共查询到20条相似文献,搜索用时 46 毫秒
1.
A useful tool for studying nonlinear differential equations is index theory. For symplectic paths on bounded intervals, the index theory has been completely established, which revealed tremendous applications in the study of periodic orbits of Hamiltonian systems. Nevertheless, analogous questions concerning homoclinic orbits are still left open. In this paper we use a geometric approach to set up Maslov index for homoclinic orbits of Hamiltonian systems. On the other hand, a relative Morse index for homoclinic orbits will be derived through Fredholm index theory. It will be shown that these two indices coincide. 相似文献
2.
In this paper we prove Morse type inequalities for the contractible 1-periodic solutions of time dependent Hamiltonian differential equations on those compact symplectic manifolds M for which the symplectic form and the first Chern class of the tangent bundle vanish over π2 (M). The proof is based on a version of infinite dimensional Morse theory which is due to Floer. The key point is an index theorem for the Fredholm operator which plays a central role in Floer homology. The index formula involves the Maslov index of nondegenerate contractible periodic solutions. This Maslov index plays the same role as the Morse index of a nondegenerate critical point does in finite dimensional Morse theory. We shall use this connection between Floer homology and Maslov index to establish the existence of infinitely many periodic solutions having integer periods provided that every 1-periodic solution has at least one Floquet multiplier which is not equal to 1. 相似文献
3.
Petra Šindelá?ová 《Topology and its Applications》2007,154(6):1097-1106
In the present paper the study of flows on n-manifolds in particular in dimension three, e.g., R3, is motivated by the following question. Let A be a compact invariant set in a flow on X. Does every neighbourhood of A contain a movable invariant set M containing A? It is known that a stable solenoid in a flow on a 3-manifold has approximating periodic orbits in each of its neighbourhoods. The solenoid with the approximating orbits form a movable set, although the solenoid is not movable. Not many such examples are known. The main part of the paper consists of constructing an example of a set in R3 that is not stable, is not a solenoid, and is approximated by Denjoy-like invariant sets instead of periodic orbits. As in the case of a solenoid, the constructed set is an inverse limit of its approximating sets. This gives a partial answer to the above question. 相似文献
4.
We first introduce an invariant index for G-equivariant elliptic differential operators on a locally compact manifold M admitting a proper cocompact action of a locally compact group G. It generalizes the Kawasaki index for orbifolds to the case of proper cocompact actions. Our invariant index is used to show that an analog of the Guillemin-Sternberg geometric quantization conjecture holds if M is symplectic with a Hamiltonian action of G that is proper and cocompact. This essentially solves a conjecture of Hochs and Landsman. 相似文献
5.
Anna Capietto Francesca Dalbono Alessandro Portaluri 《Nonlinear Analysis: Theory, Methods & Applications》2010,72(6):2874-2890
We prove a multiplicity result for a class of strongly indefinite nonlinear second order asymptotically linear systems with Dirichlet boundary conditions. The key idea for the proof is to bring together the classical shooting method and the Maslov index of the linear Hamiltonian systems associated to the asymptotic limits of the given nonlinearity. 相似文献
6.
Thierry Barbot 《Commentarii Mathematici Helvetici》1995,70(1):113-160
Let Φ′ be an Anosov flow on a (non atoroidal) 3-manifoldM. We say that an incompressible torusT embedded inM admits an optimal position with respect to Φ′ if it is isotopic to a torus transverse to Φ′ outside a finite number of periodic
orbits contained inT (there's an additional condition we dont's mention here). The first remark is that such an optimal position is quasi unique,
i.e., we prove that if two tori in optimal position are homotopics inM, then they are homotopics along the flow. Then we give some sufficient condition for a torus admiting an optimal position.
Eventually, we show that if a finite collection of disjoint tori is such that each torus admits an optimal position, then
these optimal positions can be chosen disjoints one from each other.
相似文献
7.
Michael Entov 《Inventiones Mathematicae》2001,146(1):93-141
Given a closed symplectic manifold (M,ω) we introduce a certain quantity associated to a tuple of conjugacy classes in the universal cover of the group Ham (M,ω) by means of the Hofer metric on Ham (M,ω). We use pseudo-holomorphic curves involved in the definition of the multiplicative structure on the Floer cohomology of
a symplectic manifold (M,ω) to estimate this quantity in terms of actions of some periodic orbits of related Hamiltonian flows. As a corollary we
get a new way to obtain Agnihotri-Belkale-Woodward inequalities for eigenvalues of products of unitary matrices. As another
corollary we get a new proof of the geodesic property (with respect to the Hofer metric) of Hamiltonian flows generated by
certain autonomous Hamiltonians. Our main technical tool is K-area defined for Hamiltonian fibrations over a surface with
boundary in the spirit of L. Polterovich’s work on Hamiltonian fibrations over S
2.
