Shaobo Gan, Dawei Yang
We prove that for every three-dimensional vector field, either it can be accumulated by Morse-Smale ones, or it can be accumulated by ones with a transverse homoclinic intersection of some hyperbolic periodic orbit in the $C^1$ topology.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Bifurcations of dynamical systems, described by a second order differential equations and by an impact condition are studied. It is shown that the variation of parameters when the number of impacts of a periodic solution increases, leads to the occurrence of a hyperbolic chaotic invariant set.
The paper contains a rigorous proof of existence of symbolic dynamics chaos in the generalized Hénon map's 4th iterate $H^4$, which was conjectured in the paper \textit{A 3D Smale Horseshoe in a Hyperchaotic Discrete-Time System} of Li and Yang, 2007. We prove also the uniform hyperbolicity of the invariant set with symbolic dynamics. The proofs are computer-assisted with the use of C++ library \textit{CAPD} for interval arithmetic, differentiation and integration.
An important problem in the dynamics of surface homeomorphisms is determining the forcing relation between orbits. The forcing relation between periodic orbits can be computed using standard algorithms, though this does not give much information on the structure of the forcing relation. Here we consider forcing relations between homoclinic orbits, and their relationships with periodic orbits. We outline a general procedure for computing the forcing relation, and apply this to compute the equivalence and forcing relations for homoclinic orbits of the Smale horseshoe map. We also outline a metho…
It is known that $C^r$ Morse-Smale vector fields form an open dense subset in the space of vector fields on orientable closed surfaces and are structurally stable for any $r \in \mathbb{Z}_{>0}$. In particular, $C^r$ Morse vector fields (i.e. Morse-Smale vector fields without limit cycles) form an open dense subset in the space of $C^r$ gradient vector fields on orientable closed surfaces and are structurally stable. Therefore generic time evaluations of gradient flows on orientable closed surfaces (e.g. solutions of differential equations) are described by alternating sequences of Morse flows…
In the present paper we consider preserving orientation Morse-Smale diffeomorphisms on surfaces. Using the methods of factorization and linearizing neighborhoods we prove that such diffeomorphisms have a finite number of orientable heteroclinic orbits.
We describe the topological structure of closed manifolds of dimension no less than four which admit Morse-Smale diffeomorphisms such that its non-wandering set contains any number of sink periodic points, and any number of source periodic points, and few saddle periodic points.
J.~Palis found necessary conditions for a Morse-Smale diffeomorphism on a closed $n$-dimensional manifold $M^n$ to embed into a topological flow and proved that these conditions are also sufficient for $n=2$. For the case $n=3$ a possibility of wild embedding of closures of separatrices of saddles is an additional obstacle for Morse-Smale cascades to embed into topological flows. In this paper we show that there are no such obstructions for Morse-Smale diffeomorphisms without heteroclinic intersection given on the sphere $S^n, \,n\geq 4$, and Palis's conditions again are sufficient for such di…
An N-to-1 Smale Horseshoe
该研究将Conley-Moser定理推广到N对1局部微分同胚，利用扩展符号动力学编码广义N对1马蹄映射并验证其结构稳定性。
A classical problem in dynamical systems is to measure the complexity of a map in terms of their orbits. One of the main tools we have to achieve this goal is entropy. However, many interesting families of dynamical systems have every element with zero-entropy. One of said families are the Morse-Smale diffeomorphisms. The first author and E. Pujals introduced the concept of generalized entropy that extends the classical notion of entropy. In this work, we compute the generalized entropy of Morse-Smale diffeomorphisms on surfaces.
