Degenerate lower-dimensional tori in Hamiltonian systems

We study the persistence of lower-dimensional tori in Hamiltonian systems of the form , where (x,y,z)∈Tn×Rn×R2m, ε is a small parameter, and M(ω) can be singular. We show under a weak Melnikov nonresonant condition and certain singularity-removing conditions on the perturbation that the majority of unperturbed n-tori can still survive from the small perturbation. As an application, we will consider the persistence of invariant tori on certain resonant surfaces of a nearly integrable, properly degenerate Hamiltonian system for which neither the Kolmogorov nor the g-nondegenerate condition is satisfied.     

Singularities of Lagrangian varieties and related topics

We study the classification problem for generic projections of Lagrangian submanifolds. A classification list for symmetric Lagrangian submanifolds is obtained and the generic evolutions of symmetric caustics are illustrated. We show how the singular Lagrangian varieties appear in the invariant theory of binary forms and we introduce the basic concepts of the desingularization procedure. Applications to differential geometry, geometrical optics, and mechanics are presented.     

A twist condition and periodic solutions of Hamiltonian systems

In this paper, we investigate existence of nontrivial periodic solutions to the Hamiltonian system

(HS)     

Multiple periodic solutions of Hamiltonian systems with prescribed energy

Consider the periodic solutions of autonomous Hamiltonian systems on the given compact energy hypersurface Σ=H⁻¹(1). If Σ is convex or star-shaped, there have been many remarkable contributions for existence and multiplicity of periodic solutions. It is a hard problem to discuss the multiplicity on general hypersurfaces of contact type. In this paper we prove a multiplicity result for periodic solutions on a special class of hypersurfaces of contact type more general than star-shaped ones.     

The Gauss-Bonnet theorem for vector bundles

We give a short proof of the Gauss-Bonnet theorem for a real oriented Riemannian vector bundle E of even rank over a closed compact orientable manifold M. This theorem reduces to the classical Gauss-Bonnet-Chern theorem in the special case when M is a Riemannian manifold and E is the tangent bundle of M endowed with the Levi-Civita connection. The proof is based on an explicit geometric construction of the Thom class for 2-plane bundles. Dedicated to the memory of Philip Bell Research partially supported by NSF grant DMS-9703852.     

The existence of periodic solutions of non-autonomous second-order Hamiltonian systems

The purpose of this paper is to study the existence of periodic solutions for a class of non-autonomous second-order Hamiltonian systems. Some new existence theorems are obtained by using the least action principle and the saddle point theorem.     

The compact category and multiple periodic solutions of Hamiltonian systems on symmetric starshaped energy surfaces

Dieter Puppe zum 60. Geburtstag gewidmet     

Cup-length estimate for Lagrangian intersections

In this paper, we consider the Arnold conjecture on the Lagrangian intersections of some closed Lagrangian submanifold of a closed symplectic manifold with its image of a Hamiltonian diffeomorphism. We prove that if the Hofer's symplectic energy of the Hamiltonian diffeomorphism is less than a topology number defined by the Lagrangian submanifold, then the Arnold conjecture is true in the degenerated (nontransversal) case.     

Geodesics in stationary spacetimes and classical Lagrangian systems

We state a fundamental correspondence between geodesics on stationary spacetimes and the equations of classical particles on Riemannian manifolds, accelerated by a potential and a magnetic field. By variational methods, we prove some existence and multiplicity theorems for fixed energy solutions (joining two points or periodic) of the above described Riemannian equation. As a consequence, we obtain existence and multiplicity results for geodesics with fixed energy, connecting a point to a line or periodic trajectories, in (standard) stationary spacetimes.     

Nontrivial periodic solutions for strong resonance Hamiltonian systems via a local linking theorem

In this paper, we first establish an existence result of critical points for a class of functionals defined on Hilbert spaces by using a local linking idea. Then as an application of the existence result, we obtain the existence of periodic solutions of strong resonance Hamiltonian systems which are asymptotically linear both at infinity and at origin.     

