The problem of drift and recurrence for the rolling Chaplygin ball

We investigate the motion of the point of contact (absolute dynamics) in the integrable problem of the Chaplygin ball rolling on a plane. Although the velocity of the point of contact is a given vector function of variables of the reduced system, it is impossible to apply standard methods of the theory of integrable Hamiltonian systems due to the absence of an appropriate conformally Hamiltonian representation for an unreduced system. For a complete analysis we apply the standard analytical approach, due to Bohl and Weyl, and develop topological methods of investigation. In this way we obtain conditions for boundedness and unboundedness of the trajectories of the contact point.     

Decomposing a new nonlinear differential-difference system under a Bargmann implicit symmetry constraint

Firstly, a hierarchy of integrable lattice equations and its bi-Hamilt-onian structures are established by applying the discrete trace identity. Secondly, under an implicit Bargmann symmetry constraint, every lattice equation in the nonlinear differential-difference system is decomposed by an completely integrable symplectic map and a finite-dimensional Hamiltonian system. Finally, the spatial part and the temporal part of the Lax pairs and adjoint Lax pairs are all constrained as finite dimensional Liouville integrable Hamiltonian systems.     

Multiscale KAM theorem for Hamiltonian systems

In this paper, we study the persistence of invariant tori in nearly integrable multiscale Hamiltonian systems with highorder degeneracy in the integrable part. Such Hamiltonian systems arise naturally in planar and spatial lunar problems of celestial mechanics for which the persistence problem connects closely to the stability of the systems. We introduce multiscale nondegenerate condition and multiscale Diophantine condition, comparable to the usual Diophantine condition. Using quasilinear KAM method, we prove a multiscale KAM theorem.     

WEAK CONVERGENCE FOR NON-UNIFORM φ-MIXING RANDOM FIELDS

ＷＥＡＫＣＯＮＶＥＲＧＥＮＣＥＦＯＲＮＯＮＵＮＩＦＯＲＭφＭＩＸＩＮＧＲＡＮＤＯＭＦＩＥＬＤＳＬＵＣＨＵＡＮＲＯＮＧＡｂｓｔｒａｃｔＬｅｔ｛ξｔ，ｔ∈Ｚｄ｝ｂｅａｎｏｎｕｎｉｆｏｒｍφｍｉｘｉｎｇｓｔｒｉｃｔｌｙｓｔａｔｉｏｎａｒｙｒｅａ...     

Non-symmetry constraints of the AKNS system yielding integrable Hamiltonian systems

This paper aims to show that there exist non-symmetry constraints which yield integrable Hamiltonian systems through nonlinearization of spectral problems of soliton systems, like symmetry constraints. Taking the AKNS spectral problem as an illustrative example, a class of such non-symmetry constraints is introduced for the AKNS system, along with two-dimensional integrable Hamiltonian systems generated from the AKNS spectral problem.     

The geometry of integrable and superintegrable systems

We consider the automorphism group of the geometry of an integrable system. The geometric structure used to obtain it is generated by a normal-form representation of integrable systems that is independent of any additional geometric structure like symplectic, Poisson, etc. Such a geometric structure ensures a generalized toroidal bundle on the carrier space of the system. Noncanonical diffeomorphisms of this structure generate alternative Hamiltonian structures for completely integrable Hamiltonian systems. The energy-period theorem for dynamical systems implies the first nontrivial obstruction to the equivalence of integrable systems.     

Invariant Tori in Hamiltonian Systems with High Order Proper Degeneracy

We study the existence of quasi-periodic, invariant tori in a nearly integrable Hamiltonian system of high order proper degeneracy, i.e., the integrable part of the Hamiltonian involves several time scales and at each time scale the corresponding Hamiltonian depends on only part of the action variables. Such a Hamiltonian system arises frequently in problems of celestial mechanics, for instance, in perturbed Kepler problems like the restricted and non-restricted 3-body problems and spatial lunar problems in which several bodies with very small masses are coupled with two massive bodies and the nearly integrable Hamiltonian systems naturally involve different time scales. Using KAM method, we will show under certain higher order non-degenerate conditions of Bruno–Rüssmann type that the majority of quasi-periodic, invariant tori associated with the integrable part will persist after the non-integrable perturbation. This actually concludes the KAM metric stability for such a properly degenerate Hamiltonian system.     

Algebraic integrability: The Adler-Van Moerbeke approach

In this paper, I present an overview of the active area of algebraic completely integrable systems in the sense of Adler and van Moerbeke. These are integrable systems whose trajectories are straight line motions on abelian varieties (complex algebraic tori). We make, via the Kowalewski-Painlevé analysis, a study of the level manifolds of the systems. These manifolds are described explicitly as being affine part of abelian varieties and the flow can be solved by quadrature, that is to say their solutions can be expressed in terms of abelian integrals. The Adler-Van Moerbeke method’s which will be used is devoted to illustrate how to decide about the algebraic completely integrable Hamiltonian systems and it is primarily analytical but heavily inspired by algebraic geometrical methods. I will discuss some interesting and well known examples of algebraic completely integrable systems: a five-dimensional system, the Hénon-Heiles system, the Kowalewski rigid body motion and the geodesic flow on the group SO(n) for a left invariant metric.     

A hierarchy of Liouville integrable lattice equations and its integrable coupling systems

A new discrete two-by-two matrix spectral problem with two potentials is introduced, followed by a hierarchy of integrable lattice equations obtained through discrete zero curvature equations. It is shown that the Hamiltonian structures of the resulting integrable lattice equations are established by virtue of the trace identity. Furthermore, based on a discrete four-by-four matrix spectral problem, the discrete integrable coupling systems of the resulting hierarchy are obtained. Then, with the variational identity, the Hamiltonian structures of the obtained integrable coupling systems are established. Finally, the resulting Hamiltonian systems are proved to be all Liouville integrable.     