We prove that a Morse-Smale gradient-like flow on a closed manifold has a "system of compatible invariant stable foliations" that is analogous to the object introduced by Palis and Smale in their proof of the structural stability of Morse-Smale diffeomorphisms and flows, but with finer regularity and geometric properties. We show how these invariant foliations can be used in order to give a self-contained proof of the well-known but quite delicate theorem stating that the unstable manifolds of a Morse-Smale gradient-like flow on a closed manifold $M$ are the open cells of a $CW$-decomposition …
Lyapunov graphs of Non-singular Morse-Smale flows on $S^1 \times S^2$
本文在B. Yu的研究基础上，以过滤邻域为局部数据，刻画S¹×S²上非奇异Morse-Smale流的李雅普诺夫图，并明确了可实现为该类图的有向图类型。
The set of transverse homoclinic intersections for a saddle-focus equilibrium in the planar equilateral restricted four-body problem admit certain simple homoclinic orbits which form the skeleton of the complete homoclinic intersection -- or homoclinic web. In the present work, the planar restricted four-body problem is viewed as an invariant subsystem of the spatial problem, and the influence of this planar homoclinic skeleton on the spatial dynamics is studied from a numerical point of view. Starting from the vertical Lyapunov families emanating from saddle focus equilibria, we compute the s…
We study the set of periods of the Morse--Smale diffeomorphisms on the $n$-dimensional sphere $\mathbb{S}^n$, on products of two spheres of arbitrary dimension $\mathbb{S}^m \times \mathbb{S}^n$ with $m \neq n$, on the $n$-dimensional complex projective space $\mathbb{C}\text{\textbf{P}}^n}$ and on the $n$-dimensional quaternion projective space $\mathbb{H}\text{\textbf{P}}^n}$. We classify the minimal sets of Lefschetz periods for such Morse--Smale diffeomorphisms. This characterization is done using the induced maps on the homology. The main tool used is the Lefschetz zeta function.
In this paper we build Non-singular Morse-Smale flows on S^3 with unknotted and unlinked saddle orbits by identifying fat round handles along their boundaries. This way of building the flows enables to get their phase portraits. We also show that the presence of heteroclinic trajectories imposes an order in the round handle decomposition of these flows; this order is total for NMS flows composed of one repulsive, one attractive and n unknotted saddle orbits, for n >1.
The indexed links of Non-singular Morse-Smale flows on graph manifolds
本文研究图流形上非奇异 Morse-Smale 流的闭轨道指标链环，证明其可由含奇异与特定正则 Seifert 纤维的特殊链环经有限操作构造，从而完成分类。
We study a topological structure of a closed $n$-manifold $M^n$ ($n\geq 3$) which admits a Morse-Smale diffeomorphism such that codimension one separatrices of saddles periodic points have no heteroclinic intersections different from heteroclinic points. Also we consider gradient like flow on $M^n$ such that codimension one separatices of saddle singularities have no intersection at all. We show that $M^n$ is either an $n$-sphere $S^n$, or the connected sum of a finite number of copies of $S^{n-1}\otimes S^1$ and a finite number of special manifolds $N^n_i$ admitting polar Morse-Smale systems.…
Topological classification of even the simplest Morse-Smale diffeomorphisms on 3-manifolds does not fit into the concept of singling out a skeleton consisting of stable and unstable manifolds of periodic orbits. The reason for this lies primarily in the possible "wild" behaviour of separatrices of saddle points. Another difference between Morse-Smale diffeomorphisms in dimension 3 from their surface analogues lies in the variety of heteroclinic intersections: a connected component of such an intersection may be not only a point as in the two-dimensional case, but also a curve, compact or non-c…
The topological classification of gradient like Morse-Smale vector fields and diffeomorphisms on 3-manifolds was obtained.
We consider a class $G(S^n)$ of orientation preserving Morse-Smale diffeomorphisms of the sphere $S^{n}$ of dimension $n>3$ in assumption that invariant manifolds of different saddle periodic points have no intersection. We put in a correspondence for every diffeomorphism $f\in G(S^n)$ a colored graph $Γ_f$ enriched by an automorphism $P_f$. Then we define the notion of isomorphism between two colored graphs and prove that two diffeomorphisms $f, f'\in G(S^n)$ are topologically conjugated iff the graphs $Γ_f$, $Γ_f'$ are isomorphic. Moreover we establish the existence of a linear-time algorith…
We consider general Morse-Smale diffeomorphisms on a closed orientable two-dimentional surface. In this paper it is proved that the complete topological invariant of Morse-Smale diffeomorphisms is finite, the algorithm of the construction of the complete topological invariant in explicit form is given and necessary and sufficient conditions of topological conjugacy of Morse-Smale diffeomorphisms is obtained.
