非共振倾斜翻转同宿分支

利用同宿轨附近建立的活动坐标架研究四维向量空间中的同宿轨分支. 此类同宿轨是通有的, 但它的稳定流形和不稳定流形为倾斜翻转. 给出了1-周期轨的存在条件与个数、区域, 且获得了2重1-周期轨和3重1-周期轨的分支曲面. 指出从此类同宿轨分支出的1-周期轨的个数依赖于倾斜翻转的强度.     

一类余维3的鞍-焦点异宿环分支

对余维3系统Xμ(x)具有包含一个双曲鞍-焦点O1和一个非双曲鞍-焦点O2的异宿环￡进行了研究.证明了在￡的邻域内有可数无穷条周期轨线和异宿轨线,当非粗糙异宿轨线ΓO破裂时Xμ(x)会产生同宿轨分支,并给出了相应的分支曲线和两种同宿环共存的参数值.在3参数扰动下ΓO破裂和O2点产生Hopf分支的情况下,在￡的邻域内有一条含O1点同宿环,可数无数多条的轨线同宿于O2点分支出的闭轨HO,一条或无穷多条(可数或连续统的)异宿轨线等.     

一类余维3的鞍-焦点异宿环分支

对余维3系统X_μ(x)具有包含一个双曲鞍-焦点O_1和一个非双曲鞍-焦点O_2的异宿环f进行了研究.证明了在f的邻域内有可数无穷条周期轨线和异宿轨线,当非粗糙异宿轨线Γ~0破裂时X_μ(x)会产生同宿轨分支,并给出了相应的分支曲线和两种同宿环共存的参数值.在3参数扰动下Γ~0破裂和O_2点产生Hopf分支的情况下,在f的邻域内有一条含O_1点同宿环,可数无效多条的轨线同宿于O_2点分支出的闭轨H_0,一条或无穷多条(可数或连续统的)异宿轨线等.     

非线性演化方程的显式行波解

本文系统归纳了重要的非线性演化方程的各类显式行波解.说明常微分方程中的同宿轨道和异宿轨道分别和非线性演化方程中的孤立波(或称脉冲波)和波前相对应.耗散系统也存在孤立波,二维(x,y)演化方程中的孤立波可以显示出模式(Pattern)结构.三维相空间的鞍点同宿轨道和鞍—焦点同宿轨道(称Silnikov同宿轨道)常和混沌相联系,     

具有共振特征值的轨道翻转双同宿环分支

该文研究了具有轨道翻转的双同宿环四维系统,在主特征值共振和沿轨道奇点处切方向共振下的两种分支.我们分别在系统奇点小邻域内利用规范型的解构造一个奇异映射,再在双同宿环的管状邻域内引起局部活动坐标架,利用系统线性变分方程的解定义了一个正则映射,通过复合两个映射而得到分支研究中一类重要的Poincaré映射,经过简单的计算最终得到后继函数的精确表达式.对分支方程细致地研究,我们给出了原双同宿环的保存性条件,并证明了"大"1-同宿环分支曲面,2-重"大"1-周期轨分支曲面,"大"2-同宿环分支曲面的存在性、存在区域和近似表达式,及其分支出的"大"周期轨和"大"同宿轨的存在性区域和数量.     

非扭曲退化同宿分支

本文考虑高维系统的退化同宿分支．未扰系统在平衡点ｚ＝０处Ｄｆ（０）有二重实特征根λ１和－λ２,使得 Ｄｆ（０）的其余特征根λ满足 Ｒｅλ＞λ３＞λ１＞０或者 Ｒｅλ＜－λ４＜－λ２＜０,其中λ３和λ４为某正数．利用指数二分性,在同宿轨Г的某邻域内建立适当的局部坐标系和Ｐｏｉｎｃａｒｅ映射．在非共振条件下研究了Г附近的１－同宿和１－周期轨的存在性,唯一性和不共存性．对共振同宿轨描述了更为复杂的分支．     

可反映射的性质及其应用

对可反映射的性质进行了研究 ,利用其性质对混沌系统 Hénon映射的同宿轨进行了讨论     

非扭曲退化同宿分支

本文考虑高维系统的退化同宿分支.未扰系统在平衡点z=0处Df(0)有二重实特征根λ1和-λ2,使得Df(0)的其余特征根λ满足Reλ>λ3>λl>0或者Reλ<-λ4<-λ2<0,其中λ3和λ4为某正数.利用指数二分性,在同宿轨r的某邻域内建立适当的局部坐标系和Poincaré映射.在非共振条件下研究了r附近的1-同宿和1-周期轨的存在性,唯一性和不共存性.对共振同宿轨描述了更为复杂的分支.     

一类3维反转系统中的异维环分支

研究了一类3维反转系统中包含2个鞍点的对称异维环分支问题, 且仅限于研究系统的线性对合R的不变集维数为1的情形. 给出了R-对称异宿环与R-对称周期轨线存在和共存的条件, 同时也得到了R-对称的重周期轨线存在性. 其 次, 给出了异宿环、 同宿轨线、 重同宿轨线和单参数族周期轨线的存在性、 唯一性和共存性等结论, 并且发现不可数无穷条周期轨线聚集在某一同宿轨线的小邻域内. 最后给出了相应的分支图.     

连接1阶细鞍焦点的反转同宿环附近的动态及分支问题

在反转条件下研究了连接1阶细鞍焦点的对称同宿轨线附近包括具有各种不同缠绕数的周期轨线和同宿轨线在内的极其复杂的轨线结构,并且讨论了伴随Hopf分支的同宿轨线分支情况.     

