周期扰动下卫星运动系统的动力学性质

本文运用Melnikov方法对平面卫星运动系统在周期扰动下所表现出来的动力学性质进行了探讨.首先运用次谐Melnikov方法给出了卫星轨道在周期扰动下存在次谐周期轨道的条件,并进一步运用同宿.Melnikov方法证实了该系统存在Smale马蹄意义下的混沌性质.     

三维系统周期轨道的分支

本文研究三维系统的一类非双曲周期轨道在小扰动下产生周期轨道的问题，并对一类较特殊的系统给出了判别周期轨道存在的具体条件。此外，还给出了具体的应用例子。     

三维系统余维二分支中周期轨道与不变环面的存在性

本文研究具余维二奇点的三维系统在双参数扰动下周期轨道与不变环面的分支，给出了存在不变环面的最佳条件     

梯形映射族周期轨道的计算

运用符号动力学理论,研究一种特殊的一维分段线性映射族"梯形映射族"周期轨道的计算方法,确定其周期轨道的参数范围,给出了奇的最大周期序列对应参数的精确范围,以及偶的最大周期序列参数的近似范围.该方法可应用于更一般的单峰系统.     

一类二阶非线性保守系统周期轨与同异宿轨的显式表示

给出了一类二阶非线性保守系统周期轨道族与同异宿轨道显式表示的初等积分方法;同时指出:根据周期轨道族外围分界线环类型的不同,周期轨道族需由不同的Jacobian椭圆函数来表示并揭示了其中的原因.利用文中方法,通过变量替换,旋转以及积分因子等手段,可推导获得某些更复杂非线性系统周期轨道族与同异宿轨道的显式式,因此所得结果对于非线性(扰动)系统分支与混沌的研究有帮助.     

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.     

一个非线性微分方程的周期边值问题

本文考虑作为卫星绕椭圆轨道作周期运动模型的一个二阶非线性微分方程的周期边值问题.用迭代方法证明了奇函数周期解的存在性,并且扩大了文[3]中给出的参数范围.     

一类中立型时滞系统周期运动的近似解

对一类中立型时滞系统在临界状态小扰动下的平衡点进行了稳定性分析,同时讨论了该系统在非线性扰动下的周期运动的近似解及稳定性.     

寻找周期轨道的代数分析法

本文给出了计算一般代数型（有理分式）离散动力系统周期轨道的一种分析方法－代数分析法。这种方法的优点是将非线性求解问题转化为一个线性求解问题来处理。它不仅可以准确地确定包括稳定和不稳定周期轨道的位置，而且还可以详细了解周期轨道的产生和随参数演变的分岔特性。本文利用这种方法分析了一个四维二次非线性映射，并给出了其完整的低周期轨道的分岔曲线图。     

R3中一类四次可逆系统的周期轨与极限环问题

本文研究了R3中一类四次可逆系统及其扰动系统的周期轨与极限环问题,利用Poincáre紧化理论讨论了相关平面系统的定性性质,证明了所考虑的系统存在无穷多对称周期轨的结论.然后借助一系列技巧性变换,并利用平均方法证明了R3中这类四次可逆系统的扰动系统在其某周期轨邻域至少存在两个具有不同稳定性的极限环的结论.     

EXISTENCE OF PERIODIC SOLUTIONS TO A PERTURBED FOUR-DIMENSIONAL SYSTEM

Consider a k multiple closed orbit on an invariant surface of a four dimensional system, after a suitable perturbation, the closed orbit can generate periodic orbits and double-period orbits. Using bifurcation methods and techniques, sufficient conditions for the existence of periodic solutions to the perturbed four dimensional system are obtained, and the period-doubling bifurcations is discussed.     

PERIODIC ORBITS IN PERTURBED GENERALIZED HAMILTONIAN SYSTEMS

ＰＥＲＩＯＤＩＣＯＲＢＩＴＳＩＮＰＥＲＴＵＲＢＥＤＧＥＮＥＲＡＬＩＺＥＤＨＡＭＩＬＴＯＮＩＡＮＳＹＳＴＥＭＳＺｈａｏＸｉａｏｈｕａ（赵晓华）Ｄｅｐｔ．ｏｆＭａｔｈ．ｏｆＹｕｎｎａｎＵｎｌｙ．＆Ｉｕｓｔ．ｏｆＡｐｐｌ．Ｍａｔｈ．ｏｆＹｕｎｎａｎＰｒｏｖ...     

一类共振条件下两个自由度系统的同宿轨道

研究一类在参数和强迫激励下发生共振时的两个自由度系统,利用多重尺度法证明存在锁频于Ω的周期解.在一定条件下可变换成Wiggins的系统,给出了判断这类系统同宿轨道存在的计算公式.     

