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1.
Laurent Niederman 《Regular and Chaotic Dynamics》2013,18(6):719-731
In this article, we consider linearly stable elliptic fixed points (equilibrium) for a symplectic vector field and prove generic results of super-exponential stability for nearby solutions. We will focus on the neighborhood of elliptic fixed points but the case of linearly stable isotropic reducible invariant tori in a Hamiltonian system should be similar. More specifically, Morbidelli and Giorgilli have proved a result of stability over superexponentially long times if one considers an analytic Lagrangian torus, invariant for an analytic Hamiltonian system, with a diophantine translation vector which admits a sign-definite torsion. Then, the solutions of the system move very little over times which are super-exponentially long with respect to the inverse of the distance to the invariant torus. The proof proceeds in two steps: first one constructs a high-order Birkhoff normal form, then one applies the Nekhoroshev theory. Bounemoura has shown that the second step of this construction remains valid if the Birkhoff normal form linked to the invariant torus or the elliptic fixed point belongs to a generic set among the formal series. This is not sufficient to prove this kind of super-exponential stability results in a general setting. We should also establish that the most strongly non resonant elliptic fixed point or invariant torus in a Hamiltonian system admits Birkhoff normal forms fitted for the application of the Nekhoroshev theory. Actually, the set introduced by Bounemoura is already very large but not big enough to ensure that a typical Birkhoff normal form falls into this class. We show here that this property is satisfied generically in the sense of the measure (prevalence) through infinite-dimensional probe spaces (that is, an infinite number of parameters chosen at random) with methods similar to those developed in a paper of Gorodetski, Kaloshin and Hunt in another setting. 相似文献
2.
M. A. Olshanetsky 《Theoretical and Mathematical Physics》2007,150(3):301-314
We construct a quadratic Poisson algebra of Hamiltonian functions on a two-dimensional torus compatible with the canonical
Poisson structure. This algebra is an infinite-dimensional generalization of the classical Sklyanin-Feigin-Odesskii algebras.
It yields an integrable modification of the two-dimensional hydrodynamics of an ideal fluid on the torus. The Hamiltonian
of the standard two-dimensional hydrodynamics is defined by the Laplace operator and thus depends on the metric. We replace
the Laplace operator with a pseudodifferential elliptic operator depending on the complex structure. The new Hamiltonian becomes
a member of a commutative bi-Hamiltonian hierarchy. In conclusion, we construct a Lie bialgebroid of vector fields on the
torus.
__________
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 3, pp. 355–370, March, 2007. 相似文献
3.
We prove that there is an invariant torus with the given Diophantine frequency vector for a class of Hamiltonian systems defined by an integrable large Hamiltonian function with a large non-autonomous Hamiltonian perturbation. As for application, we prove that a finite network of Duffing oscillators with periodic external forces possesses Lagrange stability for almost all initial data. 相似文献
4.
Jacky Cresson 《Journal of Differential Equations》2003,187(2):269-292
We consider hyperbolic tori of three degrees of freedom initially hyperbolic Hamiltonian systems. We prove that if the stable and unstable manifold of a hyperbolic torus intersect transversaly, then there exists a hyperbolic invariant set near a homoclinic orbit on which the dynamics is conjugated to a Bernoulli shift. The proof is based on a new geometrico-dynamical feature of partially hyperbolic systems, the transversality-torsion phenomenon, which produces complete hyperbolicity from partial hyperbolicity. We deduce the existence of infinitely many hyperbolic periodic orbits near the given torus. The relevance of these results for the instability of near-integrable Hamiltonian systems is then discussed. For a given transition chain, we construct chain of hyperbolic periodic orbits. Then we easily prove the existence of periodic orbits of arbitrarily high period close to such chain using standard results on hyperbolic sets. 相似文献
5.
We give a nonlinear symplectic coordinator transformation, which can move the normal frequencies of the lower dimensional
torus up to (k,w) where ω is the frequency vector of the torus. That means the normal frequencies with a difference (k,w)
may be regarded as the same. As an application, we derive a persistence result on lower dimensional tori of nearly integrable
Hamiltonian systems when the second Melnikov’s condition is partially violated. 相似文献
6.
For a Lagrangian torus A in a simply-connected projective symplectic manifold M, we prove that M has a hypersurface disjoint from a deformation of A. This implies that a Lagrangian torus in a compact hyperkähler manifold is a fiber of an almost holomorphic Lagrangian fibration, giving an affirmative answer to a question of Beauville’s. Our proof employs two different tools: the theory of action-angle variables for algebraically completely integrable Hamiltonian systems and Wielandt’s theory of subnormal subgroups. 相似文献
7.
