Symmetric and symplectic exponential integrators for nonlinear Hamiltonian systems

This letter studies symmetric and symplectic exponential integrators when applied to numerically computing nonlinear Hamiltonian systems. We first establish the symmetry and symplecticity conditions of exponential integrators and then show that these conditions are extensions of the symmetry and symplecticity conditions of Runge–Kutta methods. Based on these conditions, some symmetric and symplectic exponential integrators up to order four are derived. Two numerical experiments are carried out and the results demonstrate the remarkable numerical behaviour of the new exponential integrators in comparison with some symmetric and symplectic Runge–Kutta methods in the literature.     

Long-Time Oscillatory Energy Conservation of Total Energy-Preserving Methods for Highly Oscillatory Hamiltonian Systems

For an integrator when applied to a highly oscillatory system,the near conservation of the oscillatory energy over long times is an important aspect.In this paper,we study the long-time near conservation of oscillatory energy for the adapted average vector field(AAVF)method when applied to highly oscillatory Hamiltonian systems.This AAVF method is an extension of the average vector field method and preserves the total energy of highly oscillatory Hamiltonian systems exactly.This paper is devoted to analysing another important property of AAVF method,i.e.,the near conservation of its oscillatory energy in a long term.The long-time oscillatory energy conservation is obtained via constructing a modulated Fourier expansion of the AAVF method and deriving an almost invariant of the expansion.A similar result of the method in the multi-frequency case is also presented in this paper.     

辛对称Hamilton算子值域的闭性

利用扰动理论和算子矩阵的因式分解,研究了辛对称Hamilton算子值域的闭性.针对对角占优、上行占优等情形,在一定条件下给出了值域闭性的若干描述,并得到了一般情形的结果.     

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.     

对称超二次二阶哈密尔顿系统的周期解

我们利用Ambrosetti-Rabinowitz对称形式的山路引理证明了给定周期T的对称超二次二阶哈密尔顿系统具有无穷多个反T/2-周期且奇的周期解.     

Efficient Energy-preserving Methods for General Nonlinear Oscillatory Hamiltonian System

In this paper, we propose and analyze two kinds of novel and symmetric energy-preserving formulae for the nonlinear oscillatory Hamiltonian system of second-order differential equations Aq"(t)+ Bq(t)=f(q(t)), where A ∈ R^m×m is a symmetric positive definite matrix, B ∈ R^m×m is a symmetric positive semi-definite matrix that implicitly contains the main frequencies of the problem and f(q)=-▽_qV(q) for a real-valued function V(q). The energy-preserving formulae can exactly preserve the Hamiltonian H(q', q)=(1)/2q'^τ Aq' + (1)/2q^τ Bq + V(q). We analyze the properties of energy-preserving and convergence of the derived energy-preserving formula and obtain new efficient energy-preserving integrators for practical computation. Numerical experiments are carried out to show the efficiency of the new methods by the nonlinear Hamiltonian systems.     

Linearization-preserving self-adjoint and symplectic integrators

This article is concerned with geometric integrators which are linearization-preserving, i.e. numerical integrators which preserve the exact linearization at every fixed point of an arbitrary system of ODEs. For a canonical Hamiltonian system, we propose a new symplectic and self-adjoint B-series method which is also linearization-preserving. In a similar fashion, we show that it is possible to construct a self-adjoint and linearization-preserving B-series method for an arbitrary system of ODEs. Some numerical experiments on Hamiltonian ODEs are presented to test the behaviour of both proposed methods. This work was supported by the Australian Research Council and by the Marsden Fund of the Royal Society of New Zealand.     

辛映射与Hamilton控制系统

给定等价辛流形 ,即辛同态或形变等价的辛流形 ,研究了建立在这些辛流形上的Hamilton控制系统之间的一些性质的联系 ,诸如 (局部 )能观测性 ,强可接近性 ,(拟 )极小性等 .而且 ,利用Cort啨ｓ介绍的 (弱 )外等价系统的概念 ,给出使得两个Hamilton控制系统是辛同态的一个充分条件     

Numerical Energy Conservation for Multi-Frequency Oscillatory Differential Equations

The long-time near-conservation of the total and oscillatory energies of numerical integrators for Hamiltonian systems with highly oscillatory solutions is studied in this paper. The numerical methods considered are symmetric trigonometric integrators and the St?rmer–Verlet method. Previously obtained results for systems with a single high frequency are extended to the multi-frequency case, and new insight into the long-time behaviour of numerical solutions is gained for resonant frequencies. The results are obtained using modulated multi-frequency Fourier expansions and the Hamiltonian-like structure of the modulation system. A brief discussion of conservation properties in the continuous problem is also included. AMS subject classification (2000) 65L05, 65P10     

Hamiltonian Cycle Systems Which Are Both Cyclic and Symmetric

The notion of a symmetric Hamiltonian cycle system (HCS) of a graph Γ has been introduced and studied by J. Akiyama, M. Kobayashi, and G. Nakamura [J Combin Des 12 (2004), 39–45] for , by R. A. Brualdi and M. W. Schroeder [J Combin Des 19 (2011), 1–15] for , and then naturally extended by V. Chitra and A. Muthusamy [Discussiones Mathematicae Graph Theory, to appear] to the multigraphs and . In each case, there must be an involutory permutation ψ of the vertices fixing all the cycles of the HCS and at most one vertex. Furthermore, for , this ψ should be precisely the permutation switching all pairs of endpoints of the edges of I. An HCS is cyclic if it is invariant under some cyclic permutation of all the vertices. The existence question for a cyclic HCS of has been completely solved by Jordon and Morris [Discrete Math (2008), 2440–2449]—and we note that their cyclic construction is also symmetric for (mod 8). It is then natural to study the existence problem of an HCS of a graph or multigraph Γ as above which is both cyclic and symmetric. In this paper, we completely solve this problem: in the case of even order, the final answer is that cyclicity and symmetry can always cohabit when a cyclic solution exists. On the other hand, imposing that a cyclic HCS of odd order is also symmetric is very restrictive; we prove in fact that an HCS of with both properties exists if and only if is a prime.     

