Homoclinic bifurcation with nonhyperbolic equilibria

The problem of homoclinic bifurcation is studied for a high dimensional system with nonhyperbolic equilibria. By constructing local coordinate systems near the unperturbed homoclinic orbit, Poincaré maps for the new system are established. Then the persistence of the homoclinic orbit and the bifurcation of the periodic orbit for the system accompanied with pitchfork bifurcation are obtained. Some known results are extended.     

伴随超临界分支的通有同宿轨道分支

本文研究具有非双曲奇点的高维系统在小扰动下的同宿轨道分支问题,通过在未扰同宿轨道邻域建立局部坐标系,导出系统在新坐标系下的Poincare映射,对伴随超临界分支的通有同宿轨道的保存及分支出周期轨道的情况进行了讨论,推广和改进了一些文献的结果.     

快变量空间中的同宿轨道分支

本文考虑奇摄动问题的位于快变量空间中的奇异同宿轨道的保存和周期轨道分支问题.文中关于奇异同宿轨道保存的结论推广了一些已知的结果,而周期轨道产生于奇异同宿轨道的分支则提供了一种新的分支类型.     

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.     

Transversal homoclinic orbits and chaos for partial functional differential equations

The persistence of degenerate homoclinic orbit is considered for parabolic functional differential equations with small periodic perturbations. Bifurcation functions constructed between two finite-dimensional spaces are obtained. The zeros of the function correspond to the existence of the homoclinic orbit for the perturbed systems. Some applicable conditions are given to ensure that the functions are solvable. Moreover, We show that the homoclinic solution for the perturbed system is transversal and hence the perturbed system exhibits chaos.     

Transversal homoclinic orbits and chaos for partial functional differential equations

The persistence of degenerate homoclinic orbit is considered for parabolic functional differential equations with small periodic perturbations. Bifurcation functions constructed between two finite-dimensional spaces are obtained. The zeros of the function correspond to the existence of the homoclinic orbit for the perturbed systems. Some applicable conditions are given to ensure that the functions are solvable. Moreover, We show that the homoclinic solution for the perturbed system is transversal and hence the perturbed system exhibits chaos.     

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.     

快变振荡下的同宿轨道分支

该文研究带有角变量的快变振荡系统的同宿轨道分支问题，通过建立完整的Poincaré映射，讨论其同宿轨道在小扰动下的保存性, 并进一步研究其分支出周期轨道的情况，推广和改进了一些文献的结果.     

Analysis of a Shil’nikov Type Homoclinic Bifurcation

The bifurcation associated with a homoclinic orbit to saddle-focus including a pair of pure imaginary eigenvalues is investigated by using related homoclinic bifurcation theory. It is proved that, in a neighborhood of the homoclinic bifurcation value, there are countably infinite saddle-node bifurcation values, period-doubling bifurcation values and double-pulse homoclinic bifurcation values. Also, accompanied by the Hopf bifurcation, the existence of certain homoclinic connections to the periodic orbit is proved.     

On the Existence of Pulses in Reaction- Diffusion- Equations

We apply the theory of invariant manifolds for singularly perturbed ordinary differential equations and results about the persistence of homoclinic orbits in autonomous differential systems with several parameters in order to establish the existence of pulses in reaction-diffusion systems. Essential assumptions for the existence of pulses are the following: (i) Existence of a homoclinic orbit to a hyperbolic equilibrium in the corresponding reaction system. (ii) The quotient of some measure for the diffusivities and the square of the puls speed is sufficiently small. (iii) Validity of some transversality condition. The last assumption requires the occurence of parameters in the reaction term.     