On the linear level elliptic equilibria of Hamiltonian systems are mere superpositions of harmonic oscillators. Non-linear
terms can produce instability, if the ratio of frequencies is rational and the Hamiltonian is indefinite. In this paper we
study the frequency ratio ±1/2 and its unfolding. In particular we show that for the indefinite case (1:−2) the frequency
ratio map in a neighborhood of the origin has a critical point, i.e. the twist condition is violated for one torus on every
energy surface near the energy of the equilibrium. In contrast, we show that the frequency map itself is non-degenerate (i.e.
the Kolmogorov non-degeneracy condition holds) for every torus in a neighborhood of the equilibrium point. As a by product
of our analysis of the frequency map we obtain another proof of fractional monodromy in the 1:−2 resonance.
相似文献
8.
In this paper, we study the persistence of invariant tori of integrable Hamiltonian systems satisfying Rssmann's non-degeneracy condition when symplectic integrators are applied to them. Meanwhile, we give an estimate of the measure of the set occupied by the invariant tori in the phase space. On an invariant torus,numerical solutions are quasi-periodic with a diophantine frequency vector of time step size dependence. These results generalize Shang's previous ones(1999, 2000), where the non-degeneracy condition is assumed in the sense of Kolmogorov. 相似文献
9.
Andrea Giacobbe 《Geometriae Dedicata》2005,111(1):1-22
A famous theorem of Atiyah, Guillemin and Sternberg states that, given a Hamiltonian torus action, the image of the momentum map is a convex polytope. We prove that this result can be extended to the case in which the action is non-Hamiltonian. Our generalization of the theorem states that, given a symplectic torus action, the momentum map can be defined on an appropriate covering of the manifold and its image is the product of a convex polytope along a rational subspace times the orthogonal vector space. We also prove that this decomposition in direct product is stable under small equivariant perturbations of the symplectic structure; this, in particular, means that the property of being Hamiltonian is locally stable. The technique developed allows us to extend the result to any compact group action and also to deduce that any symplectic n-torus action, with fixed points, on a compact 2n-dimensional manifold, is Hamiltonian. 相似文献
10.
The aim of this paper is to prove a Kolmogorov type result for a nearly integrable Hamiltonian, quadratic in the actions, with an aperiodic time dependence. The existence of a torus with a prefixed Diophantine frequency is shown in the forced system, provided that the perturbation is real-analytic and (exponentially) decaying with time. The advantage consists in the possibility to choose an arbitrarily small decaying coefficient consistently with the perturbation size. The proof, based on the Lie series formalism, is a generalization of a work by A. Giorgilli. 相似文献
11.
M. B. Sevryuk 《Proceedings of the Steklov Institute of Mathematics》2007,259(1):167-195
Under a small perturbation of a completely integrable Hamiltonian system, invariant tori with Diophantine frequencies of motion
are not destroyed but only slightly deformed, provided that the Hessian (with respect to the action variables) of the unperturbed
Hamiltonian vanishes nowhere (the Kolmogorov nondegeneracy). The motion on every perturbed torus is quasiperiodic with the
same frequencies. In this sense the frequencies of invariant tori of the unperturbed system are preserved. Recently, it has
been found that the Kolmogorov nondegeneracy condition can be weakened so as to guarantee the preservation of only some subset
of frequencies. Such partial preservation of frequencies can also be defined for lower dimensional invariant tori, whose dimension
is less than the number of degrees of freedom. We consider a more general problem of partial preservation not only of the
frequencies of invariant tori but also of their Floquet exponents (the eigenvalues of the coefficient matrix of the variational
equation along the torus). The results are formulated for Hamiltonian, reversible, and dissipative systems (with a complete
proof for the reversible case). 相似文献
12.
Hans Koch 《Journal of Differential Equations》2010,249(8):1986-2004
We extend the renormalization scheme for vector fields on Td×Rm in order to construct lower-dimensional invariant tori with Brjuno frequency vectors for near-integrable Hamiltonian flows. For every Brjuno frequency vector ω∈Rd and every vector Ω∈RD satisfying a Diophantine condition with respect to ω, there exists an analytic manifold W of infinitely renormalizable Hamiltonian vector fields; each vector field on W is shown to have an analytic invariant torus with frequency vector ω. 相似文献
13.
