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.     

Periodic solutions with prescribed minimal period for the second order Hamiltonian systems with even potentials

In this paper, under a similar but stronger condition than that of Ambrosetti and Rabinowitz we find a T-periodic solution of the autonomous superquadratic second order Hamiltonian system with even potential for any T 〉 0; moreover, such a solution has T as its minimal period.     

一阶次二次凸Hamilton系统的最小P-边值解

利用对偶变分原理,将一阶次二次凸Hamilton系统的P-边值解问题转化为对偶变分问题,证明对偶泛函满足最小作用原理,进而推导出非平凡P-边值解的存在性结果,最后通过比较作用泛函的临界值,给出该P-边值解的最小周期估计.     

Periodic solutions of a class of superquadratic second order Hamiltonian systems

§1　IntroductionInthispaperwediscusstheexistenceofthesolutionforthefollowingsecondorderHamiltoniansystemx¨ Ax ΔF(x)=0,(1.1)whereAisann×nrealsymmetricmatrixandisnon-definite,F∈C1(Rn,R),andΔF(x)denotesthegradientofF.WhileworksforsecondorderHamiltonsystemshavemostlybeendoneundertheconditionA=0,westudythecasewhereA≠0andisnon-definiteinthepapers[1,2].DefineH=H1,2T([0,T],Rn)={x:R→Rn|xisabsolutelycontinuous,x∈L2([0,T],Rn),x(0)=x(T),x(0)=x(T)}and〈x,y〉=∫T0[(x(t),y(t)) (x…     

Maslov P-Index Theory for a Symplectic Path with Applications

The Maslov P-index theory for a symplectic path is defined. Various properties of this index theory such as homotopy invariant, symplectic additivity and the relations with other Morse indices are studied. As an application, the non-periodic problem for some asymptotically linear Hamiltonian systems is considered.     

On the minimal periodic solutions of nonconvex superlinear Hamiltonian systems

This paper proves a multiplicity result for the minimal periodic solutions of Hamiltonian system under a superquadratic hypothesis on H weaker than the Ambrosetti-Rabinowitz-type condition. We also define a class of homogeneous functions, that are more general than the classical ones, and prove the Rabinowitz's conjecture in the case of H satisfying such new homogeneity.     

A symmetry of sphere map implies its chaos*

A well-known example, given by Shub, shows that for any |d| ≥ 2 there is a self-map of the sphere Sⁿ, n ≥ 2, of degree d for which the set of non-wandering points consists of two points. It is natural to ask which additional assumptions guarantee an infinite number of periodic points of such a map. In this paper we show that if a continuous map f : Sⁿ → Sⁿ commutes with a free homeomorphism g : Sⁿ → Sⁿ of a finite order, then f has infinitely many minimal periods, and consequently infinitely many periodic points. In other words the assumption of the symmetry of f originates a kind of chaos. We also give an estimate of the number of periodic points. *Research supported by KBN grant nr 2 P03A 045 22.     

无最小正周期的周期函数的构造及其性质

从群论的角度给出周期函数的等价定义.对任意给定的非循环群GR),构造出以G为周期集的无最小正周期的周期函数.讨论无最小正周期的周期函数的性质.     

Multiple periodic solutions for non-canonical Hamiltonian systems with application to differential delay equations

We first establish Maslov index for non-canonical Hamiltonian system by using symplectic transformation for Hamiltonian system. Then the existence of multiple periodic solutions for the non-canonical Hamiltonian system is obtained by applying the Maslov index and Morse theory. As an application of the results, we study a class of non-autonomous differential delay equation which can be changed to non-canonical Hamiltonian system and obtain the existence of multiple periodic solutions for the equation by employing variational method.     

高阶积分微分方程的周期边值问题

研究一类高阶积分微分方程的周期边值问题,利用上下解和单调迭代法证得最大解和最小解的存在性。     

Periodic orbits of Hamiltonian systems on symmetric positive-type hypersurfaces

This paper discusses the existence and multiplicity of periodic orbits of Hamiltonian systems on symmetric positive-type hypersurfaces. We prove that each such energy hypersurface carries at least one symmetric periodic orbit. Under some suitable pinching conditions, we also get an existence result of multiple symmetric periodic orbits.     

一类非自治系统的周期解

本文运用重合度理论研究一类非自治非线性系统的周期解问题,得到一些有关存在性,唯一性和稳定性的新结果。