A global condition for quasi-random behavior in a class of conservative systems

Devaney has shown that an autonomous Hamiltonian system in dimension 4, with an orbit homoclinic to a saddle-focus equilibrium, admits a chaotic behavior as soon as the homoclinic orbit is the transverse intersection of the stable and unstable manifolds. In this paper we deal with two classes of saddle-focus systems: Lagrangian systems defined on a two-manifold in the presence of a gyroscopic force, and fourth-order systems arising in water-wave theory. We first establish, by a standard variational method, the existence of a homoclinic orbit. Then, under a weak nondegeneracy condition, we show that it gives rise to an infinite family of multibump homoclinic solutions and that the dynamics are chaotic. Our condition is much easier to check than transversality. For example, it is automatically satisfied for gyroscopic systems on a two-torus, for topological reasons. © 1996 John Wiley & Sons, Inc.     

四维空间中的异维环分支问题

利用局部活动坐标架法,讨论了四维空间中连接两个鞍点的异维环分支问题,在一些通有的假设下,分别得到了异维环保存、同宿环、周期轨存在的充分条件以及保存的异维环与分支出的周期轨共存(或不共存)的结果.     

Homoclinic orbits for a perturbed quintic-cubic nonlinear Schrödinger equation

The existence of homoclinic orbits for a perturbed cubic-quintic nonlinear Schrödinger equation with even periodic boundary conditions under the generalized parameters conditions is established. We combined geometric singular perturbation theory, Melnikov analysis, and integrable theory to prove the persistence of homoclinic orbits.     

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.     

The existences of transverse homoclinic solutions and chaos for parabolic equations

By using Lyapunov-Schmidt reduction and exponential dichotomies, the persistence of homoclinic orbit is considered for parabolic equations with small perturbations. Bifurcation functions are obtained, where d is the dimension of the intersection of the stable and unstable manifolds. The zeros of H correspond to the existence of the homoclinic orbit for the perturbed systems. Some applicable conditions are given to ensure that the functions are solvable. Moreover the homoclinic solution for the perturbed system is transversal under the applicable conditions and hence the perturbed system exhibits chaos. The basic tools are shadowing lemma which was obtained by Blazquez (see [C.M. Blazquez, Transverse homoclinic orbits in periodically perturbed parabolic equations, Nonlinear Anal. 10 (1986) 1277-1291]).     

Degenerate Orbit Flip Homoclinic Bifurcations with Higher Dimensions

Bifurcations of a degenerate homoclinic orbit with orbit flip in high dimensional system are studied. By establishing a local coordinate system and a Poincaré map near the homoclinic orbit, the existence and uniqueness of 1–homoclinic orbit and 1–periodic orbit are given. Also considered is the existence of 2–homoclinic orbit and 2–periodic orbit. In additon, the corresponding bifurcation surfaces are given. Project supported by the National Natural Science Foundation of China (No: 10171044), the Natural Science Foundation of Jiangsu Province (No: BK2001024), the Foundation for University Key Teachers of the Ministry of Education of China     

Homoclinic Flip Bifurcations Accompanied by Transcritical Bifurcation

The bifurcations of orbit flip homoclinic loop with nonhyperbolic equilibria are investigated. By constructing local coordinate systems near the unperturbed homoclinic orbit, Poincaré maps for the new system are established. Then the existence of homoclinic orbit and the periodic orbit is studied for the system accompanied with transcritical bifurcation.     

Persistent Homoclinic Orbits for a Perturbed Cubic-quinitic Nonlinear Schrodinger Equation

In this paper, the existence of homoclinic orbits, for a perturbed cubic-quintic nonlinear Schrödinger equation with even periodic boundary conditions, under the generalized parameters conditions is established. More specifically, we combine geometric singular perturbation theory with Melnikov analysis and integrable theory to prove the persistence of homoclinic orbits.     

Homoclinic Orbits for a Class of Fractional Hamiltonian Systems via Variational Methods

This paper is devoted to the existence and multiplicity of homoclinic orbits for a class of fractional-order Hamiltonian systems with left and right Liouville–Weyl fractional derivatives. Here, we present a new approach via variational methods and critical point theory to obtain sufficient conditions under which the Hamiltonian system has at least one homoclinic orbit or multiple homoclinic orbits. Some results are new even for second-order Hamiltonian systems.