On bi-Lyapunov stable homoclinic classes

For a C ¹ generic diffeomorphism if a bi-Lyapunov stable homoclinic class is homogeneous then it does not have weak eigenvalues. Using this, we show that such homoclinic classes are hyperbolic if it has one of the following properties: shadowing, specification or limit shadowing.     

一类二阶微分方程同宿解的存在性

讨论下列二阶微分方程(y|¨)+ay+U_y(t,y)=0.的同宿解的存在性,其中t∈R,y∈Rⁿ,n∈N,a>0是一个常数,U(t,y)∈Cn,n∈N,a>0是一个常数,U(t,y)∈C¹(R×R1(R×Rⁿ,R),U_y(t,y)表示U(t,y)关于y的梯度.引入快同宿解的概念并给出方程存在快同宿解的判定准则.     

Generic diffeomorphisms with measure-expansive homoclinic classes

We prove that for C¹ generic diffeomorphisms, every measure-expansive locally maximal homoclinic class is hyperbolic.     

SHADOWING,EXPANSIVENESS AND SPECIFICATION FOR C~1-CONSERVATIVE SYSTEMS

We prove that a C~1-generic volume-preserving dynamical system(diffeomorphism or flow) has the shadowing property or is expansive or has the weak specification property if and only if it is Anosov.Finally,as in[10,27],we prove that the C~1-robustness,within the volume-preserving context,of the expansiveness property and the weak specification property,imply that the dynamical system(diffeomorphism or flow) is Anosov.     

Global Bifurcations in Generic One-parameter Families on $$\mathbb{S}^2$$

In this paper we prove that generic one-parameter families of vector fields on \(\mathbb{S}^2\) in the neighborhood of the fields of classes AH, SN, HC, SC (Andronov–Hopf, saddle-node, homoclinic curve, saddle connection) are structurally stable. We provide a classification of bifurcations in these families.     

Forced symmetry breaking of homoclinic cycles in a PDE with O(2) symmetry

We perform a numerical study of solutions near homoclinic orbits for forced symmetry breaking of a PDE with O(2) symmetry to one with SO(2) symmetry. Taking particular care of the consequences of the continuous group action, we concentrate on the Kuramoto-Sivashinsky equation with spatially periodic boundary conditions. The breakup of structurally stable homoclinic cycles is investigated via the introduction of flux term that breaks the reflectional symmetry while retaining the translational symmetry. In particular, we note that although Chossat (1993) has proved that generic perturbations cause the appearance of quasiperiodic orbits, for the simplest possible flux terms this is not the case. We compare these results with numerical simulations of a Galerkin approximation of the equations.     

Nonuniform hyperbolicity for C1-generic diffeomorphisms

We study the ergodic theory of non-conservative C ¹-generic diffeomorphisms. First, we show that homoclinic classes of arbitrary diffeomorphisms exhibit ergodic measures whose supports coincide with the homoclinic class. Second, we show that generic (for the weak topology) ergodic measures of C ¹-generic diffeomorphisms are non-uniformly hyperbolic: they exhibit no zero Lyapunov exponents. Third, we extend a theorem by Sigmund on hyperbolic basic sets: every isolated transitive set Λ of any C ¹-generic diffeomorphism f exhibits many ergodic hyperbolic measures whose supports coincide with the whole set Λ.     

Attracting basins in ℙ2

We prove the existence of infinitely many attracting basins for some holomorphic mappings in ℙ². We also show that if a family of mappings has a complex generic homoclinic tangency, then some of the mappings in the family have an attractive periodic fixed point.     

Abundance of generic homoclinic tangencies in real-analytic families of diffeomorphisms

We consider one-parameter families of two-dimensional diffeomorphisms with homoclinic tangencies. Various authors considered the dynamical complexities due to such tangencies satisfying certain nondegeneracy conditions. In this paper we provide methods to actually verify, for real analytic families, that there are homoclinic tangencies which satisfy these (generic) nondegeneracy generic conditions.     

某类常微系统的一个基本性质

<正> 简介 考虑一n维紧致的C~∞型Riemann流形M~n(n≥2)上,由所有的C~1型常微系统就C~1度量作成的空间.这里为简便,一系统∈将暂称为A_o-系统,如果它只具有有限多个奇点和至多可数多个周期轨道;将暂称为A_1-系统,如果它只具有有限多个奇点和有限多个周期轨道.依据周知的结果,一般绝不是中每一系统都有任意小的邻     

Analytic and algebraic conditions for bifurcations of homoclinic orbits I: Saddle equilibria

We study bifurcations of homoclinic orbits to hyperbolic saddle equilibria in a class of four-dimensional systems which may be Hamiltonian or not. Only one parameter is enough to treat these types of bifurcations in Hamiltonian systems but two parameters are needed in general systems. We apply a version of Melnikov?s method due to Gruendler to obtain saddle-node and pitchfork types of bifurcation results for homoclinic orbits. Furthermore we prove that if these bifurcations occur, then the variational equations around the homoclinic orbits are integrable in the meaning of differential Galois theory under the assumption that the homoclinic orbits lie on analytic invariant manifolds. We illustrate our theories with an example which arises as stationary states of coupled real Ginzburg–Landau partial differential equations, and demonstrate the theoretical results by numerical ones.     

