Closed characteristics on non-degenerate star-shaped hypersurfaces in R<Subscript>2n</Subscript>

In this paper, firstly we establish the relation theorem between the Maslov-type index and the index defined by C. Viterbo for star-shaped Hamiltonian systems. Then we extend the iteration formula ofC. Viterbo for non-degenerate star-shaped Hamiltonian systems to the general case. Finally we prove that there exist at least two geometrically distinct closed characteristics on any non-degenerate star-shaped compact smooth hypersurface on R²ⁿ with n > 1. Here we call a hypersurface non-degenerate, if all the closed characteristics on the given hypersurface together with all of their iterations are non-degenerate as periodic solutions of the corresponding Hamiltonian system. We also study the ellipticity of closed characteristics whenn = 2     

Non-hyperbolic closed characteristics on non-degenerate star-shaped hypersurfaces in ℝ<Superscript>2<Emphasis Type="Italic">n</Emphasis></Superscript>

In this paper, we prove that for every index perfect non-degenerate compact star-shaped hypersurface ∑ ⊂ R²ⁿ, there exist at least n non-hyperbolic closed characteristics with even Maslovtype indices on ∑ when n is even. When n is odd, there exist at least n closed characteristics with odd Maslov-type indices on ∑ and at least (n-1) of them are non-hyperbolic. Here we call a compact star-shaped hypersurface ∑ ⊂ R²ⁿ index perfect if it carries only finitely many geometrically distinct prime closed characteristics, and every prime closed characteristic (τ, y) on ∑ possesses positive mean index and whose Maslov-type index i(y, m) of its m-th iterate satisfies i(y, m) ≠-1 when n is even, and i(y, m) ∉ {-2, -1, 0} when n is odd for all m ∈ N.     

本刊英文版Vol.34(2018),No.1论文摘要（英文）

正Non-hyperbolic Closed Characteristics on Non-degenerate Star-shaped Hypersurfaces in R~(2n)Hua Gui DUAN Hui LIU Yi Ming LONG Wei WANGAbstract In this paper,we prove that for every index perfect non-degenerate compact starshaped hypersurface∑(?)R~(2n),there exist at least n non-hyperbolic closed characteristics with even Maslov-type indices on E when n is even.When n is odd,there exist at least n closed     

次二次Hamilton系统的对偶Morse指标及其闭特征的稳定性

本文通过Ｇｅｌｅｒｋｉｎ逼近方法,在没有任何凸的条件下,研究了次二次Ｈａｍｉｌ－ｔｏｎ系统的ｋ对偶Ｍｏｒｓｅ指标理论．作为应用,在本文研究了Ｒ２ｎ中的凸超曲面上的闭特征的稳定性,证明了在一个较宽松的夹条件下,这类超曲面上至少有一条椭圆闭特征．     

The fixed energy problem for a class of nonconvex singular Hamiltonian systems

We consider a noncompact hypersurface H in R2N which is the energy level of a singular Hamiltonian of “strong force” type. Under global geometric assumptions on H, we prove that it carries a closed characteristic, as a consequence of a result by Hofer and Viterbo on the Weinstein conjecture in cotangent bundles of compact manifolds. Our theorem contains, as particular cases, earlier results on the fixed energy problem for singular Lagrangian systems of strong force type.     

On a theorem by Ekeland-Hofer

In [EH89, Theorem 1] Ekeland-Hofer prove that for a centrally symmetric, restricted contact type hypersurface in ℝ²ⁿ and for any global, centrally symmetric Hamiltonian perturbation there exists a leaf-wise intersection point. In this note we show that if we replace restricted contact type by star-shaped, there exist infinitely many leaf-wise intersection points or a leaf-wise intersection point on a closed characteristic.     

Hyperbolic characteristics on star-shaped hypersurfaces

In this paper, we study the stability of closed characteristics on a starshaped compact smooth hypersurface Σ in ²ⁿ. We show that the Maslov-type mean index of such a closed characteristic is independent of the choice of the Hamiltonian functions, and prove that on Σ either there are infinitely many closed characteristics, or there exists at least one nonhyperbolic closed characteristic, provided every closed characteristic possesses its Maslov-type mean index greater than 2 when n is odd, and greater than 1 when n is even.     

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.     