Chaos in PDEs and Lax Pairs of Euler Equations

Recently, the author and collaborators have developed a systematic program for proving the existence of homoclinic orbits in partial differential equations. Two typical forms of homoclinic orbits thus obtained are: (1) transversal homoclinic orbits, (2) Silnikov homoclinic orbits. Around the transversal homoclinic orbits in infinite-dimensional autonomous systems, the author was able to prove the existence of chaos through a shadowing lemma. Around the Silnikov homoclinic orbits, the author was able to prove the existence of chaos through a horseshoe construction.Very recently, there has been a breakthrough by the author in finding Lax pairs for Euler equations of incompressible inviscid fluids. Further results have been obtained by the author and collaborators.     

Chaos in a new fractional-order system without equilibrium points

Chaotic systems without equilibrium points represent an almost unexplored field of research, since they can have neither homoclinic nor heteroclinic orbits and the Shilnikov method cannot be used to demonstrate the presence of chaos. In this paper a new fractional-order chaotic system with no equilibrium points is presented. The proposed system can be considered “elegant” in the sense given by Sprott, since the corresponding system equations contain very few terms and the system parameters have a minimum of digits. When the system order is as low as 2.94, the dynamic behavior is analyzed using the predictor–corrector algorithm and the presence of chaos in the absence of equilibria is validated by applying three different methods. Finally, an example of observer-based synchronization applied to the proposed chaotic fractional-order system is illustrated.     

Homoclinic orbits and chaos in discretized perturbed NLS systems: Part II. Symbolic dynamics

Summary In Part I ([9], this journal), Li and McLaughlin proved the existence of homoclinic orbits in certain discrete NLS systems. In this paper, we will construct Smale horseshoes based on the existence of homoclinic orbits in these systems. First, we will construct Smale horseshoes for a general high dimensional dynamical system. As a result, a certain compact, invariant Cantor set Λ is constructed. The Poincaré map on Λ induced by the flow is shown to be topologically conjugate to the shift automorphism on two symbols, 0 and 1. This gives rise to deterministicchaos. We apply the general theory to the discrete NLS systems as concrete examples. Of particular interest is the fact that the discrete NLS systems possess a symmetric pair of homoclinic orbits. The Smale horseshoes and chaos created by the pair of homoclinic orbits are also studied using the general theory. As a consequence we can interpret certain numerical experiments on the discrete NLS systems as “chaotic center-wing jumping.”     

Bifurcation of Degenerate Homoclinic Orbits to Saddle-Center in Reversible Systems

The authors study the bifurcation of homoclinic orbits from a degenerate homoclinic orbit in reversible system. The unperturbed system is assumed to have saddlecenter type equilibrium whose stable and unstable manifolds intersect in two-dimensional manifolds. A perturbation technique for the detection of symmetric and nonsymmetric homoclinic orbits near the primary homoclinic orbits is developed. Some known results are extended.     

On Hamiltonian Systems with a Homoclinic Orbit to a Saddle-Center

We consider a real analytic Hamiltonian system with two degrees of freedom having a homoclinic orbit to a saddle-center equilibrium (two nonzero real and two nonzero imaginary eigenvalues). We take a two-parameter unfolding for such a system and show that in the nonresonance case, there are countable sets of multi-round homoclinic orbits to a saddle-center. We also find families of periodic orbits accumulating at homoclinic orbits. Bibliography: 6 titles.__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 300, 2003, pp. 187–193.     

Bifurcation of Degenerate Homoclinic Orbits toSaddle-Center in Reversible Systems

The authors study the bifurcation of homoclinic orbits from a degenerate homoclinic orbit in reversible system. The unperturbed system is assumed to have saddle-center type equilibrium whose stable and unstable manifolds intersect in two-dimensional manifolds. A perturbation technique for the detection of symmetric and nonsymmetric homoctinic orbits near the primary homoclinic orbits is developed. Some known results are extended.     

Bifurcation of Homoclinic Orbits with Saddle-Center Equilibrium

In this paper, the authors develop new global perturbation techniques for detecting the persistence of transversal homoclinic orbits in a more general nondegenerated system with action-angle variable. The unperturbed system is assumed to have saddle-center type equilibrium whose stable and unstable manifolds intersect in one dimensional manifold, and does not have to be completely integrable or near-integrable. By constructing local coordinate systems near the unperturbed homoclinic orbit, the conditions of existence of transversal homoclinic orbit are obtained, and the existence of periodic orbits bifurcated from homoclinic orbit is also considered.     

Homoclinic orbits for Hamiltonian systems induced by impulses

In this paper, we consider a class of impulsive Hamiltonian systems with a p‐Laplacian operator. Under certain conditions, we establish the existence of homoclinic orbits by means of the mountain pass theorem and an approximation technique. In some special cases, the homoclinic orbits are induced by the impulses in the sense that the associated non‐impulsive systems admit no non‐trivial homoclinic orbits. Copyright © 2015 John Wiley & Sons, Ltd.     

同宿流形中的奇异轨道

研究较一般的高维退化系统的同宿、异宿轨道分支问题．利用推广的Melnikov函数、横截性理论及奇摄动理论，对具有鞍—中心型奇点的带有角变量的奇摄动系统，在角变量频率产生共振的情况下，讨论其同宿、异缩轨道的扰动下保存和横截的条件．推广和改进了一些文献的结果。