Emmanuel Dufraine
We consider an open and dense subset of the set of all connections (heteroclinic orbits) between two saddles of a vector field in dimension three. We give a complete classification up to topological equivalence of the dynamics in a neighbourhood of a connection in this subset.
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For almost every flat surface the sequence of saddle connection lengths listed in increasing order is uniformly distributed mod one.
The goal of this note is to study the action of the backward Rauzy-Veech algorithm on the translation surfaces with horizontal saddle connections. In particular, we prove that the orbit of a translation surface via the aforementioned algorithm is $\infty$-complete if and only if the surface does not posses horizontal saddle connections. Moreover, we show that appearance of all saddle connections as sides of the polygonal representations of the translations surface along the backward Rauzy-Veech induction orbit is equivalent to the minimality of the horizontal translation flow.
Uniform distribution of saddle connection lengths in all $\mathsf{SL}(2,\mathbb{R})$ orbits
研究显示SL(2,R)轨道中平坦表面的鞍点连接长度序列在单位区间内呈现均匀分布特性。
An analytic reversible Hamiltonian system with two degrees of freedom is studied in a neighborhood of its symmetric heteroclinic connection made up of a symmetric saddle-center, a symmetric orientable saddle periodic orbit lying in the same level of a Hamiltonian and two non-symmetric heteroclinic orbits permuted by the involution. This is a codimension one structure and therefore it can be met generally in one-parameter families of reversible Hamiltonian systems. There exist two possible types of such connections in dependence on how the involution acts near the equilibrium. We prove a series…
In a smooth dynamical system, a homoclinic connection is a closed orbit returning to a saddle equilibrium. Under perturbation, homoclinics are associated with bifurcations of periodic orbits, and with chaos in higher dimensions. Homoclinic connections in nonsmooth systems are complicated by their interaction with discontinuities in their vector fields. A connection may involve a regular saddle outside a discontinuity set, or a pseudo-saddle on a discontinuity set, with segments of the connection allowed to cross or slide along the discontinuity. Even the simplest case, that of connection to a …
We prove that the asymptotic number of pairs of saddle connections with length smaller than $L$ with bounded virtual area is quadratic for almost every translation surface with respect to any ergodic $SL(2,\mathbb{R})$-invariant measure. A key tool of the proof is that Siegel-Veech transforms of bounded functions with compact supports are in $L^{2+κ}$ for every $SL(2,\mathbb{R})$-invariant measure.
Counting pairs of saddle connections
研究平移曲面中具特定叉积大小的鞍点连接对的数量，结合经典计数结果与西格尔 - 维奇变换\(L^2\)结果，证明其数量渐近增长如\(cR^2\)，格曲面平行对也有此二次增长，\(c\)因情况而异。
Asymptotic quadratic growth rates of saddle connections and families of periodic cylinders on translation tori with n marked points are studied. For any marking the existence of limits of the quadratic growth rate is shown using elementary methods (not Ratners theorem). We study the growth rate limit as function of the marking. We give precise formulas for this function in the case of two marked points and describe the sets where the growth function is maximal and continuous in any case. For rational two markings we calculate the index of the Veech group in SL(2,Z) using two different ways.
We show that heterodimensional cycles can be born at the bifurcations of a pair of homoclinic loops to a saddle-focus equilibrium for flows in dimension 4 and higher. In addition to the classical heterodimensional connection between two periodic orbits, we found two new types of heterodimensional connections: one is a heteroclinic between a periodic orbit of index 2 and a homoclinic loop, and the other connects a periodic orbit of index 3 to the saddle-focus equilibrium.
Unbounded bunching of saddle connections on the golden L
证明黄金L平移曲面具有无界聚集性质：对任意正整数K，存在半径为1的球包含至少K个该曲面上鞍连接的周期向量。
Counting Saddle Connections on Hyperelliptic Translation Surfaces with a Slit
该研究通过水平周期重正化方法，证明了超椭圆翻译表面中特定鞍点连接的增长速率符合 $L (\log L)^{d-2}$ 的下界。
Motivated by the study of billiards in polygons, we prove fine results for the distribution of gaps of directions of saddle connections on translation surfaces. As an application we prove that for almost every holomorphic differential $ω$ on a Riemann surface of genus $g \geq 2$ the smallest gap between saddle connection directions of length at most a fixed length decays faster than quadratically in the length. We also characterize the exceptional set: the decay rate is not faster than quadratic if and only if $ω$ is a lattice surface.
