees of freedom, when the unperturbed Hamiltonian is not necessarily convex and the induced inner dynamics is not necessarily a twist map, and the perturbation satisfies explicit conditions that are generic. We also mention several other applications where this mechanism is easy to verify (analytically or numerically), such as the planar elliptic restricted three-body problem and the spatial circular restricted three-body problem. Our method differs, in several crucial aspects, from earlier works. Unlike the well-known “two-dynamics” approach, the method we present here relies on the outer dynamics alone. There are virtually no assumptions on the inner dynamics, such as on existence of its invariant objects (e.g., primary and secondary tori, lower-dimensional hyperbolic tori, and their stable/unstable manifolds, Aubry-Mather sets), which are not used at all. © 2019 Wiley Periodicals, Inc.     

Hamilton-Jacobi methods for Vakonomic Mechanics

We extend the theory of Aubry-Mather measures to Hamiltonian systems that arise in vakonomic mechanics and sub-Riemannian geometry. We use these measures to study the asymptotic behavior of (vakonomic) action-minimizing curves, and prove a bootstrapping result to study the partial regularity of solutions of convex, but not strictly convex, Hamilton-Jacobi equations.      

The Mather problem for lower semicontinuous Lagrangians

In this paper we develop the Aubry-Mather theory for Lagrangians in which the potential energy can be discontinuous. Namely we assume that the Lagrangian is lower semicontinuous in the state variable, piecewise smooth with a (smooth) discontinuity surface, as well as coercive and convex in the velocity. We establish existence of Mather measures, various approximation results, partial regularity of viscosity solutions away from the singularity, invariance by the Euler–Lagrange flow away from the singular set, and further jump conditions that correspond to conservation of energy and tangential momentum across the discontinuity.     

不相交斯泰纳和柯克曼三元系大集两定理(一)

给定一个斯泰纳或柯克曼三元系,介绍其大集的生成方法;得到其大集存在条件的判据是一个位差各异的循环数;柯克曼三元系其大集判据是,不能分解的柯克曼三元系有大集,能分解的无大集.     

Regions of Applicability of Aubry-Mather Theory for Non-convex Hamiltonian

Herman constructed an autonomous system of two degrees of freedom which says that in non-convex situations, oscillations do happen and Aubry-Mather Theory cannot apply (see the results due to W. F. Chen in 1992). In this paper, it is shown that although the orbits could visit a region far away from the initial point in phase space, they can only exist in some fixed regions in I = (I ₁, I ₂) plane. Moreover, Aubry-Mather Theory can be applied outside the regions.     

Weighted Variational Inequalities

In this paper, we introduce weighted variational inequalities over product of sets and system of weighted variational inequalities. It is noted that the weighted variational inequality problem over product of sets and the problem of system of weighted variational inequalities are equivalent. We give a relationship between system of weighted variational inequalities and systems of vector variational inequalities. We define several kinds of weighted monotonicities and establish several existence results for the solution of the above-mentioned problems under these weighted monotonicities. We introduce also the weighted generalized variational inequalities over product of sets, that is, weighted variational inequalities for multivalued maps and systems of weighted generalized variational inequalities. Extensions of weighted monotonicities for multivalued maps are also considered. The existence of a solution of weighted generalized variational inequalities over product of sets is also studied. The existence results for a solution of weighted generalized variational inequality problem give also the existence of solutions of systems of generalized vector variational inequalities. The first and third author express their thanks to the Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia for providing excellent research facilities. The authors are also grateful to the referees for comments and suggestions improving the final draft of this paper.     

Existence of infinite non-Birkoff periodic orbits for area-preserving monotone twist maps of cylinders

The exact monotone twist map of infinite cylinders in the Birkhoff region of instability is studied. A variational method based on Aubry-Mather theory is used to discover infinitely many non-Birkhoff periodic orbits of fixed rotation number sufficiently close to some irrational number for which the angular invariant circle does not exist.     

On the possibility of globally stabilizing invariant sets

In this paper we study the possibility of globally stabilizinginvariant sets of autonomous continuous nonlinear systems bythe state (output) feedback control law. We show that the topologicalstructure of invariant sets can give rise to obstruction tothe existence of a continuous control law for global stabilizationof invariant sets.     

Integral sets of systems of difference equations

The problem of existence of integral sets of systems of difference equations is studied. We establish sufficient conditions for the existence of these sets and their stability. For the system under consideration, the behavior of trajectories that originate in a sufficiently small neighborhood of integral sets is investigated.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 12, pp. 1613–1621, December, 1993.     

Invariant sets and Knaster-Tarski principle

Our aim is to point out the applicability of the Knaster-Tarski fixed point principle to the problem of existence of invariant sets in discrete-time (multivalued) semi-dynamical systems, especially iterated function systems.     

Ensembles invariants non enlacés des difféomorphismes du tore et de l’anneau

We introduce the notion of unlinked invariant set for a diffeomorphism of the 2-torus or the closed annulus and we look at the properties of these sets which generalize the Aubry-Mather sets. We prove that for any irrational number in the rotation set of an area-preserving diffeomorphism of the annulus, there exists an unliked invariant set whose rotation set is reduced to . In the same way we prove that any minimal diffeomorphism of the 2-torus homotopic to the identity may be approximated in the C⁰-topology by a periodic diffeomorphism.

---

---

Mathematics Subject Classification (2000) 37E30, 37E45, 37J10     

Quasiperiodic Dynamics in Bose-Einstein Condensates in Periodic Lattices and Superlattices

We employ KAM theory to rigorously investigate quasiperiodic dynamics in cigar-shaped Bose-Einstein condensates (BEC) in periodic lattices and superlattices. Toward this end, we apply a coherent structure ansatz to the Gross-Pitaevskii equation to obtain a parametrically forced Duffing equation describing the spatial dynamics of the condensate. For shallow-well, intermediate-well, and deep-well potentials, we find KAM tori and Aubry-Mather sets to prove that one obtains mostly quasiperiodic dynamics for condensate wave functions of sufficiently large amplitude, where the minimal amplitude depends on the experimentally adjustable BEC parameters. We show that this threshold scales with the square root of the inverse of the two-body scattering length, whereas the rotation number of tori above this threshold is proportional to the amplitude. As a consequence, one obtains the same dynamical picture for lattices of all depths, as an increase in depth essentially affects only scaling in phase space. Our approach is applicable to periodic superlattices with an arbitrary number of rationally dependent wave numbers.     

On the existence of periodic solutions for differential inclusions

In this paper we study the existence of periodic solutions for differential inclusions. We prove existence theorems under various sets of hypotheses for both the nonconvex and convex problems. Also we show the existence of extreme solutions. Some feedback control systems are also considered.     

The large-time behavior of solutions of the Cauchy-Dirichlet problem for Hamilton-Jacobi equations

We investigate the large-time behavior of viscosity solutions of the Cauchy-Dirichlet problem (CD) for Hamilton-Jacobi equations on bounded domains. We establish general convergence results for viscosity solutions of (CD) by using the Aubry-Mather theory.