We construct a topological invariant for a Morse-Smale flow on a 3-manifold and prove that the flows are topologically equivalent iff their invariants are same.
In this paper we give a complete topological classification of orientation preserving Morse-Smale diffeomorphisms on orientable closed surfaces. For MS diffeomorphisms with relatively simple behaviour it was known that such a classification can be given through a directed graph, a three-colour directed graph or by a certain topological object, called a scheme. Here we will assign to any MS surface diffeomorphism a finite amount of data which completely determines its topological conjugacy class. Moreover, we show that associated to any abstract version of this data, there exists a unique conju…
Morse-Smale 3-diffeomorphisms with saddles of the same unstable manifold dimension
本文研究闭三维流形上所有鞍点具有相同不稳定流形维数的Morse-Smale微分同胚，推广Reeb的结果，证明此类微分同胚仅能在三维球面上定义。
Lens spaces are the only 3-manifolds that admit gradient-like flows with four fixed points. This is an immediate corollary of Morse inequality and of the Morse function with four critical points existence. A similar question for gradient-like diffeomorphisms is open. Solution can be approached by describing a complete topological conjugacy invariant of the class of considered diffeomorphisms and constructing of representative diffeomorphism for every conjugacy class by the abstract invariant. Ch. Bonnati and V. Z. Grines proved that the topological conjugacy class of Morse-Smale flows with uni…
The classical approach to the study of dynamical systems consists in representing the dynamics of the system in the form of a "source-sink", that means identifying an attractor-repeller pair, which are attractor-repellent sets for all other trajectories of the system. If there is a way to choose this pair so that the space orbits in its complement (the characteristic space of orbits) is connected, this creates prerequisites for finding complete topological invariants of the dynamical system. It is known that such a pair always exists for arbitrary Morse-Smale diffeomorphisms given on any manif…
The complete invariant for gradient like Morse-Smale dynamical systems (vector fields and diffeomorphisms) on closed 4-manifolds are constructed. It is same as Kirby diagram in a case of polar vector field without fixed points of index 3.
Let $f : M \rightarrow M$ be a Morse-Smale diffeomorphism defined on a compact and connected manifold without boundary. Let $C(M)$ denote the hyperspace of all subcontinua of M endowed with the Hausdorff metric and $C(f) : C(M) \rightarrow C(M)$ denote the induced homeomorphism of $f$. We show in this paper that if $M$ is the unit circle $S^1$ then the induced map $C(f)$ has not the shadowing property. Also we show that the topological entropy of $C(f)$ has only two possible values: $0$ or $\infty$. In particular, we show that the entropy of $C(f)$ is $0$ when $M$ is the unit circle $S^1$ and …
Morse-Smale dynamics in discrete Huygens networks
本文研究离散惠更斯网络的相位振子动力学，推导映射z↦z+hK sin z的莫尔斯-斯梅尔准则，构造含非横截异宿交点的树，拓展了同步网络的动力学分析。
For a closed oriented 3-manifold $Y$ we define $n(Y)$ to be the minimal non-negative number such that in each homotopy class of non-singular vector fields of $Y$ there is a Morse-Smale vector field with less or equal to $n(Y)$ periodic orbits. We combine the construction process of Morse-Smale flows given in [2] with handle decompositions of compact orientable surfaces to provide an upper bound to the number $n(Y)$ for oriented Seifert manifolds and oriented graph manifolds prime to $\stwo\times\sone$.
We analyse the topological (knot-theoretic) features of a certain codimension-one bifurcation of a partially hyperbolic fixed point in a flow on $\real^3$ originally described by Shil'nikov. By modifying how the invariant manifolds wrap around themselves, or ``pleat,'' we may apply the theory of templates, or branched two-manifolds, to capture the topology of the flow. This analysis yields a class of flows which bifurcate from a Morse-Smale flow to a Smale flow containing periodic orbits of all knot and link types.
