指数二分性与奇异摄动系统的横戳异宿轨道

本文研究奇异摄动系统的横截异宿轨道的存在性,利用指数二分性理论和Liapunov－Schmidt方法,获得了判断奇异摄动系统存在横截异宿轨道的Melnikov型函数,因而推广了一些文献的结果．     

自治奇摄动系统的同、异宿轨道

利用指数二分性理论和泛函分析方法来处理第一变分方程在R上有多于一个非平凡有界解下的奇摄动系统的同宿轨道分支问题.利用此方法我们给出了判断奇摄动系统在退化情形下存在同、异宿轨道的Melnikov向量函数并给出了存在同宿轨道的参数估计范围.     

二阶渐近周期奇异哈密顿系统同宿轨道

运用集中紧性方法和Ekeland变分原理研究R^2中二阶渐近周期奇异Hamilton系统ue (1 g(t))V‘u(t,u)=0的极小问题，并证明该系统具有两条非平凡同宿轨道。     

正二阶快-慢系统中的奇性同宿轨道和极限环

本文研究一类正二阶快-慢系统中奇性同宿轨道和极限环,并且给出了此系统存在奇性同宿轨道和极限环的充分条件.     

指数型二分性与拟自治奇异摄动系统

通过对Ｋ．Ｊ．Ｐａｌｍｅｒ￣［４］中方法的改进，本文讨论了摄动系统有界解的存在性，得到了不同于［４］中的结果，并把改进了的方法运用到拟自治奇异摄动系统，得到了拟自治奇异摄动系统存在有界解的一个简洁的充分条件。本文的方法还提供了一个处理摄动项含小参数的方法。     

指数二分性与自治奇摄动系统的退化同宿分支

利用指数二分性理论和泛函分析方法，我们研究了自治奇摄动系统的同，异宿轨道的存在性，给出了高维奇摄动系统从退化系统分支出同异宿轨道的Ｍｅｌ－ｎｉｋｏｖ型函数。     

奇摄动下的共振不变环面分支

本文研究高维退化系统在小扰动下的动力学行为,在共振的情况下,利用延拓的方法,讨论了扰动 系统不变环面的保存性,并利用推广的Melnikov函数、横截性理论讨论了同宿于不变环面的横截同宿 轨道存在的条件,推广和改进了一些文献的结果.     

奇摄动下的共振不变环面分支

本文研究高维退化系统在小扰动下的动力学行为,在共振的情况下,利用延拓的方法,讨论了扰动系统不变环面的保存性,并利用推广的Melnikov函数、横截性理论讨论了同宿于不变环面的横截同宿轨道存在的条件,推广和改进了一些文献的结果.     

不满足Gordon-强力条件的奇异二阶周期Hamilton系统同宿轨道的存在性

考虑奇异的二阶周期 Hamilton 系统这里q=(q_1,q_2,…,q_n)∈R~n,n>2,V(t,q):R×R~n\{e}→R 是一个奇异的位势函数,e≠0.当V(t,q)具有唯一最大值,但不满足 Gordon- 强力条件时,我们证明了(HS)至少具有一条非平凡的同宿轨道。     

奇摄动问题中的异宿流形

应用指数2分性和横截性理论等动力系统方法来处理奇摄动问题中的同宿、异宿轨道的存在性和横截性问题，对具有较高退化程度的所谓奇异同宿轨道和奇异异宿轨道（见定义1.1）在奇摄动下何时变为同宿、异宿轨道给出了用Melnikov向量来刻划的判据和实例．     

Global bifurcations and single‐pulse homoclinic orbits of a plate subjected to the transverse and in‐plane excitations

The Shilnikov‐type single‐pulse homoclinic orbits and chaotic dynamics of a simply supported truss core sandwich plate subjected to the transverse and the in‐plane excitations are investigated in detail. The resonant case considered here is principal parametric resonance and 1:2 internal resonance. Based on the normal form theory, the desired form for the global perturbation method is obtained. By using the global perturbation method developed by Kovacic and Wiggins, explicit sufficient conditions for the existence of a Shilnikov‐type homoclinic orbit are obtained, which implies that chaotic motions may occur for this class of truss core sandwich plate in the sense of Smale horseshoes. Numerical results obtained by using the fourth‐order Runge–Kutta method agree with theoretical analysis at least qualitatively. Copyright © 2017 John Wiley & Sons, Ltd.     