It is shown that plane wave solutions to the cubic nonlinear Schrödinger equation on a torus behave orbitally stable under generic perturbations of the initial data that are small in a high-order Sobolev norm, over long times that extend to arbitrary negative powers of the smallness parameter. The perturbation stays small in the same Sobolev norm over such long times. The proof uses a Hamiltonian reduction and transformation and, alternatively, Birkhoff normal forms or modulated Fourier expansions in time. 相似文献
14.
Henri Anciaux 《Mathematische Zeitschrift》2002,241(3):639-664
We compute loops integrals on Hamiltonian stationary Lagrangian tori in which are symplectic invariants, then we show an isoperimetric inequality involving these invariants and the area. Finally,
we show that the flat torus has least area among Hamiltonian stationary Lagrangian tori of its isotopy class.
Received: 4 December 2000; in final form: 18 January 2002 / Published online: 5 September 2002 相似文献
15.
M. B. Sevryuk 《Proceedings of the Steklov Institute of Mathematics》2007,259(2):167-195
Under a small perturbation of a completely integrable Hamiltonian system, invariant tori with Diophantine frequencies of motion
are not destroyed but only slightly deformed, provided that the Hessian (with respect to the action variables) of the unperturbed
Hamiltonian vanishes nowhere (the Kolmogorov nondegeneracy). The motion on every perturbed torus is quasiperiodic with the
same frequencies. In this sense the frequencies of invariant tori of the unperturbed system are preserved. Recently, it has
been found that the Kolmogorov nondegeneracy condition can be weakened so as to guarantee the preservation of only some subset
of frequencies. Such partial preservation of frequencies can also be defined for lower dimensional invariant tori, whose dimension
is less than the number of degrees of freedom. We consider a more general problem of partial preservation not only of the
frequencies of invariant tori but also of their Floquet exponents (the eigenvalues of the coefficient matrix of the variational
equation along the torus). The results are formulated for Hamiltonian, reversible, and dissipative systems (with a complete
proof for the reversible case).
Original Russian Text ? M.B. Sevryuk, 2007, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2007, Vol.
259, pp. 174–202. 相似文献
16.
This paper is concerned with the Hess case in the Euler–Poisson equations and with its generalization on the pencil of Poisson brackets. It is shown that in this case the problem reduces to investigating the vector field on a torus and that the graph showing the dependence of the rotation number on parameters has horizontal segments (limit cycles) only for integer values of the rotation number. In addition, an example of a Hamiltonian system is given which possesses an invariant submanifold (similar to the Hess case), but on which the dependence of the rotation number on parameters is a Cantor ladder. 相似文献
17.
This paper deals with Lagrangian bundles which are symplectic torus bundles that occur in integrable Hamiltonian systems.
We review the theory of obstructions to triviality, in particular monodromy, as well as the ensuing classification problems
which involve the Chern and Lagrange class. Our approach, which uses simple ideas from differential geometry and algebraic
topology, reveals the fundamental role of the integer affine structure on the base space of these bundles. We provide a geometric
proof of the classification of Lagrangian bundles with fixed integer affine structure by their Lagrange class.
相似文献
18.
Exponential stability for time dependent potentials 总被引:2,自引:0,他引:2
Antonio Giorgilli Eduard Zehnder 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》1992,43(5):827-855
For a classical Hamiltonian system on a torus defined by a time dependent, bounded and analytic potential we establish global and quantitative bounds for the solutions over an exponentially long interval of time by using techniques which go back to Nekhoroshev. 相似文献
19.
Yves Benoist 《Geometriae Dedicata》2002,89(1):177-241
For any symplectic action of a compact connected group on a compact connected symplectic manifold, we show that the intersection
of the Weyl chamber with the image of the moment map is a closed convex polyhedron. This extends Atiyah–Guillemin–Sternberg–Kirwan's
convexity theorems to non-Hamiltonian actions. As a consequence, we describe those symplectic actions of a torus which are
coisotropic (or multiplicity free), i.e. which have at least one coisotropic orbit: they are the product of an Hamiltonian
coisotropic action by an anhamiltonian one. The Hamiltonian coisotropic actions have already been described by Delzant thanks
to the convex polyhedron. The anhamiltonian coisotropic actions are actions of a central torus on a symplectic nilmanifold.
This text is written as an introduction to the theory of symplectic actions of compact groups since complete proofs of the
preliminary classical results are given.
An erratum to this article is available at . 相似文献
20.