We give effective estimates for the number of saddle connections on a translation surface that have length $\leq L$ and are in a prescribed homology class modulo $q$. Our estimates apply to almost all translation surfaces in a stratum of the moduli space of translation surfaces, with respect to the Masur-Veech measure on the stratum.
In this work we consider formal singular vector fields in $ C^{3}$with an isolated and doubly-resonant singularity of saddle-node typeat the origin. Such vector fields come from irregular two-dimensionalsystems with two opposite non-zero eigenvalues, and appear for instancewhen studying the irregular singularity at infinity in Painlev{é} equations$(P\_{j})\_{j\in(I,II,III,IV,V)}$, for generic values of the parameters.Under generic assumptions we give a complete formal classificationfor the action of formal diffeomorphisms (by changes of coordinates)fixing the origin and fibered in the independ…
In this work, following [Bit15] and [Bit16a], we consider analytic singular vector fields in $(\mathbb{C}^{3},0)$ with an isolated and doubly-resonant singularity of saddle-node type at the origin. Such vector fields come from irregular two-dimensional differential systems with two opposite non-zero eigenvalues, and appear for instance when studying the irregular singularity at infinity in Painlev{é} equations (P\_j), j=I...V , for generic values of the parameters. Under suitable assumptions, we provide an analytic classification under the action of fibered diffeomorphisms, based on the study …
We compute the gap distribution of directions of saddle connections for two classes of translation surfaces. One class will be the translation surfaces arising from gluing two identical tori along a slit. These yield the first explicit computations of gap distributions for non-lattice translation surfaces. We show that this distribution has support at 0 and quadratic tail decay. We also construct examples of translation surfaces in any genus $d>1$ that have the same gap distribution as the gap distribution of two identical tori glued along a slit. The second class we consider are twice-marked …
Slope Gap Distribution of Saddle Connections on the 2n-gon
本文显式计算任意2n边形上鞍连接的极限斜率间隙分布，证明其不总为单峰，并给出不可微点数量随n增长的线性上下界。
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…
Classification of NMS-flows with unique twisted saddle orbit on orientable 4-manifolds
本文研究闭可定向4-流形上带唯一扭鞍点轨道的NMS流分类，证明仅$\u2118^3\times\u2118^1$容许这类流，且其恰好分为八个等价类并给出代表流构造，完善了NMS流拓扑等价分类研究。
Dilation surfaces are generalizations of translation surfaces where the geometric structure is modelled on the complex plane up to affine maps whose linear part is real. They are the geometric framework to study suspensions of affine interval exchange maps. However, though the $SL(2,\mathbb{R})$-action is ergodic in connected components of strata of translation surfaces, there may be mutually disjoint $SL(2,\mathbb{R})$-invariant open subsets in components of dilation surfaces. A first distinction is between triangulable and non-triangulable dilation surfaces. For non-triangulable surfaces, th…
We explicitly compute the limiting gap distribution for slopes of saddle connections on the flat surface associated to the regular octagon with opposite sides identified. This is the first such computation where the Veech group of the translation surface has multiple cusps. We also show how to parametrize a Poincaré section for the horocycle flow on $SL(2,\mathbb{R})/SL(X,ω)$ associated to an arbitrary Veech surface $(X, ω)$. As a corollary, we show that the associated gap distribution is piecewise real analytic.
In this work, we consider germs of analytic singular vector elds in (C^3,0) with an isolated and doubly-resonant singularity of saddle-node type at the origin. Such vector elds come from irregular two-dimensional dierential systems with two opposite non-zero eigenvalues, and appear for instance when studying the irregular singularity at innity in Painlev{é} equations (P j) j=I,...,V for generic values of the parameters. Under suitable assumptions, we prove a theorem of analytic normalization over sectorial domains, analogous to the classical one due to Hukuhara-Kimura-Matuda for saddle-nodes i…
Measure bound for translation surfaces with short saddle connections
本文证明了平移曲面分层上遍历 $SL_2(R)$-不变概率测度的强正则性，利用多尺度紧化和代数性结果给出了短鞍连接集合的测度界。
Equidistribution of saddle connections on translation surfaces
本文证明了平移曲面上长度至多为$R$的鞍连接平均测度在$R\to\infty$时弱收敛于Lebesgue测度，并利用角度计数测度的弱极限性质与Kerckhoff-Masur-Smillie唯一遍历结果完成证明。
In this paper, a modified van der Pol oscillator equation is considered which appears in several heart action models. We study its global dynamics and verify many interesting bifurcations such as a Hopf bifurcation, a heteroclinic saddle connection, and a homoclinic saddle connection. Some of these bifurcations are detected by using Conley index methods. We demonstrate how the study of connection matrices and transition matrices shows how to select interesting parameter values for simulations.