Transversal homoclinic orbits for higher dimensional difference equations

In this paper, we use the functional analytic method (theory of exponential dichotomies and Liapunov-Schmidt method) to study the homoclinic bifurcations of higher dimensional difference equations in a degenerate case. We obtain a Melnikov vector mapping for difference equations with the help of which the existence of transversal homoclinic orbits can be detected.     

A new method to find homoclinic and heteroclinic orbits

A new series method is provided for continuous-time autonomous dynamical systems, which can find exact orbits as opposed to approximate ones. The method can reduce the connecting orbit problem as a boundary value problem in an infinite time domain to the initial value problem. It consists of transforming time to the logarithmic scale, substituting a power series around each fixed point of interest for each of the unknown functions into the system, and equating the corresponding coefficients. When solving for the power series coefficients, additional parameters are used in order to find the intersections of the unstable manifold and the stable manifold of the equilibria. This paper demonstrates how the new method allows to obtain heteroclinic and homoclinic orbits in some well-known cases, such as Nagumo system, stretch-twist-fold flow or mathematical pendulum.     

Computation of bifurcation manifolds of linearly independent homoclinic orbits

Regarding the small perturbation as a parameter in an appropriate space of functions, we can discuss co-existence of homoclinic orbits for non-autonomous perturbations of an autonomous system in Rn and describe conditions of parameters for such degenerate homoclinic bifurcations with some bifurcation manifolds of infinite dimension. Since those manifolds determine the relation among parameters for such bifurcations, in this paper we give an algorithm to compute approximately those manifolds and concretely obtain their first order approximates.     

Hunting for homoclinic orbits in reversible systems: A shooting technique

A dynamical system is said to be reversible if there is an involution of phase space that reverses the direction of the flow. Examples are Hamiltonian systems with quadratic potential energy. In such systems, homoclinic orbits that are invariant under the reversible transformation are typically not destroyed as a parameter is varied. A strategy is proposed for the direct numerical approximation to paths of such homoclinic orbits, exploiting the special properties of reversible systems. This strategy incorporates continuation using a simplification of known methods and a shooting approach, based on Newton's method, to compute starting solutions for continuation. For Hamiltonian systems, the shooting uses symplectic numerical integration. Strategies are discussed for obtaining initial guesses for the unknown parameters in Newton's method. An example system, for which there is an infinity of symmetric homoclinic orbits, is used to test the numerical techniques. It is illustrated how the orbits can be systematically located and followed. Excellent agreement is found between theory and numerics.This paper is presented as an outcome of the LMS Durham Symposium convened by Professor C.T.H. Baker on 4–14 July 1992 with support from the SERC under grant reference number GR/H03964.     

Existence of infinitely many homoclinic orbits to Aubry sets for positive definite Lagrangian systems

In this paper, we study the problem of homoclinic orbits to Aubry sets for time-periodic positive definite Lagrangian systems. We show that there are infinitely many homoclinic orbits to some Aubry set under the conditions that the associated Mather set is uniquely ergodic and the first relative homology group of the projection of this Aubry set is nonzero.     

Codimension 3 nonresonant bifurcations of homoclinic orbits with two inclination flips

Homoclinic bifurcations in four-dimensional vector fields are investigated by setting up a local coordinate near a homoclinic orbit. This homoclinic orbit is principal but its stable and unstable foliations take inclination flip. The existence, nonexistence, and uniqueness of the 1-homoclinic orbit and 1-periodic orbit are studied. The existence of the two-fold 1 -periodic orbit and three-fold 1 -periodic orbit are also obtained. It is indicated that the number of periodic orbits bifurcated from this kind of homoclinic orbits depends heavily on the strength of the inclination flip.