Morse-Bott-Smale chain complex
本文统一了Morse-Bott-Smale链复形的五个退化条件为单一条件，解决其良定义性问题，并证明与Morse-Smale-Witten链复形拟同构，从而给出Morse同调定理的新证明。
The paper is devoted to finding conditions to the existence of a self-indexing energy function for Morse-Smale diffeomorphisms on a 3-manifold. These conditions involve how the stable and unstable manifolds of saddle points are embedded in the ambient manifold. We also show that the existence of a self-indexing energy function is equivalent to the existence of a Heegaard splitting of a special type with respect to the considered diffeomorphism.
Every finite set of natural numbers is realizable as algebraic periods of a Morse$\unicode{x2013}$Smale diffeomorphism
本文解决了代数周期实现问题，证明了自然数的任意有限集合均可作为某可定向或不可定向曲面上 Morse–Smale 微分同胚的代数周期，并给出了反转定向情形的限制及映射类共轭类数目的指数估计。
Canonical chain complexes for Morse-Smale vector fields
本文提出Morse-Smale向量场的规范链复形，通过Čech同调谱序列技术，证明其在拓扑等价下不变，并代数推导Morse不等式。
A new geometric criterion is derived for the existence of chaos in continuous-time autonomous systems in three-dimensional Euclidean spaces, where a type of Smale horseshoe in a subshift of finite type exists, but the intersection of stable and unstable manifolds of two points on a hyperbolic periodic orbit does not imply the existence of a Smale horseshoe of the same type on cross-sections of these two points. This criterion is based on the existence of a hyperbolic periodic orbit, differing from the classical equilibrium-based Shilnikov criterion and the condition of transversal homoclinic o…
We introduce a Milnor metric on the determinant line of the cohomology of the underlying closed manifold with coefficients in a flat vector bundle, by means of interactions between the fixed points and the closed orbits of a Morse-Smale flow. This allows us to generalise the notion of the absolute value at zero point of the Ruelle dynamical zeta function, even in the case where this value is not well defined in the classical sense. We give a formula relating the Milnor metric and the Ray-Singer metric. An essential ingredient of our proof is Bismut-Zhang's Theorem.
In the first part of this paper, we consider smooth maps from a compact orientable 3-manifold without boundary to the 2-sphere. We give a geometric criterion to decide whether two given maps are homotopic, based on the sets of points where the maps are equal or antipodal. We extend this criterion to non-singular vector fields (and co-oriented plane fields) on 3-manifolds. In the second part, we make use of this criterion to study non-singular Morse-Smale vector fields on Seifert manifolds, giving a simple proof of a result of Yano.
Let f:E -> E be a homeomorphism of the plane E. We define open sets P, called {\em pruning fronts} after the work of Cvitanović, for which it is possible to construct an isotopy H: E x [0,1] -> E with open support contained in the union of f^{n}(P), such that H(*,0)=f(*) and H(*,1)=f_P(*), where f_P is a homeomorphism under which every point of P is wandering. Applying this construction with f being Smale's horseshoe, it is possible to obtain an uncountable family of homeomorphisms, depending on infinitely many parameters, going from trivial to chaotic dynamic behaviour. This family is a 2-dim…
Shadowing property on hyperspace of continua induced by Morse gradient system
本文研究了由Morse梯度系统诱导的连续体超空间的影子性质，证明了此类映射不具有影子性质。
We give a new proof of the Morse Homology Theorem by constructing a chain complex associated to a Morse-Bott-Smale function that reduces to the Morse-Smale-Witten chain complex when the function is Morse-Smale and to the chain complex of smooth singular $N$-cube chains when the function is constant. We show that the homology of the chain complex is independent of the Morse-Bott-Smale function by using compactified moduli spaces of time dependent gradient flow lines to prove a Floer-type continuation theorem.
In the present paper the exhaustive topological classification of nonsingular Morse-Smale flows of $n$-manifolds with two limit cycles is presented. Hyperbolicity of periodic orbits implies that among them one is attracting and another is repelling. Due to Poincare-Hopf theorem Euler characteristic of closed manifold $M^n$ which admits the considered flows is equal to zero. Only torus and Klein bottle can be ambient manifolds for such flows in case of $n=2$. Authors established that there exist exactly two classes of topological equivalence of such flows of torus and three of the Klein bottle.…
In this paper using approach of 1-D auxiliary maps we prove the existence of trapping domains containing attractors of the multidimensional Henon-like maps. For both of quadratic and cubic nonlinearities we obtain sufficient conditions of topological Smale horseshoes existence. The complex structure of attractors is discussed in the case of small coupling parameter.