We explore sequential escape behaviour of coupled bistable systems under the influence of stochastic perturbations. We consider transient escapes from a marginally stable "quiescent" equilibrium to a more stable "active" equilibrium. The presence of coupling introduces dependence between the escape processes: for diffusive coupling there is a strongly coupled limit (fast domino regime) where the escapes are strongly synchronised while for intermediate coupling (slow domino regime) without partially escaped stable states, there is still a delayed effect. These regimes can be associated with bif…
Shrinking rates of horizontal gaps for generic translation surfaces
本文研究一般平移曲面上鞍形连接的分布，通过分析平面离散子集，得到了长度至多为R时两条几乎水平鞍形连接间角度差随R趋于无穷的精确衰减速率。
We show that Y. Cheung's general $Z$-continued fractions can be adapted to give approximation by saddle connection vectors for any compact translation surface. That is, we show the finiteness of his Minkowski constant for any compact translation surface. Furthermore, we show that for a Veech surface in standard form, each component of any saddle connection vector dominates its conjugates. The saddle connection continued fractions then allow one to recognize certain transcendental directions by their developments.
We consider a one-parameter family $(f_λ)_{λ\, \geqslant \, 0}$ of symmetric vector fields on the three-dimensional sphere $\mathbb{S}^3\subset\mathbb{R}^4$ whose flows exhibit a heteroclinic network between two saddle-foci inside a global attracting set. More precisely, when $λ= 0$, there is an attracting heteroclinic cycle between the two equilibria which is made of two $1$-dimensional connections together with a $2$-dimensional sphere which is both the stable manifold of one saddle-focus and the unstable manifold of the other. After slightly increasing the parameter while keeping the $1$-di…
Effective shadowing of transition maps for a Hamiltonian saddle-center-center equilibrium
该研究针对鞍-中心-中心哈密顿平衡点的法向双曲不变流形，提出基于伯克霍夫正则形与转移映射的轨迹跟踪方法，在空间限制性三体问题天平动点案例中验证了方法的高效性。
Smooth planar vector fields containing two hyperbolic saddles may possess contours formed by heteroclinic connections between these saddles. We present an overview of the bifurcations of these contours based on papers by J.W. Reyn and A.V. Dukov. Additionally, two new explicit polynomial systems containing such contours are derived, which are studied using the bifurcation software MatCont and are shown to exhibit the theoretically predicted phenomena, including series of heteroclinic connections.
Non-singular flows with twisted saddle orbit on orientable 3-manifolds
本文分类了闭可定向3-流形上仅含一个扭转鞍形轨道的非奇异Morse-Smale流，确定了容许流形类型并计算了等价类数量。
The structure of Morse flows and co-dimension one gradient flows on the sphere with holes
本文研究了带孔2球面上梯度流的拓扑结构，通过分离图分析奇点的分岔特性，揭示了最大六个奇点下的结构特征。
A holomorphic 1-form on a compact Riemann surface S naturally defines a flat metric on S with cone-type singularities. We present the following surprising phenomenon: having found a geodesic segment (saddle connection) joining a pair of conical points one can find with a nonzero probability another saddle connection on S having the same direction and the same length as the initial one. The similar phenomenon is valid for the families of parallel closed geodesics. We give a complete description of all possible configurations of parallel saddle connections (and of families of parallel closed geo…
On a classification of axiom A diffeomorphisms with codimension one basic sets and isolated saddles
本文对闭可定向流形上具有余维一基本集和孤立鞍点的公理A微分同胚，基于支撑流形的拓扑结构，在非游荡集上按全局共轭进行了分类。
Torus and hyperchaos in 3D Lotka-Volterra map
本研究探讨了三维 Lotka-Volterra 映射中三频准周期环面与超混沌的出现，通过计算流形和分岔分析揭示了通向准周期结构与超混沌的多种路径及转变机制。
In this paper we prove the breakdown of the two-dimensional stable and unstable manifolds associated to two saddle-focus points which appear in the unfoldings of the Hopf-zero singulariry. The method consists in obtaining an asymptotic formula for the difference between this manifolds which turns to be exponentially small respect to the unfolding parameter. The formula obtained is explicit but depends on the so-called Stokes constants, which arise in the study of original vector field and which corresponds to the so called inner equation in singular perturbation theory.