Oblatum 23-II-2001 & 9-V-2001?Published online: 20 July 2001 相似文献
8.
Weinan E 《纯数学与应用数学通讯》1999,52(7):811-828
Consider a Hamiltonian system with Hamiltonian of the form H(x, t, p) where H is convex in p and periodic in x, and t and x ∈ ℝ1. It is well‐known that its smooth invariant curves correspond to smooth Z2‐periodic solutions of the PDE ut + H(x, t, u)x = 0. In this paper, we establish a connection between the Aubry‐Mather theory of invariant sets of the Hamiltonian system and Z2‐periodic weak solutions of this PDE by realizing the Aubry‐Mather sets as closed subsets of the graphs of these weak solutions. We show that the complement of the Aubry‐Mather set on the graph can be viewed as a subset of the generalized unstable manifold of the Aubry‐Mather set, defined in (2.24). The graph itself is a backward‐invariant set of the Hamiltonian system. The basic idea is to embed the globally minimizing orbits used in the Aubry‐Mather theory into the characteristic fields of the above PDE. This is done by making use of one‐ and two‐sided minimizers, a notion introduced in [12] and inspired by the work of Morse on geodesics of type A [26]. The asymptotic slope of the minimizers, also known as the rotation number, is given by the derivative of the homogenized Hamiltonian, defined in [21]. As an application, we prove that the Z2‐periodic weak solution of the above PDE with given irrational asymptotic slope is unique. A similar connection also exists in multidimensional problems with the convex Hamiltonian, except that in higher dimensions, two‐sided minimizers with a specified asymptotic slope may not exist. © 1999 John Wiley & Sons, Inc. 相似文献
9.
We show that certain mechanical systems, including a geodesic flow in any dimension plus a quasi-periodic perturbation by a potential, have orbits of unbounded energy.The assumptions we make in the case of geodesic flows are:
- (a)
- The metric and the external perturbation are smooth enough.
- (b)
- The geodesic flow has a hyperbolic periodic orbit such that its stable and unstable manifolds have a tranverse homoclinic intersection.
- (c)
- The frequency of the external perturbation is Diophantine.
- (d)
- The external potential satisfies a generic condition depending on the periodic orbit considered in (b).
10.
Chungen LIU 《数学年刊B辑(英文版)》2006,27(4):441-458
Abstract
The Maslov P-index theory for a symplectic path is defined. Various properties of this index theory such as homotopy invariant, symplectic
additivity and the relations with other Morse indices are studied. As an application, the non-periodic problem for some asymptotically
linear Hamiltonian systems is considered.
*Project supported by the National Natural Science Foundation of China (No.10531050) and FANEDD. 相似文献
11.
Shanzhong Sun 《Journal of Fixed Point Theory and Applications》2017,19(1):299-343
Gutzwiller’s famous semiclassical trace formula plays an important role in theoretical and experimental quantum mechanics with tremendous success. We review the physical derivation of this deep periodic orbit theory in terms of the phase space formulation with a view toward the Hamiltonian dynamical systems. The Maslov phase appearing in the trace formula is clarified by Meinrenken as Conley–Zehnder index for periodic orbits of Hamiltonian systems. We also survey and compare various versions of Maslov indices to establish this fact. A refinement and improvement to Conley–Zehnder’s index theory in which we will recall all essential ingredients is the Maslov-type index theory for symplectic paths developed by Long and his collaborators. It would shed new light on the computations and understandings of the semiclassical trace formula. The insights in Gutzwiller’s work also seems plausible for the studies of Hamiltonian systems. 相似文献
12.
L. H. Eliasson 《Bulletin of the Brazilian Mathematical Society》1994,25(1):57-76
We prove that small perturbations of a real analytic integrable Hamiltonian system ind degrees of freedom generically have biasymptotic orbits which are obtained as intersections of the stable and unstable manifolds of invariant hyperbolic tori of dimensiond–1. Hence, these solutions will be forward and backward asymptotic to such a torus and not to a periodic solution. The generic condition, which is open and dense, is given by an explicit condition on the averaged perturbation. 相似文献
13.
Diogo Aguiar Gomes 《NoDEA : Nonlinear Differential Equations and Applications》2007,14(3-4):233-257
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.
相似文献
14.