Abundance of Smale's horseshoes and ergodic measures via multifractal analysis and various quantitative spectrums
本文结合密度、熵和多重分形分析研究遍历测度结构，证明了传递Anosov系统中遍历测度的稠密$G_δ$性质，建立了“多马蹄”熵稠密性质的一般框架，并推广至非双曲系统与多种定量谱分析。
Dilation surfaces are generalizations of translation surfaces where the transition maps of the atlas are translations and homotheties with a positive ratio. In contrast with translation surfaces, the directional flow on dilation surfaces may contain trajectories accumulating on a limit cycle. Such a limit cycle is called hyperbolic because it induces a nontrivial homothety. It has been conjectured that a dilation surface with no actual hyperbolic closed geodesic is in fact a translation surface. Assuming that a dilation surface contains a horizon saddle connection, we prove that the directions…
In this paper, we first develope the concept of Lyapunov graph to weighted Lyapunov graph (abbreviated as WLG) for nonsingular Morse-Smale flows (abbreviated as NMS flows) on $S^3$. WLG is quite sensitive to NMS flows on $S^3$. For instance, WLG detect the indexed links of NMS flows. Then we use WLG and some other tools to describe nonsingular Morse-Smale flows without heteroclinic trajectories connecting saddle orbits (abbreviated as depth $0$ NMS flows). It mainly contains the following several directions: \begin{enumerate} \item we use WLG to list depth $0$ NMS flows on $S^3$; \item with th…
On a smooth, compact and oriented manifold without boundary, we give a complete description of the correlation function of a Morse-Smale gradient flow satisfying a certain nonresonance assumption. This is done by analyzing precisely the spectrum of the generator of such a flow acting on certain anisotropic spaces of currents. In particular, we prove that this dynamical spectrum is given by linear combinations with integer coefficients of the Lyapunov exponents at the critical points of the Morse function. Via this spectral analysis and in analogy with Hodge-de Rham theory, we give an interpret…
We analyze the chain recurrent set of skew product semiflows obtained from nonautonomous differential equations -- ordinary differential equations or semilinear parabolic differential equations. For many gradient-like dynamical systems, Morse-Smale dynamical systems e.g., the chain recurrent set contains only isolated equilibria. The structure in the asymptotically autonomous setting is richer but still close to to the structure of a Morse-Smale dynamical system. The main tool used in this paper is a nonautonomous flavour of Conley index theory developed by the author. We will see that for a c…
We generalise the semi-Riemannian Morse index theorem to elliptic systems of partial differential equations on star-shaped domains. Moreover, we apply our theorem to bifurcation from a branch of trivial solutions of semilinear systems, where the bifurcation parameter is introduced by shrinking the domain to a point. This extends recent results of the authors for scalar equations.
In this paper, we show that, for scalar reaction-diffusion equations $u_t=u_{xx}+f(x,u,u_x)$ on the circle $S^1$, the Morse-Smale property is generic with respect to the non-linearity $f$. In \cite{CR}, Czaja and Rocha have proved that any connecting orbit, which connects two hyperbolic periodic orbits, is transverse and that there does not exist any homoclinic orbit, connecting a hyperbolic periodic orbit to itself. In \cite{JR}, we have shown that, generically with respect to the non-linearity $f$, all the equilibria and periodic orbits are hyperbolic. Here we complete these results by showi…
For Morse-Smale pairs on a smooth, closed manifold the Morse-Smale-Witten chain complex can be defined. The associated Morse homology is isomorphic to the singular homology of the manifold and yields the classical Morse relations for Morse functions. A similar approach can be used to define homological invariants for isolated invariant sets of flows on a smooth manifold, which gives an analogue of the Conley index and the Morse-Conley relations. The approach will be referred to as Morse-Conley-Floer homology.