A discontinuous change in the size of an attractor is the most easily observed type of global bifurcation. More generally, an explosion is a discontinuous change in the set of recurrent points. An explosion often results from heteroclinic and homoclinic tangency bifurcations. Newhouse and Palis conjectured in 1976 that planar explosions are generically the result of either tangency or saddle node bifurcations. In this paper, we prove this conjecture for one-dimensional maps. Furthermore, we give a full classification for all possible tangency bifurcations and whether they lead to explosions.
Topology and dynamics of a flow that has a non-saddle set or a $W$-set
本文研究流形中孤立非鞍集或$W$-集影响区域内流的动力学与拓扑性质，建立非鞍集与流形间的上同调关系，进而获得曲面的动力学分类，并考察非鞍集的鲁棒性、分岔性质及$W$-集在二维流形中的行为。
Three-dimensional convex bodies can be classified in terms of the number and stability types of critical points on which they can balance at rest on a horizontal plane. For typical bodies these are nondegenerate maxima, minima, and saddle-points, the numbers of which provide a primary classification. Secondary and tertiary classifications use graphs to describe orbits connecting these critical points in the gradient vector field associated with each body. In previous work it was shown that these classifications are complete in that no class is empty. Here we construct 1- and 2-parameter famili…
Explicit formulae are given for the saddle connection for an integrable family of standard maps studied by Suris. A generalization of Melnikov's method shows that, upon perturbation, this connection is destroyed. We give explicit formula for the first order approximation of the area of the lobes of the resultant turnstile. It is shown that the lobe area is exponentially small in the limit when the Suris map approaches the trivial twist map.
In translation surfaces of finite area (corresponding to holomorphic differentials), directions of saddle connections are dense in the unit circle. On the contrary, saddle connections are fewer in translation surfaces with poles (corresponding to meromorphic differentials). The Cantor-Bendixson rank of their set of directions is a measure of descriptive set-theoretic complexity. Drawing on a previous work of David Aulicino, we prove a sharp upper bound that depends only on the genus of the underlying topological surface. The proof uses a new geometric lemma stating that in a sequence of three …
We study perturbations of diffeomorphisms that have a saddle connection between a pair of normally hyperbolic invariant manifolds. We develop a first-order deformation calculus for invariant manifolds and show that a generalized Melnikov function or Melnikov displacement can be written in a canonical way. This function is defined to be a section of the normal bundle of the saddle connection. We show how our definition reproduces the classical methods of Poincaré and Melnikov and specializes to methods previously used for exact symplectic and volume-preserving maps. We use the method to detect …
Heteroclinic cycles involving two saddle-foci, where the saddle-foci share both invariant manifolds, occur persistently in some symmetric differential equations on the 3-dimensional sphere. We analyse the dynamics around this type of cycle in the case when trajectories near the two equilibria turn in the same direction around a 1-dimensional connection - the saddle-foci have the same chirality. When part of the symmetry is broken, the 2-dimensional invariant manifolds intersect transversely creating a heteroclinic network of Bykov cycles. We show that the proximity of symmetry creates heterocl…
Real-time blow-up and connection graphs of rational vector fields on the Riemann sphere
该研究探讨复时间下有理向量场的全局有界性与有限时间爆破问题，通过分析黎曼球面上的爆破轨道连接图，实现了全局动力学的分类，提供了仅依赖多项式次数的相图组合计数方法。
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…
Classification of connection graphs of global attractors for $S^1$-equivariant parabolic equations
针对$S^1$上带可逆非线性项的一维半线性抛物方程，证明同lap signature的可逆非线性项间存在保持异宿连接的可逆同伦，完成其全局吸引子连接图的分类，并给出相关分岔图。
Bifurcations of mode-locked periodic orbits in three-dimensional maps
研究三维及更高维映射中锁模周期轨道的两类“加倍”分岔，验证其类型由第三特征值符号决定，并揭示 Neimark-Sacker 分岔与非同步鞍–结连接结构。
Topology of 4-manifolds that admit non-singular flows with saddle orbits of the same index
本文证明在4维情况下，非奇异流只能在特定的斜积或直积流形上存在，且其鞍形轨道具有不同的Morse索引。
Intersection theory and Siegel-Veech constants for Prym eigenform loci in $Ω\mathcal{M}_3(2,2)^{\rm odd}$
本文研究了Prym特征形式在特定模空间中的鞍连接的Siegel-Veech常数，涉及交集理论和实二次域的非平方判别式特征。
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