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.     

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.     

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.     

时不变分数阶系统反周期解的存在性

反周期解问题是非线性微分系统动力学的重要特征.近年来,非线性整数阶微分系统的反周期解问题得到了广泛的研究,非线性分数阶微分系统的反周期解问题也得到了初步的讨论.不同于已有的工作,该文研究时不变分数阶系统反周期解的存在性问题.证明了时不变分数阶系统在有限时间区间内不存在反周期解,而当分数阶导数的下限趋近于无穷大时,时不变分数阶系统却存在反周期解.     

具有扩散率Ayala模型的概周期解

本文研究了非自治Ayala模型的概周期和周期系统,我们得到在一定条件下,其概周期系统存在唯一全局吸引的概周期解且其概周期解在壳扰动下是稳定的。在与概周期情形类似的条件下我们得到其w-周期系统存在唯一全局吸引的w-周期解。     

扰动系统概周期解和有界解的存在性

该文研究扰动系统．在适当条件下给出扰动系统概周期解和有界解的存在性证明．     

一类多滞量周期扰动非线性系统的周期解

研究一类具有多个滞量的周期扰动非线性系统的T周期解.利用拓扑度的方法得到了系统存在T周期解的充分条件.作为应用,证明了具有滞后的单种群对数模型在一定条件下存在正周期解.     

带小扰动的随机发展方程的不变测度

本文考虑带小扰动的随机发展方程,证明如何建立此方程的耦合解.作为应用,我们证明解的Feller连续性和不变测度的存在唯一性.还进一步建立了当扰动趋于零时,关于这族不变测度的大偏差原理.     

一类四阶离散哈密顿系统周期解的存在性

本文研究了一类四阶离散哈密顿系统周期解的存在性问题.利用临界点理论中的极大极小方法,通过引入两个不同的控制函数,得到了新的可解性条件,并且利用这些条件得到了新的存在性结果.     

一类二阶哈密顿系统的周期解

本文考虑一类二阶哈密顿系统,利用临界点理论中的极小极大理论和局部环绕理论,在一类新的次二次增长条件下,研究得到了非平凡周期解的存在性.证明过程不依赖于经典的Palias-Smale紧性.     

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.     

Invariant tori of locally Hamiltonian systems close to conditionally integrable systems

We study the problem of perturbations of quasiperiodic motions in the class of locally Hamiltonian systems. By using methods of the KAM-theory, we prove a theorem on the existence of invariant tori of locally Hamiltonian systems close to conditionally integrable systems. On the basis of this theorem, we investigate the bifurcation of a Cantor set of invariant tori in the case where a Liouville-integrable system is perturbed by a locally Hamiltonian vector field and, simultaneously, the symplectic structure of the phase space is deformed. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 1, pp. 71–98, January, 2007.     

具拟周期摄动的近可积Hamiltonian系统的KAM型定理

In this paper we consider the persistence of invariant tori of an integrable Hamiltonian system with a quasiperiodic perturbation. It is proved that if the unperturbed system satisfies the Rtissmann non-degenerate condition and the perturbed system satisfies the co-linked non-resonant condition, then the majority of invariant tori is persistent under the perturbation.     

Persistence of Invariant Tori on Submanifolds in Hamiltonian Systems

Summary. Generalizing the degenerate KAM theorem under the Rüssmann nondegeneracy and the isoenergetic KAM theorem, we employ a quasilinear iterative scheme to study the persistence and frequency preservation of invariant tori on a smooth submanifold for a real analytic, nearly integrable Hamiltonian system. Under a nondegenerate condition of Rüssmann type on the submanifold, we shall show the following: (a) the majority of the unperturbed tori on the submanifold will persist; (b) the perturbed toral frequencies can be partially preserved according to the maximal degeneracy of the Hessian of the unperturbed system and be fully preserved if the Hessian is nondegenerate; (c) the Hamiltonian admits normal forms near the perturbed tori of arbitrarily prescribed high order. Under a subisoenergetic nondegenerate condition on an energy surface, we shall show that the majority of unperturbed tori give rise to invariant tori of the perturbed system of the same energy which preserve the ratio of certain components of the respective frequencies.     

A KAM-type Theorem for Generalized Hamiltonian Systems

In this paper we mainly concern the persistence of invariant tori in 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, system under consideration can be odd dimensional. Under the Riissmann type non-degenerate condition, we proved that the majority of the lower-dimension invariant tori of the integrable systems in generalized Hamiltonian system are persistent under small perturbation. The surviving lower-dimensional tori might be elliptic, hyperbolic, or of mixed type.     

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.     

Partial preservation of frequencies and floquet exponents in KAM theory

Under a small perturbation of a completely integrable Hamiltonian system, invariant tori with Diophantine frequencies of motion are not destroyed but only slightly deformed, provided that the Hessian (with respect to the action variables) of the unperturbed Hamiltonian vanishes nowhere (the Kolmogorov nondegeneracy). The motion on every perturbed torus is quasiperiodic with the same frequencies. In this sense the frequencies of invariant tori of the unperturbed system are preserved. Recently, it has been found that the Kolmogorov nondegeneracy condition can be weakened so as to guarantee the preservation of only some subset of frequencies. Such partial preservation of frequencies can also be defined for lower dimensional invariant tori, whose dimension is less than the number of degrees of freedom. We consider a more general problem of partial preservation not only of the frequencies of invariant tori but also of their Floquet exponents (the eigenvalues of the coefficient matrix of the variational equation along the torus). The results are formulated for Hamiltonian, reversible, and dissipative systems (with a complete proof for the reversible case).     

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 Isoenergetic KAM-Type Theorem at Resonant Case for Nearly Integrable Hamiltonian Systems

In this paper, we study the persistence of resonant invariant tori on energy surfaces for nearly integrable Hamiltonian systems under the usual R$\ddot{u}$ssmann nondegenerate condition. By a quasilinear iterative scheme, we prove the following things: (1) The majority of resonant tori on a given energy surface will be persisted under R$\ddot{u}$ssmann nondegenerate condition. (2) The maximal number about the preserved frequency components on a perturbed torus is characterized by the smaller of the maximal rank of the Hessian matrices of the unperturbed system and the nondegeneracy of resonance. (3) If unperturbed systems admit subisoenergetic nondegeneracy on an energy surface, then the majority of the unperturbed resonant tori on the energy surface will be persisted and give rise to a family of perturbed tori with the same energy, whose frequency ratios among respective ''nondegenerate'' components are preserved.     

Investigation of invariant deformations of integral manifolds of adiabatically perturbed completely integrable Hamiltonian systems. II

By using the Cartan differential-geometric theory of integral submanifolds (invariant tori) of completely Liouville—Arnold integrable Hamiltonian systems on the cotangent phase space, we consider an algebraic-analytical method for the investigation of the corresponding mapping of imbedding of an invariant torus into the phase space. This enables one to describe analytically the structure of quasiperiodic solutions of the Hamiltonian system under consideration. We also consider the problem of existence of adiabatic invariants associated with a slowly perturbed Hamiltonian system. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 11, pp. 1513–1528, November, 1999.