Energy and Quadratic Invariants Preserving Methods for Hamiltonian Systems with Holonomic Constraints

We introduce a new class of parametrized structure--preserving partitioned Runge-Kutta ($\alpha$-PRK) methods for Hamiltonian systems with holonomic constraints. The methods are symplectic for any fixed scalar parameter $\alpha$, and are reduced to the usual symplectic PRK methods like Shake-Rattle method or PRK schemes based on Lobatto IIIA-IIIB pairs when $\alpha=0$. We provide a new variational formulation for symplectic PRK schemes and use it to prove that the $\alpha$-PRK methods can preserve the quadratic invariants for Hamiltonian systems subject to holonomic constraints. Meanwhile, for any given consistent initial values $(p_{0}, q_0)$ and small step size $h>0$, it is proved that there exists $\alpha^*=\alpha(h, p_0, q_0)$ such that the Hamiltonian energy can also be exactly preserved at each step. Based on this, we propose some energy and quadratic invariants preserving $\alpha$-PRK methods. These $\alpha$-PRK methods are shown to have the same convergence rate as the usual PRK methods and perform very well in various numerical experiments.     

一族离散的可积Hamilton方程与可积的辛映射

该文讨论一个新的离散特征值问题，导出了相应的离散的Hamilton系统的保谱族，并且证明了它们是Liouville可积系。通过谱问题的双非线性化，导出一个新的可积的辛映射 。      

基于修正的Magnus方法的高振荡动力系统的数值积分方法

基于建立于一般线性动力系统上的Magnus数值积分方法,针对随时间而高频率振荡的二阶动力系统,给出了有效的修正Magnus数值积分算法．首先,将二阶动力系统重新表示为一阶系统的形式,通过引进新变量进行参考坐标变换,使动力系统的高振荡性质保留在新形式内;进而基于局部线性化技术用修正的Magnus方法求解新形式下的系统方程;最后,通过一系列数值实验说明了文中方法的有效性．     

SBA splitting for approximation of invariants in time-dependent Hamiltonian systems

Most physical phenomena are described by time-dependent Hamiltonian systems with qualitative features that should be preserved by numerical integrators used for approximating their dynamics. The initial energy of the system together with the energy added or subtracted by the outside forces, represent a conserved quantity of the motion. For a class of time-dependent Hamiltonian systems [8] this invariant can be defined by means of an auxiliary function whose dynamics has to be integrated simultaneously with the system’s equations. We propose splitting procedures featured by a SB³A property that allows to construct composition methods with a reduced number of determining order equations and to provide the same high accuracy for both the dynamics and the preservation of the invariant quantity.     

Symmetric period solutions with prescribed minimal period for even autonomous semipositive Hamiltonian systems

We study some monotonicity and iteration inequality of the Maslov-type index i-1of linear Hamiltonian systems.As an application we prove the existence of symmetric periodic solutions with prescribed minimal period for first order nonlinear autonomous Hamiltonian systems which are semipositive,even,and superquadratic at zero and infinity.This result gives a positive answer to Rabinowitz’s minimal period conjecture in this case without strictly convex assumption.We also give a different proof of the existence of symmetric periodic solutions with prescribed minimal period for classical Hamiltonian systems which are semipositive,even,and superquadratic at zero and infinity which was proved by Fei,Kim and Wang in 2001.     

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.     

Conservation properties of numerical integrators for highly oscillatory Hamiltonian systems

^** Email: David.Cohen{at}math.unige.ch. Present address: Mathematisches Institut, Universität Tübingen, D-72076 Tübingen, Germany (cohen{at}na.uni-tuebingen.de) Modulated Fourier expansion is used to show long-time near-conservationof the total and oscillatory energies of numerical methods forHamiltonian systems with highly oscillatory solutions. The numericalmethods considered are an extension of the trigonometric methods.A brief discussion of conservation properties in the continuousproblem and in the multi-frequency case is also given.     

对称不定系统的一类预处理子的注记

由于对称正定系统已有很多有效的求解方法,因此将对称的、或者非对称的不定系统转化为对称正定系统就成为解决这类问题的方法之一构造了一类简洁有效的预处理子,将对称不定系统转化为对称正定型,研究了所得预处理系统的谱性质,估计了其谱条件数,推广了现有结论.     

KAM-Type Theorem on Resonant Surfaces for Nearly Integrable Hamiltonian Systems

Summary. In this paper, we consider analytic perturbations of an integrable Hamiltonian system in a given resonant surface. It is proved that, for most frequencies on the resonant surface, the resonant torus foliated by nonresonant lower dimensional tori is not destroyed completely and that there are some lower dimensional tori which survive the perturbation if the Hamiltonian satisfies a certain nondegenerate condition. The surviving tori might be elliptic, hyperbolic, or of mixed type. This shows that there are many orbits in the resonant zone which are regular as in the case of integrable systems. This behavior might serve as an obstacle to Arnold diffusion. The persistence of hyperbolic lower dimensional tori has been considered by many authors [5], [6], [15], [16], mainly for multiplicity one resonant case. To deal with the mechanisms of the destruction of the resonant tori of higher multiplicity into nonhyperbolic lower dimensional tori, we have to deal with some small coefficient matrices that are the generalization of small divisors. Received December 18, 1997; revised December 30, 1998; accepted June 21, 1999