An integrable coupling hierarchy of the Mkdv_integrable systems, its Hamiltonian structure and corresponding nonisospectral integrable hierarchy

A four-by-four matrix spectral problem is introduced, locality of solution of the related stationary zero curvature equation is proved. An integrable coupling hierarchy of the Mkdv_integrable systems is presented. The Hamiltonian structure of the resulting integrable coupling hierarchy is established by means of the variational identity. It is shown that the resulting integrable couplings are all Liouville integrable Hamiltonian systems. Ultimately, through the nonisospectral zero curvature representation, a nonisospectral integrable hierarchy associated with the resulting integrable couplings is constructed.     

Persistence of hyperbolic tori in Hamiltonian systems

We generalize the well-known result of Graff and Zehnder on the persistence of hyperbolic invariant tori in Hamiltonian systems by considering non-Floquet, frequency varying normal forms and allowing the degeneracy of the unperturbed frequencies. The preservation of part or full frequency components associated to the degree of non-degeneracy is considered. As applications, we consider the persistence problem of hyperbolic tori on a submanifold of a nearly integrable Hamiltonian system and the persistence problem of a fixed invariant hyperbolic torus in a non-integrable Hamiltonian system.     

一种构造可积Hamilton系统的新方法

建议了一种新的构造可积Hamilton系统的方法。对于给定的Poisson流形,本文利用Dirac-Poisson结构构造其上的新Poisson括号[1],进而获得了新的可积Hamilton系统。构造的Poisson括号一般是非线的,并且这种方法也不同于通常的方法[2～4]。本文还给出了两个实例。     

Orthogonal separable Hamiltonian systems on T~2

In this paper we characterize the Liouvillian integrable orthogonal separable Hamiltonian systems on T~2 for a given metric,and prove that the Hamiltonian flow on any compact level hypersurface has zero topological entropy.Furthermore,by examples we show that the integrable Hamiltonian systems on T~2 can have complicated dynamical phenomena.For instance they can have several families of invariant tori,each family is bounded by the homoclinic-loop-like cylinders and heteroclinic-loop-like cylinders.As we know,it is the first concrete example to present the families of invariant tori at the same time appearing in such a complicated way.     

Geometric and dynamical invariants of integrable Hamiltonian and dissipative systems

This paper presents results concerning the geometric invariant theory of completely integrable Hamiltonian systems and also the classification of integrable cases of low-dimensional and high-dimensional rigid body dynamics in a nonconservative force field. The latter problems are described by dynamical systems with variable dissipation. The first part of the work is the basis for the doctorial dissertation of V. V. Trofimov (1953–2003), which was already published in parts. However, in the present complete form, it has not appeared, and we decided to fill in this gap. The second part is a development of the results presented in the doctoral dissertation of M. V. Shamolin and has not appeared in the present variant. These two parts complement one another well, which initiated this work (its sketches already appeared in 1997).     

利用李代数构造非线性可积及双可积哈密顿耦合

With the help of a Lie algebra,two kinds of Lie algebras with the forms of blocks are introduced for generating nonlinear integrable and bi-integrable couplings.For illustrating the application of the Lie algebras,an integrable Hamiltonian system is obtained,from which some reduced evolution equations are presented.Finally,Hamiltonian structures of nonlinear integrable and bi-integrable couplings of the integrable Hamiltonian system are furnished by applying the variational identity.The approach presented in the paper can also provide nonlinear integrable and bi-integrable couplings of other integrable system.     

A HIERARCHY OF INTEGRABLE HAMILTONIAN SYSTEMS WITH NEUMANN TYPE CONSTRAINT

A hierarchy of integrable Hamiltonian systems with Neumann type constraint isobtained by restricting a hierarchy of evolution equations associated with λφ_(xx)+u_iλ~iφ=λ~mφ to aninvariant subspace of their recursion operator.The independentintegrals of motion and Hamiltonian functions for these Hamiltonian systems areconstructed by using relevant reeursion formula and are shown to be in involution.Thusthese Hamiltonian systems are completely integrable and commute with each other.     

Nekhoroshev estimates for finitely differentiable quasi-convex Hamiltonians

A major result about perturbations of integrable Hamiltonian systems is the Nekhoroshev theorem, which gives exponential stability for all solutions provided the system is analytic and the integrable Hamiltonian is generic. In the particular but important case where the latter is quasi-convex, these exponential estimates have been generalized by Marco and Sauzin if the Hamiltonian is Gevrey regular, using a method introduced by Lochak in the analytic case. In this paper, using the same approach, we investigate the situation where the Hamiltonian is assumed to be only finitely differentiable, for which it is known that exponential stability does not hold but nevertheless we prove estimates of polynomial stability.     

Symplectic Topology of Integrable Hamiltonian Systems, II: Topological Classification

The main purpose of this paper is to give a topological and symplectic classification of completely integrable Hamiltonian systems in terms of characteristic classes and other local and global invariants.     

Decomposition of a hierarchy of nonlinear evolution equations

The generalized Hamiltonian structures for a hierarchy of nonlinear evolution equations are established with the aid of the trace identity. Using the nonlinearization approach, the hierarchy of nonlinear evolution equations is decomposed into a class of new finite-dimensional Hamiltonian systems. The generating function of integrals and their generator are presented, based on which the finite-dimensional Hamiltonian systems are proved to be completely integrable in the Liouville sense. As an application, solutions for the hierarchy of nonlinear evolution equations are reduced to solving the compatible Hamiltonian systems of ordinary differential equations.     

The KAM theorem with a large perturbation and application to the network of Duffing oscillators

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.