The Brake Orbits of Hamiltonian Systems on Positive-type Hypersurfaces

This paper deals with the brake orbits of Hamiltonian system on given energy hypersurfaces Σ = H ⁻¹(1). We introduce a class of contact type but not necessarily star-shaped hypersurfaces in ℝ²ⁿ and call them normalized positive-type hypersurfaces. By using of the critical point theory, we prove that if Σ is a partially symmetric normalized positive-type hypersurface, it must carries a brake orbit of (HS). Furthermore, we obtain some multiplicity results under certain pinching conditions. Our results include the earlier works on this subject given by P. Rabinowitz and A. Szulkin in star-shaped case. An example of partially symmetric normalized positive-type hypersurface in ℝ⁴ that is not star-shaped is also presented Partially supported by NNSF of China (10571085) and Science Foundation of Hohai University.     

KAM theorem of symplectic algorithms for Hamiltonian systems

Summary. In this paper we prove that an analog of the celebrated KAM theorem holds for symplectic algorithms, which Channel and Scovel (1990), Feng Kang (1991) and Sanz-Serna and Calvo (1994) suggested a few years ago. The main results consist of the existence of invariant tori, with a smooth foliation structure, of a symplectic numerical algorithm when it applies to a generic integrable Hamiltonian system if the system is analytic and the time-step size of the algorithm is s ufficiently small. This existence result also implies that the algorithm, when it is applied to a generic integrable system, possesses n independent smooth invariant functions which are in involution and well-defined on the set filled by the invariant tori in the sense of Whitney. The invariant tori are just the level sets of these functions. Some quantitative results about the numerical invariant tori of the algorithm approximating the exact ones of the system are also given. Received December 27, 1997 / Revised version received July 15, 1998 / Published online: July 7, 1999     

Birkhoff's theorem and multidimensional numerical range

We show that, under certain conditions, Birkhoff's theorem on doubly stochastic matrices remains valid for countable families of discrete probability spaces which have nonempty intersections. Using this result, we study the relation between the spectrum of a self-adjoint operator A and its multidimensional numerical range. It turns out that the multidimensional numerical range is a convex set whose extreme points are sequences of eigenvalues of the operator A. Every collection of eigenvalues which can be obtained by the Rayleigh-Ritz formula generates an extreme point of the multidimensional numerical range. However, it may also have other extreme points.     

Geodesic rays,Busemann functions and monotone twist maps

We study the action-minimizing half-orbits of an area-preserving monotone twist map of an annulus. We show that these so-called rays are always asymptotic to action-minimizing orbits. In the spirit of Aubry-Mather theory which analyses the set of action-minimizing orbits we investigate existence and properties of rays. By analogy with the geometry of the geodesics on a Riemannian 2-torus we define a Busemann function for every ray. We use this concept to prove that the minimal average action A() is differentiable at irrational rotation numbers while it is generically non-differentiable at rational rotation numbers (cf. also [18]). As an application of our results in the geometric framework we prove that a Riemannian 2-torus which has the same marked length spectrum as a flat 2-torus is actually isometric to this flat torus.     

The hyperbolic invariant tori of symplectic mappings

In this paper we study the persistence of lower dimensional hyperbolic invariant tori for nearly integrable twist symplectic mappings. Under a Rüssmann-type non-degenerate condition, by introducing a modified KAM iteration scheme, we proved that nearly integrable twist symplectic mappings admit a family of lower dimensional hyperbolic invariant tori as long as the symplectic perturbation is small enough.     

Degenerate bifurcation points of periodic solutions of autonomous Hamiltonian systems

We study connected branches of nonconstant 2π-periodic solutions of the Hamilton equation

     

On the number of limit cycles in small perturbations of a class of hyper-elliptic Hamiltonian systems with one nilpotent saddle

In this paper, we make a complete study on small perturbations of Hamiltonian vector field with a hyper-elliptic Hamiltonian of degree five, which is a Liénard system of the form x^′=y, y^′=Q₁(x)+εyQ₂(x) with Q₁ and Q₂ polynomials of degree respectively 4 and 3. It is shown that this system can undergo degenerated Hopf bifurcation and Poincaré bifurcation, which emerges at most three limit cycles in the plane for sufficiently small positive ε. And the limit cycles can encompass only an equilibrium inside, i.e. the configuration (3,0) of limit cycles can appear for some values of parameters, where (3,0) stands for three limit cycles surrounding an equilibrium and no limit cycles surrounding two equilibria.