PERIODIC SOLUTIONS OF ASYMPTOTICALLY LINEA RHAMILTONIAN SYSTEMS

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Invariant measures of Hamiltonian systems with prescribed asymptotic Maslov index

We study the properties of the asymptotic Maslov index of invariant measures for time-periodic Hamiltonian systems on the cotangent bundle of a compact manifold M. We show that if M has finite fundamental group and the Hamiltonian satisfies some general growth assumptions on the momenta, then the asymptotic Maslov indices of periodic orbits are dense in the half line [0,+∞). Furthermore, if the Hamiltonian is the Fenchel dual of an electromagnetic Lagrangian, then every non-negative number r is the limit of the asymptotic Maslov indices of a sequence of periodic orbits which converges narrowly to an invariant measure with asymptotic Maslov index r. We discuss the existence of minimal ergodic invariant measures with prescribed asymptotic Maslov index by the analogue of Mather’s theory of the beta function, the asymptotic Maslov index playing the role of the rotation vector. Dedicated to Vladimir Igorevich Arnold     

Morse theory for periodic solutions of hamiltonian systems and the maslov index

In this paper we prove Morse type inequalities for the contractible 1-periodic solutions of time dependent Hamiltonian differential equations on those compact symplectic manifolds M for which the symplectic form and the first Chern class of the tangent bundle vanish over π₂ (M). The proof is based on a version of infinite dimensional Morse theory which is due to Floer. The key point is an index theorem for the Fredholm operator which plays a central role in Floer homology. The index formula involves the Maslov index of nondegenerate contractible periodic solutions. This Maslov index plays the same role as the Morse index of a nondegenerate critical point does in finite dimensional Morse theory. We shall use this connection between Floer homology and Maslov index to establish the existence of infinitely many periodic solutions having integer periods provided that every 1-periodic solution has at least one Floquet multiplier which is not equal to 1.     

非自治时滞微分系统的周期解

本文将一类非自治时滞微分方程变换为等价的哈密顿系统.这样求解此时滞微分系统可以等价于求解相应的哈密顿系统.运用Floquet变换和辛变换方法,建立了此类微分系统多重周期解的存在性定理,此结果推广了先前文献中的一些结果.     

Maslov index, Poincaré-Birkhoff Theorem and Periodic Solutions of Asymptotically Linear Planar Hamiltonian Systems

We study the relationship between the twist condition in the Poincaré-Birkhoff fixed point theorem and the assumptions on the Maslov index for asymptotically linear planar Hamiltonian systems. For this aim, we develop a variant of the Poincaré-Birkhoff theorem which, together with its classical version, allows to obtain a lower bound for the number of nontrivial periodic solutions in terms of the gap between the Maslov indexes associated to the linearizations of the planar system at zero and at infinity.     

一类非线性哈密顿系统的周期解

结合Maslov指标理论,利用环绕定理证明了一类非线性哈密顿系统的周期解的存在性,而这类哈密顿系统所对应的作用泛函可能不满足Palais-Smale条件.     

Morse-type index theory for flows and periodic solutions for Hamiltonian Equations

An index theory for flows is presented which extends the classical Morse theory for gradient flows on compact manifolds. The theory is used to prove a Morse-type existence statement for periodic solutions of a time-dependent (periodic in time) and asymptotically linear Hamiltonian equation.     

Gutzwiller’s semiclassical trace formula and Maslov-type index theory for symplectic paths

Gutzwiller’s famous semiclassical trace formula plays an important role in theoretical and experimental quantum mechanics with tremendous success. We review the physical derivation of this deep periodic orbit theory in terms of the phase space formulation with a view toward the Hamiltonian dynamical systems. The Maslov phase appearing in the trace formula is clarified by Meinrenken as Conley–Zehnder index for periodic orbits of Hamiltonian systems. We also survey and compare various versions of Maslov indices to establish this fact. A refinement and improvement to Conley–Zehnder’s index theory in which we will recall all essential ingredients is the Maslov-type index theory for symplectic paths developed by Long and his collaborators. It would shed new light on the computations and understandings of the semiclassical trace formula. The insights in Gutzwiller’s work also seems plausible for the studies of Hamiltonian systems.     