The stable and unstable manifolds of a saddle fixed point (SFP) of the Bonhoeffer-van der Pol oscillator are numerically studied. A correspondence between the existence of homoclinic tangencies (whic are related to the creation or destruction of Smale horseshoes) and the chaos observed in the bifurcation diagram is described. It is observed that in the non-chaotic zones of the bifurcation diagram, there may or may not be Smale horseshoes, but there are no homoclinic tangencies.
Takens' Last Problem and strong pluripotency
本文针对双曲不变集动力系统的强多重势能问题，提出了一种组合方法来识别Newhouse域中的强多重势能微分同胚，并证明了其$C^r$-鲁棒性，从而解决了Takens最后问题中的相关存在性证明。
Quasi-invariant measures for non-discrete groups of diffeomorphisms containing a Morse-Smale dynamics are studied. The assumption concerning the presence of a Morse-Smale dynamics allows us to extend to higher dimensions a number of recently established results for non-discrete groups acting on the circle. These results are also applied to show that, for many groups as above, every continuous orbit equivalence must coincide almost everywhere with a diffeomorphism of the corresponding manifold.
The Morse Smale property for time-periodic scalar reaction-diffusion equation on the circle
本文研究圆周上时间周期标量半线性抛物方程的Morse-Smale性质，证明其Poincaré映射双曲不动点无同宿连接、异宿不动点稳定与不稳定流形横截相交，且在所有ω-极限集双曲时方程具备Morse-Smale性质。
Let $f:M \rightarrow \mathbb{R}$ be a Morse-Bott function on a finite dimensional closed smooth manifold $M$. Choosing an appropriate Riemannian metric on $M$ and Morse-Smale functions $f_j:C_j \rightarrow \mathbb{R}$ on the critical submanifolds $C_j$, one can construct a Morse chain complex whose boundary operator is defined by counting cascades \cite{FraTheA}. Similar data, which also includes a parameter $ε> 0$ that scales the Morse-Smale functions $f_j$, can be used to define an explicit perturbation of the Morse-Bott function $f$ to a Morse-Smale function $h_ε:M \rightarrow \mathbb{R}$ \…
Floer Homology: From Generalized Morse-Smale Dynamical Systems to Forman's Combinatorial Vector Fields
本文通过计数闭轨道与孤立临界点间的流线，构造广义Morse-Smale系统的Floer型边缘算子，并推广到Forman组合向量场，从而直接恢复光滑与离散结构的$\mathbb{Z}_2$同调。
Transient rebellions in the Kuramoto oscillator: Morse-Smale structural stability and connection graphs of finite 2-shift type
本文研究了库莫托振子模型中瞬态反叛现象，揭示其结构稳定性及有限2-移型连接图特性。
In this paper, we study the number of ergodic measures of maximal entropy for partially hyperbolic diffeomorphisms defined on $3-$torus with compact center leaves. Assuming the existence of a periodic leaf with Morse-Smale dynamics we prove a sharp upper bound for the number of maximal measures in terms of the number of sources and sinks of Morse-Smale dynamics. A well-known class of examples for which our results apply are the so-called Kan-type diffeomorphisms admitting physical measures with intermingled basins.
Singular Morse-Smale Flows on Pseudomanifolds with Spherical-Cone Singularities: Conley Theory and Intersection Homology
将Morse-Conley理论推广到具有球锥奇点的伪流形，通过引入奇异Morse-Smale流，建立了奇点Conley指数的局部动力学公式，证明了全局Lyapunov函数的存在性，并利用磨光化方法关联奇异与光滑情形，最后通过动力系统方法计算交合同调。
On a smooth compact Riemannian manifold without boundary, we construct a finite dimensional cohomological complex of currents that are invariant by an Axiom A flow verifying Smale's transversality assumptions. The cohomology of that complex is isomorphic to the De Rham cohomology via certain spectral projectors. This construction is achieved by defining anisotropic Sobolev spaces adapted to the global dynamics of Axiom A flows. In the particular case of Morse-Smale gradient flows, this complex coincides with the classical Morse complex.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。