Explicit K-symplectic methods for nonseparable non-canonical Hamiltonian systems

We propose efficient numerical methods for nonseparable non-canonical Hamiltonian systems which are explicit,K-symplectic in the extended phase space with long time energy conservation properties. They are based on extending the original phase space to several copies of the phase space and imposing a mechanical restraint on the copies of the phase space. Explicit K-symplectic methods are constructed for two non-canonical Hamiltonian systems. Numerical tests show that the proposed methods exhibit good numerical performance in preserving the phase orbit and the energy of the system over long time, whereas higher order Runge–Kutta methods do not preserve these properties. Numerical tests also show that the K-symplectic methods exhibit better efficiency than that of the same order implicit symplectic, explicit and implicit symplectic methods for the original nonseparable non-canonical systems. On the other hand, the fourth order K-symplectic method is more efficient than the fourth order Yoshida’s method, the optimized partitioned Runge–Kutta and Runge–Kutta–Nystr ¨om explicit K-symplectic methods for the extended phase space Hamiltonians, but less efficient than the the optimized partitioned Runge–Kutta and Runge–Kutta–Nystr ¨om extended phase space symplectic-like methods with the midpoint permutation.

HOMOCLINIC SOLUTIONS NEAR THE ORIGIN FOR A CLASS OF FIRST ORDER HAMILTONIAN SYSTEMS

In this paper,we study the existence of infinitely many homoclinic solutions for a class of first order Hamiltonian systems ż=J H_(z)(t,z),where the Hamiltonian function H possesses the form H(t,z)=1/2L(t)z⋅z+G(t,z),and G(t,z)is only locally defined near the origin with respect to z.Under some mild conditions on L and G,we show that the existence of a sequence of homoclinic solutions is actually a local phenomenon in some sense,which is essentially forced by the subquadraticity of G near the origin with respect to z.

Crossing Limit Cycles of Planar Piecewise Hamiltonian Systems with Linear Centers Separated by Two Parallel Straight Lines

In this paper, we have studied several classes of planar piecewise Hamiltonian systems with three zones separated by two parallel straight lines. Firstly, we give the maximal number of limit cycles in these classes of systems with a center in two zones and without equilibrium points in the other zone (or with a center in one zone and without equilibrium points in the other zones). In addition, we also give examples to illustrate that it can reach the maximal number.

Noether conserved quantities and Lie point symmetries for difference nonholonomic Hamiltonian systems in irregular lattices

The Noether conserved quantities and the Lie point symmetries for difference nonholonomic Hamiltonian systems in irregular lattices are studied. The generalized Hamiltonian equations of the systems are given on the basis of the transformation operators in the space of discrete Hamiltonians. The Lie transformations acting on the lattice, as well as the equations and the determining equations of the Lie symmetries are obtained for the nonholonomic Hamiltonian systems. The discrete analogue of the Noether conserved quantity is constructed by using the Lie point symmetries. An example is discussed to illustrate the results.

THE EXISTENCE OF NONTRIVIAL SOLUTIONS OF HAMILTONIAN SYSTEMS WITH LAGRANGIAN BOUNDARY CONDITIONS

Some theorems are obtained for the existence of nontrivial solutions of Hamiltonian systems with Lagrangian boundary conditions by the minimax methods.

HOMOCLINIC ORBITS FOR A CLASS OF THE SECOND ORDER HAMILTONIAN SYSTEMS

The existence of homoclinic orbits is obtained by the variational approach for a class of second order Hamiltonian systems q（t） ＋ ↓△V（t, q（t）） = 0, where V（t, x） = -K（t, x） ＋ W（t, x）, K（t, x） is neither a quadratic form in x nor periodic in t and W（t, x） is superquadratic in x.

Passive robust control for uncertain Hamiltonian systems by using operator theory

In this study,the passivity-based robust control and tracking for Hamiltonian systems with unknown perturbations by using the operator-based robust right coprime factorisation method is concerned.For the system with unknown perturbations,a design scheme is proposed to guarantee the uncertain non-linear systems to be robustly stable while the equivalent non-linear systems is passive,meanwhile the asymptotic tracking property of the plant output is discussed.Moreover,the design scheme can be also used into the general Hamiltonian systems while the simulation is used to further demonstrate the effectiveness of the proposed method.

Symmetries and variational calculation of discrete Hamiltonian systems

We present a numerical simulation method of Noether and Lie symmetries for discrete Hamiltonian systems. The Noether and Lie symmetries for the systems are proposed by investigating the invariance properties of discrete Lagrangian in phase space. The numerical calculations of a two-degree-of-freedom nonlinear harmonic oscillator show that the difference discrete variational method preserves the exactness and the invariant quantity.

NONTRIVIAL SOLUTIONS FOR ASYMPTOTICALLY LINEAR HAMILTONIAN SYSTEMS WITH LAGRANGIAN BOUNDARY CONDITIONS

In this article, we study the existence of nontrivial solutions for a class of asymptotically linear Hamiltonian systems with Lagrangian boundary conditions by the Galerkin approximation methods and the L-index theory developed by the first author.

