Error analysis of symplectic Lanczos method for Hamiltonian eigenvalue problem

A rounding error analysis for the symplectic Lanczos method is given to solve the large-scale sparse Hamiltonian eigenvalue problem. If no breakdown occurs in the method, then it can be shown that the Hamiltonian structure preserving requirement does not destroy the essential feature of the nonsymmetric Lanczos algorithm. The relationship between the loss of J-orthogonality among the symplectic Lanczos vectors and the convergence of the Ritz values in the symmetric Lanczos algorithm is discussed. It is demonstrated that under certain assumptions the computed J-orthogonal Lanczos vectors lose the J-orthogonality when some Ritz values begin to converge. Our analysis closely follows the recent works of Bai and Fabbender. Selected from Journal of Mathematical Research and Exposition, 2004, 24(1): 91–106     

Maslov P-Index Theory for a Symplectic Path with Applications

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.     

Symplectic convexity for orbifolds

Let a connected compact Lie group G act on a connected symplectic orbifold of orbifold fundamental group Г. If the action preserves the symplectic structure and there is a G-equivariant and mod-Г proper momentum map for the lifted action on the universal branch covering orbifold, and if the lifted G-action commutes with that of Г, then the symplectic convexity theorem is still true for this kind of lifted Hamiltonian action.     

Hamiltonian Systems on Time Scales

Linear and nonlinear Hamiltonian systems are studied on time scales . We unify symplectic flow properties of discrete and continuous Hamiltonian systems. A chain rule which unifies discrete and continuous settings is presented for our so-called alpha derivatives on generalized time scales. This chain rule allows transformation of linear Hamiltonian systems on time scales under simultaneous change of independent and dependent variables, thus extending the change of dependent variables recently obtained by Do lý and Hilscher. We also give the Legendre transformation for nonlinear Euler–Lagrange equations on time scales to Hamiltonian systems on time scales.     

Two Novel Classes of Arbitrary High-Order Structure-Preserving Algorithms for Canonical Hamiltonian Systems

In this paper, we systematically construct two classes of structure-preserving schemes with arbitrary order of accuracy for canonical Hamiltonian systems. The one class is the symplectic scheme, which contains two new families of parameterized symplectic schemes that are derived by basing on the generating function method and the symmetric composition method, respectively. Each member in these schemes is symplectic for any fixed parameter. A more general form of generating functions is introduced, which generalizes the three classical generating functions that are widely used to construct symplectic algorithms. The other class is a novel family of energy and quadratic invariants preserving schemes, which is devised by adjusting the parameter in parameterized symplectic schemes to guarantee energy conservation at each time step. The existence of the solutions of these schemes is verified. Numerical experiments demonstrate the theoretical analysis and conservation of the proposed schemes.     

Symplectic Structure of Discrete Hamiltonian Systems

This paper is concerned with the symplectic structure of discrete nonlinear Hamiltonian systems. The results are related to an open problem that was first proposed by C. D. Ahlbrandt [J. Math. Anal. Appl.180 (1993), 498-517] discussed elsewhere in the literature. But we give a different statement and different proof. Under a solvable condition, we show that the solution operator of a discrete nonlinear Halmiltonian system is symplectic. Then its phase flow is a discrete one-parameter family of symplectic transformations and preserves the phase volume.     

VARIATIONS ON A THEME BY EULER

1.IntroductionAHallliltolliansystemofdifferentialequationsonRZnisgivedbyP~~H,(P,q),q=HP(P,q),(1)wherep=(pl,'.,P.),q=(ql,',q.)eR"arethegeneralizedcoordinatesandmolllentarespectivelyandH(P,q)istheellergyofthesystem.Thesystem(1)canberewrittenasthecompactf…     

ADIABATIC INVARIANTS OF SLOWLY VARYING THREE－DIMENSIONAL SYSTEMS AND EXISTENCE OF INVARIANT TORI OF LOTKA－VOLTERRA EQUATION

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Preservation of equilibria for symplectic methods applied to Hamiltonian systems

In this paper, the linear stability of symplectic methods for Hamiltonian systems is studied. In par- ticular, three classes of symplectic methods are considered： symplectic Runge-Kutta （SRK） methods, symplectic partitioned Runge-Kutta （SPRK） methods and the composition methods based on SRK or SPRK methods. It is shown that the SRK methods and their compositions preserve the ellipticity of equilibrium points uncondi- tionally, whereas the SPRK methods and their compositions have some restrictions on the time-step.     

Strong Lefschetz property under reduction

Let n > 1 and G be the group SU(n) or Sp(n). This paper constructs compact symplectic manifolds whose symplectic quotient under a Hamiltonian G-action does not inherit the strong Lefschetz property.     

Stochastic discrete Hamiltonian variational integrators

Variational integrators are derived for structure-preserving simulation of stochastic Hamiltonian systems with a certain type of multiplicative noise arising in geometric mechanics. The derivation is based on a stochastic discrete Hamiltonian which approximates a type-II stochastic generating function for the stochastic flow of the Hamiltonian system. The generating function is obtained by introducing an appropriate stochastic action functional and its corresponding variational principle. Our approach permits to recast in a unified framework a number of integrators previously studied in the literature, and presents a general methodology to derive new structure-preserving numerical schemes. The resulting integrators are symplectic; they preserve integrals of motion related to Lie group symmetries; and they include stochastic symplectic Runge–Kutta methods as a special case. Several new low-stage stochastic symplectic methods of mean-square order 1.0 derived using this approach are presented and tested numerically to demonstrate their superior long-time numerical stability and energy behavior compared to nonsymplectic methods.     