Glenn Rice 《代数通讯》2013,41(8):3047-3055
Let (R, 𝔪) be a Noetherian local ring and M be a submodule of the free module F = R r with height(I r (M)) > 0. Asymptotic sequences over M will be defined analogous to Rees’ definition of asymptotic sequences over an ideal. It is then shown that all maximal asymptotic sequences over M have the same length. This length gives a bound on the analytic spread of M. Namely, if s is the length of a maximal asymptotic sequence over M then l(M) ≤dim R + rank M ? 1 ? s. Equality holds if R is quasi-unmixed. 相似文献
15.
Let f : M → M be an Anosov diffeomorphism on a nilmanifold. We consider Birkhoff sums for a Holder continuous observation along periodic orbits. We show that if there are two Birkhoff sums distributed at both sides of zero, then the set of Birkhoff sums of all the periodic points is dense in R. 相似文献
16.
We study the long time behavior of viscosity solutions of the Cauchy problem for Hamilton–Jacobi equations in ? n . We prove that if the Hamiltonian H(x, p) is coercive and strictly convex in a mild sense in p and upper semi-periodic in x, then any solution of the Cauchy problem “converges” to an asymptotic solution for any lower semi-almost periodic initial function. 相似文献
17.
Let M be a normally hyperbolic symplectic critical manifold of a Hamiltonian system. Suppose M consists of equilibria with real eigenvalues. We prove an analog of the Shilnikov lemma (strong version of the λ-lemma) describing the behavior of trajectories near M. Using this result, trajectories shadowing chains of homoclinic orbits to M are represented as extremals of a discrete variational problem. Then the existence of shadowing periodic orbits is proved. This paper is motivated by applications to the Poincaré’s second species solutions of the 3 body problem with 2 masses small of order µ. As µ → 0, double collisions of small bodies correspond to a symplectic critical manifold M of the regularized Hamiltonian system. Thus our results imply the existence of Poincaré’s second species (nearly collision) periodic solutions for the unrestricted 3 body problem. 相似文献
18.
《Communications in Nonlinear Science & Numerical Simulation》2014,19(8):2660-2675
The numerical study of Dynamical Systems leads to obtain invariant objects of the systems such as periodic orbits, invariant tori, attractors and so on, that helps to the global understanding of the problem. In this paper we focus on the rigorous computation of periodic orbits and their distribution on the phase space, which configures the so called skeleton of the system. We use Computer Assisted Proof techniques to make a rigorous proof of the existence and the stability of families of periodic orbits in two-degrees of freedom Hamiltonian systems, which provide rigorous skeletons of periodic orbits. To that goal we show how to prove the existence and stability of a huge set of discrete initial conditions of periodic orbits, and later, how to prove the existence and stability of continuous families of periodic orbits. We illustrate the approach with two paradigmatic problems: the Hénon–Heiles Hamiltonian and the Diamagnetic Kepler problem. 相似文献
19.
Ali Ghaffari 《Semigroup Forum》2008,76(1):95-106
Let S be a foundation locally compact topological semigroup. Two new topologies τ
c
and τ
w
are introduced on M
a
(S)*. We introduce τ
c
and τ
w
almost periodic functionals in M
a
(S)*. We study these classes and compare them with each other and with the norm almost periodic and weakly almost periodic functionals.
For f∈M
a
(S)*, it is proved that T
f
∈ℬ(M
a
(S),M
a
(S)*) is strong almost periodic if and only if f is τ
c
-almost periodic. Indeed, we have obtained a generalization of a well known result of Crombez for locally compact group to
a more general setting of foundation topological semigroups. Finally if P(S) (the set of all probability measures in M
a
(S)) has the semiright invariant isometry property, it is shown that the set of τ
w
-almost periodic functionals has a topological left invariant mean. 相似文献
20.
Siniša Slijepčevič 《Mathematische Zeitschrift》2001,237(3):469-504
If F is an exact symplectic map on the{\it d}-dimensional cylinder , with a generating function h having superlinear growth and uniform bounds on the second derivative, we construct a strictly gradient semiflow on the space of shift-invariant probability measures on the space of configurations . Stationary points of are invariant measures of F, and the rotation vector and all spectral invariants are invariants of . Using and the minimisation technique, we construct minimising measures with an arbitrary rotation vector , and with an additional assumption that F is strongly monotone, we show that the support of every minimising measure is a graph of a Lipschitz function. Using and the relaxation technique, assuming a weak condition on (satisfied e.g. in Hedlund's counter-example, and in the anti-integrable limit) we show existence of double-recurrent orbits
of F (and F-ergodic measures) with an arbitrary rotation vector , and action arbitrarily close to the minimal action .
Received November 4, 1999; in final form July 29, 2000 / Published online April 12, 2001 相似文献