Codimension 3 Non-resonant Bifurcations of Rough Heteroclinic Loops with One Orbit Flip

Heteroclinic bifurcations in four dimensional vector fields are investigated by setting up a local coordinates near a rough heteroclinic loop. This heteroclinic loop has a principal heteroclinic orbit and a non-principal heteroclinic orbit that takes orbit flip. The existence, nonexistence, coexistence and uniqueness of the 1-heteroclinic loop, 1-homoclinic orbit and 1-periodic orbit are studied. The existence of the two-fold or three-fold 1-periodic orbit is also obtained.     

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.     

Orbit configuration spaces of small covers and quasi-toric manifolds

In this article,we investigate the orbit configuration spaces of some equivariant closed manifolds over simple convex polytopes in toric topology,such as small covers,quasi-toric manifolds and(real)moment-angle manifolds;especially for the cases of small covers and quasi-toric manifolds.These kinds of orbit configuration spaces have non-free group actions,and they are all noncompact,but still built via simple convex polytopes.We obtain an explicit formula of the Euler characteristic for orbit configuration spaces of small covers and quasi-toric manifolds in terms of the h-vector of a simple convex polytope.As a by-product of our method,we also obtain a formula of the Euler characteristic for the classical configuration space,which generalizes the Félix-Thomas formula.In addition,we also study the homotopy type of such orbit configuration spaces.In particular,we determine an equivariant strong deformation retraction of the orbit configuration space of 2 distinct orbit-points in a small cover or a quasi-toric manifold,which allows to further study the algebraic topology of such an orbit configuration space by using the Mayer-Vietoris spectral sequence.     

RESONANT BIFURCATIONS OF THE HOMOCLINIC MANIFOLD FOR FOURTH-DIMENSIONAL SYSTEM

The homoclinic bifurcations under resonant conditions are considered in the ho- moclinic manifold consisting of a series of homoclinic orbits for the fourth-dimensional system.The existence,coexistence and uniqueness of 1-homoclinic orbit,1-periodic orbit and 2-fold 1-periodic orbit are obtained under resonant condition,the correspon- ding bifurcation surfaces and existing regions are also given.     

SLANT SUBMANIFOLDS OF A RIEMANNIAN PRODUCT MANIFOLD

In this article, the geometry of the slant submanifolds of a Riemannian product manifold is studied. Some necessary and sufficient conditions on slant, bi-slant and semi-slant submanifolds are given. We research fundamental properties of the distributions which are involved in definitions of semi- and bi-slant submanifolds in a Riemannian product manifold.     

图式流形

The discovery of manifold is a major advance in modern mathematics.In 1974,J.W.Morgan and D.P.Sullivan Surggest the concept of a Z/n manifold.We think that,it is more useful to suggest the concept of a graphlike manifold and research related topics.And these works will be done in this paper.     

CODIMENSION 3 BIFURCATIONS OFHOMOCLINIC ORBITS WITH ORBITFLIPS AND INCLINATION FLIPS

The homoclinic bifurcations in four dimensional vector fields are investigated by setting up a local coordinates near the homoclinic orbit. This homoclinic orbit is non-principal in the meanings that its positive semi-orbit takes orbit flip and its unstable foliation takes inclination flip. The existence, nonexistence, uniqueness and coexistence of the 1-homoclinic orbit and the 1-periodic orbit are studied. The existence of the twofold periodic orbit and three-fold periodic orbit are also obtained.     

A Review of the Index Method in Closed Geodesic Problem

In this paper we review and systematize the index method in closed geodesic problem.As we know,the closed geodesic problem on compact Riemannian or Finsler manifold is a famous problem,and has far from been resolved.In recent years,the Maslov-type index theory for symplectic path has been applied to studying the closed geodesic problem,and has induced a great number of results on the multiplicity and stability of closed geodesics.We will systematically introduce these progresses in this review.     

STRANGE ATTRACTORS IN SYMMETRIC UNFOLDINGS OF A SINGULARITY WITH THREE-FOLD ZERO EIGENVALUE

In this paper,we study the Sil'nikov heteroclinic bifurcations,which display strange attractors,for the symmetric versal unfoldings of the singularity at the origin with a nilpotent linear part and 3-jet,using the normal form,the blow-up and the generalized Mel'nikov methods of heteroclinic orbits to two hyperbolic or nonhyperbolic equilibria in a high-dimensional space.     

THE CRITERION FOR THE STABILITY OF A MULTIPLE LIMIT CYCLE UNDER HIGHER DEGENERATIONS

In this paper, we obtain the exact computation formulae to determine the stability of a multiple limit cycle with the third or fourth order degenerations. We employ the method of computing the expansion of the Poincar′e map around the closed orbit using "normal bundle" coordinates parameterized by time variable in a neighborhood of the closed orbit. An example is given to show the feasibility of our results.