线性Hamilton系统边值问题的保辛数值方法

以Hamilton系统的正则变换和生成函数为基础研究线性时变Hamilton系统边值问题的保辛数值求解算法.根据第二类生成函数系数矩阵与状态传递矩阵的关系,构造了生成函数系数矩阵的区段合并递推算法,并进一步将递推算法推广到线性非齐次边值问题中;然后利用生成函数的性质将边值问题转化为初值问题,最后采用初值问题的保辛算法求解以达到整个Hamilton系统保辛的目的.数值算例表明该方法能够有效地求解线性齐次与非齐次问题,并能很好地保持Hamilton系统的固有特性.     

A Generalized W-Transformation for Constructing Symplectic Partitioned Runge-Kutta Methods

This paper deals with the construction of implicit symplectic partitioned Runge–Kutta methods (PRKM) of high order for separable and general partitioned Hamiltonian systems. The main tool is a generalized W-transformation for PRKM based on different quadrature formulas. Methods of high order and special properties can be determined using the transformed coefficient matrices. Examples are given.     

Semisimple Symplectic Symmetric Spaces

A symplectic symmetric space is a connected affine symmetric manifold M endowed with a symplectic structure which is invariant under the geodesic symmetries. When the transvection group G⁰ of such a symmetric space M is semisimple, its action on (M,) is strongly Hamiltonian; a classical theorem due to Kostant implies that the moment map associated to this action realises a G⁰-equivariant symplectic covering of a coadjoint orbit O in the dual of the Lie algebra of G⁰. We show that this orbit itself admits a structure of symplectic symmetric space whose transvection algebra is . The main result of this paper is the classification of symmetric orbits for any semisimple Lie group. The classification is given in terms of root systems of transvection algebras and therefore provides, in a symplectic framework, a theorem analogous to the Borel–de Siebenthal theorem for Riemannian symmetric spaces. When its dimension is greater than 2, such a symmetric orbit is not regular and, in general, neither Hermitian nor pseudo-Hermitian.     

可允许度量下有限能量辛涡旋的渐近行为献给钱敏教授90华诞

假设(X,ω)是一个具有紧致单连通Lie群G Hamilton作用的紧致光滑辛流形.本文证明只要Riemann面的柱形端口具有一个比标准柱形度量增长速度快的线性度量,那么任何一个有限能量辛涡旋将以指数衰减的速度收敛到辛流形X在正则值辛约化的扭曲分支或非扭曲分支上.本文结果无需假设群G在正则水平集上的作用是自由的.因此,它直接推广了Ziltener在群作用自由的假设下得出的相关结果.本文结果在作者关于量子化Kirwan同态的系列工作中有重要应用.     

On theoretical and numerical aspects of symplectic Gram–Schmidt-like algorithms

Gram–Schmidt-like orthogonalization process with respect to a given skew-symmetric scalar product is a key step in model reduction methods, structure-preserving, for large sparse Hamiltonian eigenvalue problem. Theoretical as well as numerical aspects of this step do not benefit of enough attention, compared to the one allowed to the classical Gram–Schmidt algorithm and its modified version. The aim of this paper is to revisit the symplectic Gram–Schmidt algorithms, to built some modified versions and to deal with their theoretical and numerical features. AMS subject classification 65F15, 65F50     

Symplectic invariants and covariants of matrices

This paper deals with symplectic algebraic invariants and covariants of matrices. We adapt the Aronhold symbolic method to characterize and compute the generators of algebras of symplectic invariants and covariants. New symplectic identities are obtained and applied to construct minimal generator systems of a matrix of order 4 and a Hamiltonian matrix of order 2n.     

On κ—ordered Graphs Involved Degree Sum

A graph G is κ-ordered Hamiltonian 2≤κ≤n,if for every ordered sequence S of κ distinct vertices of G,there exists a Hamiltonian cycle that encounters S in the given order,In this article,we prove that if G is a graph on n vertices with degree sum of nonadjacent vertices at least n 3κ-9/2,then G is κ-ordered Hamiltonian for κ=3,4,…,[n/19].We also show that the degree sum bound can be reduced to n 2[κ/2]-2 if κ(G）≥3κ-1/2 or δ（G）≥5κ-4.Several known results are generalized.     

Application of the GALI method to localization dynamics in nonlinear systems

We investigate localization phenomena and stability properties of quasiperiodic oscillations in N$N$ degree of freedom Hamiltonian systems and N$N$ coupled symplectic maps. In particular, we study an example of a parametrically driven Hamiltonian lattice with only quartic coupling terms and a system of N$N$ coupled standard maps. We explore their dynamics using the Generalized Alignment Index (GALI), which constitutes a recently developed numerical method for detecting chaotic orbits in many dimensions, estimating the dimensionality of quasiperiodic tori and predicting slow diffusion in a way that is faster and more reliable than many other approaches known to date.