ON STABILITY BOUNDARY OF LINEAR MULTI-PARAMETER HAMILTONIAN SYSTEMS

In this paper an approximate equation is derived to describe smooth parts of the stability boundary for linear Hamiltonian systems, depending on arbitrary number of parameters. With this equation, we can obtain parameters corresponding to the stability boundary, as well as to the stability and instability domains, provided that one point on the stability boundary is known. Then differential equations describing the evolution of eigenvalues and eigenvectors along a curve on the stability boundary surface are derived. These equations also allow us to obtain curves belonging to the stability boundary. Applications to linear gyroscopic systems are considered and studied with examples.

MULTIPLICITY OF PERIODIC SOLUTIONS FOR SECOND ORDER HAMILTONIAN SYSTEMS WITH MIXED NONLINEARITIES

The multiplicity of periodic solutions for a class of second order Hamiltonian system with superquadratic plus subquadratic nonlinearity is studied in this paper.Obtained via the Symmetric Mountain Pass Lemma,two results about infinitely many periodic solutions of the systems extend some previously known results.

Continuous finite element methods for Hamiltonian systems

By applying the continuous finite element methods of ordinary differential equations, the linear element methods are proved having second-order pseudo-symplectic scheme and the quadratic element methods are proved having third-order pseudo- symplectic scheme respectively for general Hamiltonian systems, and they both keep energy conservative. The finite element methods are proved to be symplectic as well as energy conservative for linear Hamiltonian systems. The numerical results are in agree-ment with theory.

Multiplicity of Solutions for Fractional Hamiltonian Systems under Local Conditions

Under some local superquadratic conditions on W (t, u) with respect to u, the existence of infinitely many solutions is obtained for the nonperiodic fractional Hamiltonian systems, where L (t) is unnecessarily coercive.

MULTIPLE SOLUTIONS FOR HAMILTONIAN SYSTEMS WITH PERIODIC NONLINEARITY AND STRONG RESONANCE

I. Introduction In this paper we are looking for solutions of the following Hamiltonian system of second order: where x= （x1, x2） and V satisfies （V. 1） V: R×R2→R is a C1-function, 1-periodic In t, （V.2） V is periodic in x1 with the period T>0, （V. 3） V→O, Vx→O as |x2|→∞, uniformly in （t, x1）.

NEW METHOD FOR THE CONSTRUCTION OF INTEGRABLE HAMILTONIAN SYSTEMS

A new method.for the construction of integrable Hamiltonian system is proposed.For a given Poisson manifold the present paper constructs new Poisson brackets on it by making use of the Dirac-Poisson structure[1],and obtains .further new integrable Hamiltonian systems The constructed Poisson bracket is usual non-linear, and this new method is also different from usual ones[2-4].Two examples are given.

Homoclinic Solutions for a Class of Perturbed Fractional Hamiltonian Systems with Subquadratic Conditions

In this paper,we consider the following perturbed fractional Hamiltonian systems{tD_(∞)^(α)(_(-∞)D_(t)^(α)u(t))+L(t)u(t)=■_(u)W(t,u(t))+■(u)G(t,u(t)),t∈R,u∈H^(α)(R,R^(N)),whereα∈(1/2,1],L∈C(R,R^(N×N))is symmetric and not necessarily required to be positive definite,W∈C1(R×R^(N,R))is locally subquadratic and locally even near the origin,and perturbed term G∈C1(R×R^(N,R))maybe has no parity in u.Utilizing the perturbed method improved by the authors,a sequence of nontrivial homo clinic solutions is obtained,which generalizes previous results.

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.

ENERGY AND QUADRATIC INVARIANTS PRESERVING METHODS FOR HAMILTONIAN SYSTEMS WITH HOLONOMIC CONSTRAINTS

We introduce a new class of parametrized structure–preserving partitioned RungeKutta(α-PRK)methods for Hamiltonian systems with holonomic constraints.The methods are symplectic for any fixed scalar parameterα,and are reduced to the usual symplectic PRK methods like Shake-Rattle method or PRK schemes based on Lobatto IIIA-IIIB pairs whenα=0.We provide a new variational formulation for symplectic PRK schemes and use it to prove that theα-PRK methods can preserve the quadratic invariants for Hamiltonian systems subject to holonomic constraints.Meanwhile,for any given consistent initial values(p0,q0)and small step size h>0,it is proved that there existsα∗=α(h,p0,q0)such that the Hamiltonian energy can also be exactly preserved at each step.Based on this,we propose some energy and quadratic invariants preservingα-PRK methods.Theseα-PRK methods are shown to have the same convergence rate as the usual PRK methods and perform very well in various numerical experiments.

PERIODIC SOLUTIONS OF ASYMPTOTICALLY LINEA RHAMILTONIAN SYSTEMS

The authors establish the existence of nontrival periodic solntions of the asymptotically linear Hamiltonian systems in the general case that the asymptotic matrix may be degenerate and time-dependent.This is done by using the critical point theory,Galerkin approximation procedure and the Maslov-type index theory introduced and generalized by Conley,Zehnder and Long.

FINITE ELEMENT METHOD WITH SUPERCONVERGENCE FOR NONLINEAR HAMILTONIAN SYSTEMS

This paper is concerned with the finite element method for nonlinear Hamiltonian systems from three aspects： conservation of energy, symplicity, and the global error. To study the symplecticity of the finite element methods, we use an analytical method rather than the commonly used algebraic method. We prove optimal order of convergence at the nodes tn for mid-long time and demonstrate the symplecticity of high accuracy. The proofs depend strongly on superconvergence analysis. Numerical experiments show that the proposed method can preserve the energy very well and also can make the global trajectory error small for long time.