On the existence of periodic orbits for the fixed homogeneous circle problem

We prove the existence of some types of periodic orbits for a particle moving in Euclidean three-space under the influence of the gravitational force induced by a fixed homogeneous circle. These types include periodic orbits very far and very near the homogeneous circle, as well as eight and spiral periodic orbits.     

Dynamical behaviour and exact solutions of thirteenth order derivative nonlinear Schr\"{o}dinger equation

In this paper, we considered the model of the thirteenth order derivatives of nonlinear Schr\"{o}dinger equations. It is shown that a wave packet ansatz inserted into these equations leads to an integrable Hamiltonian dynamical sub-system. By using bifurcation theory of planar dynamical systems, in different parametric regions, we determined the phase portraits. In each of these parametric regions we obtain possible exact explicit parametric representation of the traveling wave solutions corresponding to homoclinic, hetroclinic and periodic orbits.     

碰振系统中的共存周期轨道

提出一种寻找分段线性碰振系统中的多个周期轨道共存的分析方法,这些单碰周期轨道包含稳定的和不稳定的轨道。给出了单碰周期轨道存在性或不存在性的解析判别式,特别是对如何保证在单碰周期运动中不会发生其它的碰撞的问题作了比较深入的研究,得到若干定理。最后讨论了所得共存周期轨道的稳定性问题,获得了稳定性的判别式。还以数值模拟结果验证了理论分析的结论。     

Numerical Continuation of Hamiltonian Relative Periodic Orbits

The bifurcation theory and numerics of periodic orbits of general dynamical systems is well developed, and in recent years, there has been rapid progress in the development of a bifurcation theory for dynamical systems with structure, such as symmetry or symplecticity. But as yet, there are few results on the numerical computation of those bifurcations. The methods we present in this paper are a first step toward a systematic numerical analysis of generic bifurcations of Hamiltonian symmetric periodic orbits and relative periodic orbits (RPOs). First, we show how to numerically exploit spatio-temporal symmetries of Hamiltonian periodic orbits. Then we describe a general method for the numerical computation of RPOs persisting from periodic orbits in a symmetry breaking bifurcation. Finally, we present an algorithm for the numerical continuation of non-degenerate Hamiltonian relative periodic orbits with regular drift-momentum pair. Our path following algorithm is based on a multiple shooting algorithm for the numerical computation of periodic orbits via an adaptive Poincaré section and a tangential continuation method with implicit reparametrization. We apply our methods to continue the famous figure eight choreography of the three-body system. We find a relative period doubling bifurcation of the planar rotating eight family and compute the rotating choreographies bifurcating from it.      

First kind symmetric periodic solutions and their stability for the Kepler problem and anisotropic Kepler problem plus generalized anisotropic perturbation

Motivated by some problems in Celestial Mechanics that combines quasihomogeneous potential in the anisotropic space, we investigate the existence of several families of first kind symmetric periodic solutions for a family of planar perturbed Kepler problem. In addition, we give sufficient conditions for the existence of first kind periodic solutions and also we characterize its type of stability. As an application of this general situation, we discuss the existence of symmetric periodic solutions for the anisotropic Kepler problem plus a generalized anisotropic perturbation, (shortly, p-AKPQ problem) and for the Kepler problem plus a generalized anisotropic perturbation (shortly, p-KPQ problem), as continuation of circular orbits of the two-dimensional Kepler problem. To get this objective, we consider different types of perturbations and then we apply our main result.     

EXISTENCE AND STABILITY OF PERIODIC ORBITS FOR A PERTURBED FOUR-DIMENSIONAL SYSTEM

The purpose of this paper is to study periodic orbits of a perturbed four- dimensional system.Using bifurcation methods and the integral manifold theory,sufficient conditions for the existence and stability of periodic orbits of the perturbed four-dimensional system are obtained.     

Replicate periodic windows in the parameter space of driven oscillators

In the bi-dimensional parameter space of driven oscillators, shrimp-shaped periodic windows are immersed in chaotic regions. For two of these oscillators, namely, Duffing and Josephson junction, we show that a weak harmonic perturbation replicates these periodic windows giving rise to parameter regions correspondent to periodic orbits. The new windows are composed of parameters whose periodic orbits have the same periodicity and pattern of stable and unstable periodic orbits already existent for the unperturbed oscillator. Moreover, these unstable periodic orbits are embedded in chaotic attractors in phase space regions where the new stable orbits are identified. Thus, the observed periodic window replication is an effective oscillator control process, once chaotic orbits are replaced by regular ones.