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1.
Different methods are proposed and tested for transforming a nonlinear differential system, and more particularly a Hamiltonian one, into a map without having to integrate the whole orbit as in the well known Poincaré map technique . We construct a synthetic map, based on a global fitting, which satisfies the symplectic condition. Taking the Standard Map as a model problem we point our attention on methods suitable for comparing the model map and its synthetic counterpart. We test the agreement of the fitting on finer scales through the visual representation, the computation of the rotation number and the measure of the local distribution of the Lyapunov characteristic exponents. Comparing these results with those obtained by Froeschlé and Petit using a method based on Taylor interpolation, we show that the symplectic character is a crucial condition for the recovering of the finest details of a dynamical system. On the other hand the global character of our method makes the generalization to any system of differential equations difficult. 相似文献
2.
We provide the direct proof of the Nekhoroshev theorem on the stability of nearly integrable analytic symplectic maps. Specifically, we prove the stability of the actions for a number of iterations which grows exponentially with an inverse power of the norm of the perturbation by conjugating the generating function of the map to suitable normal forms with exponentially small remainder.Communicated by Eduard Zehndersubmitted 16/06/03, accepted 31/03/04 相似文献
3.
本文主要研究具有衰减记忆的多维映射的逼近,考虑了因果和非因果两种情形.在实际应用上,可以用来识别一类因果和非因果系统. 相似文献
4.
Dissipative perturbations of strongly nonlinear oscillators that correspond to slowly varying double-well potentials are considered. The method of averaging, which describes the solution as nearly periodic, fails as the trajectory approaches the unperturbed separatrix, a homoclinic orbit of the saddle point, significantly before it is captured in either well. Nevertheless, perturbed initial conditions corresponding to the boundary of the basin of attraction for each well, which are the perturbed stable manifolds of the saddle point, are accurately determined using only the method of averaging modified by Melnikov energy ideas near the separatrix. To determine the amplitude and phase of the captured oscillations after crossing the separatrix, a transition region is constructed consisting of a large sequence of nearly solitary pulses along the separatrix. The amplitude and phases of the slowly varying nonlinear oscillations away from the separatrix, both before and after capture, are matched to this transition region. In this way, analytic connection formulas across the separatrix are obtained and are shown to depend on the perturbed initial conditions. 相似文献
5.
A symplectic module is a finite group with a regular antisymmetric form. The paper determines sufficient conditions for the
invariants of the maximal isotropic subgroups (Lagrangians), and asymptotic values for a lower bound of a group which contains
Lagrangians of all symplectic modules of a fixed orderp
n. These results have application to the splitting fields of universal division algebras. 相似文献
6.
Simeon Ball John Bamberg Michel Lavrauw Tim Penttila 《Designs, Codes and Cryptography》2004,32(1-3):9-14
We construct an infinite family of symplectic spreads in spaces of odd rank and characteristic. 相似文献
7.
Victor Guillemin Lisa Jeffrey Jeffrey Reyer Sjamaar Sjamaar 《Transformation Groups》2002,7(2):155-185
Let K be a compact Lie group. We introduce the process of symplectic implosion, which associates to every Hamiltonian K-manifold a stratified space called the imploded cross-section. It bears a resemblance to symplectic reduction, but instead of quotienting by the entire group, it cuts the symmetries down to a maximal torus of K. We examine the nature of the singularities and describe in detail the imploded cross-section of the cotangent bundle of K, which turns out to be identical to an affine variety studied by Gelfand, Popov, Vinberg, and others. Finally we show that "quantization commutes with implosion". 相似文献
8.
Alexandre V. Borovik Israel Gelfand Neil White 《Journal of Algebraic Combinatorics》1998,8(3):235-252
A symplectic matroid is a collection B of k-element subsets of J = {1, 2, ..., n, 1*, 2*, ...; n*}, each of which contains not both of i and i* for every i n, and which has the additional property that for any linear ordering of J such that i j implies j* i* and i j* implies j i* for all i, j n, B has a member which dominates element-wise every other member of B. Symplectic matroids are a special case of Coxeter matroids, namely the case where the Coxeter group is the hyperoctahedral group, the group of symmetries of the n-cube. In this paper we develop the basic properties of symplectic matroids in a largely self-contained and elementary fashion. Many of these results are analogous to results for ordinary matroids (which are Coxeter matroids for the symmetric group), yet most are not generalizable to arbitrary Coxeter matroids. For example, representable symplectic matroids arise from totally isotropic subspaces of a symplectic space very similarly to the way in which representable ordinary matroids arise from a subspace of a vector space. We also examine Lagrangian matroids, which are the special case of symplectic matroids where k = n, and which are equivalent to Bouchet's symmetric matroids or 2-matroids. 相似文献
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10.
《Acta Appl Math》1992,28(1):95-97
Book Review
Symplectic GeometryA. T. Fomenko: Advanced Studies in Contemporary Mathematics, Volume 5, Gordon and Breach Science Publishers, 1988, 387 pp 相似文献11.
Every complex projective space of odd dimension carries a natural contact structure. We give first steps towards the enumeration
of curves in ℙ3 tangent to the contact structure. Such a curve is involutive in the sense that its homogeneous ideal is closed under Poisson
bracket. Involutive curves in ℙ3 contained in a plane split as a union of concurrent lines. We give a formula for the number of plane involutive curves of
a given degree in ℙ3 meeting the appropriate number of lines. We also discuss strategies to deal with the enumeration of involutive rational curves. 相似文献
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本文证明了三次系统不可能同时存在三个三曲线x^3-3xy^2-1=0分界线环,但是可以同时存在两个三曲线分界线环,给出了同时存在两个三曲线分界线环的充要条件。 相似文献
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Robert Piziak 《Journal of Combinatorial Theory, Series A》1974,16(1):87-96
Let E be a finite dimensional vector space over the Galois field GF(2). Let lin(E) denote the set of one-dimensional subspaces of E. Let Φ be the symplectic inner product on E. Consider the elements of lin(E) as vertices of a graph, two vertices being connected exactly when they are distinct and orthogonal with respect to Φ. This graph is characterized abstractly. 相似文献
17.
Izu Vaisman 《Monatshefte für Mathematik》1985,100(4):299-327
In the paper, one establishes the decomposition of the space of tensors which have the symmetries of the curvature of a torsionless symplectic connection into Sp (n)-irreducible components. This leads to three interesting classes of symplectic connections: flat, Ricci flat, and similar to the Levi-Civita connections of Kähler manifolds with constant holomorphic sectional curvature (we call them connections with reducible curvature). A symplectic manifold with two transversal polarizations has a canonical symplectic connection, and we study properties that are encountered if this canonical connection belongs to the classes mentioned above. For instance, in the reducible case we can compute the Pontrjagin classes, and these will be zero if the polarizations are real, etc. If the polarizations are real and there exist points where they are either singular or nontransversal, one has residues in the sense ofLehmann [L], which should be expected to play an interesting role in symplectic geometry. 相似文献
18.
Kang Feng & Dao-Liu Wang 《计算数学(英文版)》1991,9(1):86-96
We consider the construction of phase flow generating functions and symplectic difference schemes for Hamiltonian systems in general symplectic structure with variable coefficients. 相似文献
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We study polar representations in the sense of Dadok and Kac which are symplectic. We show that such representations are coisotropic and use this fact to give a classification. We also study their moment maps and prove that they separate closed orbits. Our work can also be seen as a specialization of some of the results of Knop on multiplicity free symplectic representations to the polar case. 相似文献