On maximal transitive sets of generic diffeomorphisms

Abstract. – We construct locally generic C¹-diffeomorphisms of 3-manifolds with maximal transitive Cantor sets without periodic points. The locally generic diffeomorphisms constructed also exhibit strongly pathological features generalizing the Newhouse phenomenon (coexistence of infinitely many sinks or sources). Two of these features are: coexistence of infinitely many nontrivial (hyperbolic and nonhyperbolic) attractors and repellors, and coexistence of infinitely many nontrivial (nonhyperbolic) homoclinic classes.?We prove that these phenomena are associated to the existence of a homoclinic class H(P,f) with two specific properties:?– in a C¹-robust way, the homoclinic class H(P,f) does not admit any dominated splitting,?– there is a periodic point P^′ homoclinically related to P such that the Jacobians of P^′ and P are greater than and less than one, respectively. Manuscrit reĉu le 13 décembre 2000. RID="*" ID="*"This paper was partially supported by CNPq, Faperj, and Pronex Dynamical Systems (Brazil), PICS-CNRS and the Agreement Brazil-France in Mathematics. The authors acknowledge to IMPA and Laboratoire de Topologie, Université de Bourgogne, for the warm hospitality during their visits while preparing this paper. We also acknowledge M.-C. Arnaud, F. Béguin and the referees for their comments on the first version of this paper.     

On fundamental groups related to the Hirzebruch surface F_1

Given a projective surface and a generic projection to the plane,the braid monodromy factorization(and thus,the braid monodromy type)of the complement of its branch curve is one of the most important topological invariants,stable on deformations.From this factorization,one can compute the fundamental group of the complement of the branch curve,either in C~2 or in CP~2.In this article,we show that these groups,for the Hirzebruch surface F_1,(a,b),are almost-solvable.That is, they are an extension of a solvable group,which strengthen the conjecture on degeneratable surfaces.     

Lyapunov-Schmidt reduction for unfolding heteroclinic networks of equilibria and periodic orbits with tangencies

This article concerns arbitrary finite heteroclinic networks in any phase space dimension whose vertices can be a random mixture of equilibria and periodic orbits. In addition, tangencies in the intersection of un/stable manifolds are allowed. The main result is a reduction to algebraic equations of the problem to find all solutions that are close to the heteroclinic network for all time, and their parameter values. A leading order expansion is given in terms of the time spent near vertices and, if applicable, the location on the non-trivial tangent directions. The only difference between a periodic orbit and an equilibrium is that the time parameter is discrete for a periodic orbit. The essential assumptions are hyperbolicity of the vertices and transversality of parameters. Using the result, conjugacy to shift dynamics for a generic homoclinic orbit to a periodic orbit is proven. Finally, equilibrium-to-periodic orbit heteroclinic cycles of various types are considered.     

Smooth diffeomorphisms of the plane with stable periodic points in a neighborhood of a homoclinic point

We consider self-diffeomorphisms of the plane of the class C ^r (1 ?? r < ??) with a fixed hyperbolic point and a nontransversal point homoclinic to it. We present a method for constructing a set of diffeomorphisms for which the neighborhood of a homoclinic point contains countably many stable periodic points with characteristic exponents bounded away from zero.     

Stability and uniqueness of periodic orbits produced during homoclinic bifurcation

Under a generic assumption, the existence and the uniqueness of the periodic orbit generating from a homoclinic bifurcation are shown, and the dimensions of its stable and unstable manifolds are given. In the case of a 3-dimensional system, our result revises the stability criterion given in [4,5].Supported by the National Natural Science Foundation of China.     

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.     

Codimension one generic homoclinic classes with interior

We study C ¹-generic diffeomorphisms with a homoclinic class with non empty interior and in particular those admitting a codimension one dominated splitting. We prove that if in the finest dominated splitting the extreme subbundles are one dimensional then the diffeomorphism is partially hyperbolic and from this we deduce that the diffeomorphism is transitive.     

Resonance bifurcation from homoclinic cycles

Differential equations that are equivariant under the action of a finite group can possess robust homoclinic cycles that can moreover be asymptotically stable. For differential equations in R⁴ there exists a classification of different robust homoclinic cycles for which moreover eigenvalue conditions for asymptotic stability are known. We study resonance bifurcations that destroy the asymptotic stability of robust ‘simple homoclinic cycles’ in four-dimensional differential equations. We establish that typically a periodic trajectory near the cycle is created, asymptotically stable in the supercritical case.