离散哈密顿系统多个周期解的存在性

本文利用临界点理论,建立了一类离散哈密顿系统存在多个周期解的一些充分条件.     

Traveling wave solutions for Schrödinger equation with distributed delay

The paper is devoted to study of traveling waves of nonlinear Schrödinger equation with distributed delay by applying geometric singular perturbation theory, differential manifold theory and the regular perturbation analysis for a Hamiltonian system. Under the assumptions that the distributed delay kernel is strong general delay kernel and the average delay is small, we first investigate the existence of solitary wave solutions by differential manifold theory. Then by utilizing the regular perturbation analysis for a Hamiltonian system, we explore the periodic traveling wave solutions.     

Relative Morse index and multiple brake orbits of asymptotically linear Hamiltonian systems in the presence of symmetries

In this paper, we define a relative Morse index for two continuous symmetric matrices paths in R2n satisfying condition (B1) and study its relation with the Maslov-type indices under brake orbit boundary value of these two symmetric matrices paths. As applications, using this relation we obtain a multiple existence of periodic brake orbit solutions of asymptotically linear Hamiltonian system in the presence of symmetries.     

二阶离散Hamiltonian系统的多重周期解

利用变分原理和Clark定理,研究了带参数的二阶离散Hamiltonian系统的多重周期解,得到了此类方程周期解个数的下界估计.     

Maslov index for homoclinic orbits of Hamiltonian systems

A useful tool for studying nonlinear differential equations is index theory. For symplectic paths on bounded intervals, the index theory has been completely established, which revealed tremendous applications in the study of periodic orbits of Hamiltonian systems. Nevertheless, analogous questions concerning homoclinic orbits are still left open. In this paper we use a geometric approach to set up Maslov index for homoclinic orbits of Hamiltonian systems. On the other hand, a relative Morse index for homoclinic orbits will be derived through Fredholm index theory. It will be shown that these two indices coincide.     

一类带阻尼项的二阶Hamilton系统的多重周期解

利用临界点理论研究带阻尼项的二阶Hamilton系统周期解的存在性.在具有部分周期位势和线性增长非线性项时,根据广义鞍点定理定理,得到了系统多重周期解存在的充分条件.     

Multiple periodic solutions for Hamiltonian systems with not coercive potential

Under an appropriate oscillating behavior of the nonlinear term, the existence of infinitely many periodic solutions for a class of second order Hamiltonian systems is established. Moreover, the existence of two non-trivial periodic solutions for Hamiltonian systems with not coercive potential is obtained, and the existence of three periodic solutions for Hamiltonian systems with coercive potential is pointed out. The approach is based on critical point theorems.     

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 for some nonautonomous p(t)-Laplacian Hamiltonian systems

In this paper, we deal with the existence of periodic solutions of the p(t)-Laplacian Hamiltonian system . Some new existence theorems are obtained by using the least action principle and minimax methods in critical point theory, and our results generalize and improve some existence theorems.     

EXISTENCE OF MULTIPLE POSITIVE PERIODIC SOLUTIONS TO A CLASS OF INTEGRO-DIFFERENTIAL EQUATION

In this paper,by the Avery-Henderson fixed point theorem,we investigate the existence of multiple positive periodic solutions to a class of integro-differential equation. Some suficient conditions are obtained for the existence of multiple positive periodic solutions.     

The iteration formula of the Maslov-type index theory with applications to nonlinear Hamiltonian systems

In this paper, the iteration formula of the Maslov-type index theory for linear Hamiltonian systems with continuous, periodic, and symmetric coefficients is established. This formula yields a new method to determine the minimality of the period for solutions of nonlinear autonomous Hamiltonian systems via their Maslov-type indices. Applications of this formula give new results on the existence of periodic solutions with prescribed minimal period for such systems, and unify known results under various convexity conditions.

     

Small-divisor equation of higher order with large variable coefficient and application to the coupled KdV equation

In this paper, we establish an estimate for the solutions of small-divisor equation of higher order with large variable coefficient. Then by formulating an infinite-dimensional KAM theorem which allows for multiple normal frequencies and unbounded perturbations, we prove that there are many periodic solutions for the coupled KdV equation subject to small Hamiltonian perturbations.