Hamiltonian dynamics on convex symplectic manifolds

We study the dynamics of Hamiltonian diffeomorphisms on convex symplectic manifolds. To this end we first establish an explicit isomorphism between the Floer homology and the Morse homology of such a manifold, and then use this isomorphism to construct a biinvariant metric on the group of compactly supported Hamiltonian diffeomorphisms analogous to the metrics constructed by Viterbo, Schwarz and Oh. These tools are then applied to prove and reprove results in Hamiltonian dynamics. Our applications comprise a uniform lower estimate for the slow entropy of a compactly supported Hamiltonian diffeomorphism, the existence of infinitely many non-trivial periodic points of a compactly supported Hamiltonian diffeomorphism of a subcritical Stein manifold, new cases of the Weinstein conjecture, and, most noteworthy, new existence results for closed trajectories of a charge in a magnetic field on almost all small energy levels. We shall also obtain some new Lagrangian intersection results. Partially supported by the Swiss National Foundation. Supported by the Swiss National Foundation and the von Roll Research Foundation.     

Indexing domains of instability for Hamiltonian systems

Based upon the understanding of the global topologies of the singular subset, its complement, and the hyperbolic subset in the symplectic group, in this paper we study the domains of instability for hyperbolic Hamiltonian systems and define a characteristic index for such domains. This index is defined via the Maslov-type index theory for symplectic paths starting from the identity defined by C. Conley, E. Zehnder, and Y. Long, and the hyperbolic index of symplectic matrices. The old problem of the relation between the non-degenerate local minimality and the instability of hyperbolic extremal loops in the calculus of variation is also studied via this new index for the domains of instability. Received July 4, 1997     

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.     

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.     

正定型超曲面上双曲闭特征的Maslov型指标的迭代公式

本文给出了R~(2n)中规范正定型超曲面上双曲闭特征的Maslov型指标的迭代公式.结果包含了凸超曲面和星形超曲面上已有的相应结论。     

Persistance d’orbites périodiques relatives dans les systèmes hamiltoniens symétriques

It was known to Poincaré that a non-degenerate periodic orbit in a Hamiltonian system persists to nearby energy-levels. In this Note, we consider the analogous problem for relative periodic orbits in symmetric Hamiltonian systems. We show that non-degenerate relative periodic orbits also persist when shifting to nearby values of the energy-momentum map, under the hypothesis that the group of symmetries acts freely.     

Persistence of lower-dimensional hyperbolic invariant tori for generalized Hamiltonian systems

In this paper we study the persistence of lower dimensional hyperbolic invariant tori for generalized Hamiltonian systems. Here the generalized Hamiltonian systems refer to the systems which may admit a distinct number of action and angle variables. In particular, systems under consideration can be odd-dimensional. Under Rüssmann-type non-degenerate condition, by introducing a modified linear KAM iterative scheme, we proved that the majority of the lower-dimensional hyperbolic invariant tori persist under small perturbations for generalized Hamiltonian systems.     

The Stability of Subharmonic Solutions for Hamiltonian Systems

In this paper, by using the dual Morse index theory, we study the stability of subharmonic solutions of the non-autonomous Hamiltonian systems. We obtain a (infinite) sequence of geometrically distinct periodic solutions such that every element has at most one direction of instability (i.e., it has at least 2n − 2 Floquet multipliers lying on the unit circle in the complex plane if the periodic solution is non-degenerate) or it is elliptic (all its 2n Floquet multipliers are lying on the unit circle) if the periodic solution is degenerate.     

Explicit Hamiltonian realizations of non-degenerate three-dimensional real linear differential systems

The purpose of this work is to give explicit Hamiltonian realizations for all non-degenerate real three-dimensional linear differential systems.     

Hörmander index in finite-dimensional case

We calculate the Hörmander index in the finite-dimensional case. Then we use the result to give some iteration inequalities, and prove almost existence of mean indices for given complete autonomous Hamiltonian system on compact symplectic manifold with symplectic trivial tangent bundle and given autonomous Hamiltonian system on regular compact energy hypersurface of symplectic manifold with symplectic trivial tangent bundle.     

The Existence of Two Closed Characteristics on Every Compact Star-shaped Hypersurface in R~4

Recently, Cristofaro-Gardiner and Hutchings proved that there exist at least two closed characteristics on every compact star-shaped hypersuface in R~4. Then Ginzburg, Hein, Hryniewicz,and Macarini gave this result a second proof. In this paper, we give it a third proof by using index iteration theory, resonance identities of closed characteristics and a remarkable theorem of Ginzburg et al.     

The existence of two closed characteristics on every compact star-shaped hypersurface in ℝ4

Recently, Cristofaro-Gardiner and Hutchings proved that there exist at least two closed characteristics on every compact star-shaped hypersuface in R⁴. Then Ginzburg, Hein, Hryniewicz, and Macarini gave this result a second proof. In this paper, we give it a third proof by using index iteration theory, resonance identities of closed characteristics and a remarkable theorem of Ginzburg et al.