No elliptic islands for the universal area-preserving map

Unique resonant normal forms for area preserving maps at an elliptic fixed point

We construct a resonant normal form for an area-preserving map near a generic resonant elliptic fixed point. The normal form is obtained by a simplification of a formal interpolating Hamiltonian. The resonant normal form is unique and therefore provides the formal local classification for area-preserving maps with the elliptic fixed point. The total number of formal invariants is infinite. We consider the cases of weak (of order $n\ge5$) and strong (of order $n=3,4$) resonances. We also construct unique normal forms for analytic families of area-preserving maps. We note that our constructions …

Simplified normal forms near a degenerate elliptic fixed point in two-parametric families of area-preserving maps

We derive simplified normal forms for an area-preserving map in a neighbourhood of a degenerate resonant elliptic fixed point. Such fixed points appear in generic two-parameter families of area-preserving maps. We also derive a simplified normal form for a generic two-parametric unfolding. The normal forms are used to analyse bifurcations of $n$-periodic orbits. In particular, for $n\ge6$ we find regions of parameters where the normal form has "meandering'' invariant curves.

On dynamics and bifurcations of area-preserving maps with homoclinic tangencies

We study bifurcations of area-preserving maps, both orientable (symplectic) and non-orientable, with quadratic homoclinic tangencies. We consider one and two parameter general unfoldings and establish results related to the appearance of elliptic periodic orbits. In particular, we find conditions for such maps to have infinitely many generic (KAM-stable) elliptic periodic orbits of all successive periods starting at some number.

Global structure of regular tori in a generic 4D symplectic map

For the case of generic 4D symplectic maps with a mixed phase space we investigate the global organization of regular tori. For this we compute elliptic 1-tori of two coupled standard maps and display them in a 3D phase-space slice. This visualizes how all regular 2-tori are organized around a skeleton of elliptic 1-tori in the 4D phase space. The 1-tori occur in two types of one-parameter families: (a) Lyapunov families emanating from elliptic-elliptic periodic orbits, which are observed to exist even far away from them and beyond major resonance gaps, and (b) families originating from rank-1…

On bifurcations of area-preserving and non-orientable maps with quadratic homoclinic tangencies

We study bifurcations of non-orientable area-preserving maps with quadratic homoclinic tangencies. We study the case when the maps are given on non-orientable two-dimensional surfaces. We consider one and two parameter general unfoldings and establish results related to the emergence of elliptic periodic orbits.

Oscillating mushrooms: adiabatic theory for a non-ergodic system

Can elliptic islands contribute to sustained energy growth as parameters of a Hamiltonian system slowly vary with time? In this paper we show that a mushroom billiard with a periodically oscillating boundary accelerates the particle inside it exponentially fast. We provide an estimate for the rate of acceleration. Our numerical experiments confirms the theory. We suggest that a similar mechanism applies to general systems with mixed phase space.

On stochastic sea of the standard map

Consider a generic one-parameter unfolding of a homoclinic tangency of an area preserving surface diffeomorphism. We show that for many parameters (residual subset in an open set approaching the critical value) the corresponding diffeomorphism has a transitive invariant set $Ω$ of full Hausdorff dimension. The set $Ω$ is a topological limit of hyperbolic sets and is accumulated by elliptic islands. As an application we prove that stochastic sea of the standard map has full Hausdorff dimension for sufficiently large topologically generic parameters.

Growth of heights in piecewise-affine planar maps

We consider the growth of heights of the points of the orbits of (piecewise) affine maps of the plane, with rational parameters. We analyse the asymptotic growth rate of both global and local ($p$-adic) heights, for the primes $p$ that divide the parameters. We show that almost all the points in a domain of linearity (such as an elliptic island in an area-preserving map) have the same exponential growth rate. We also show that the convergence of the $p$-adic height may be non-uniform, with arbitrarily large fluctuations occurring arbitrarily close to any point. We explore numerically the behav…

A dichotomy in area-preserving reversible maps

In this paper we study R-reversible area-preserving maps f on a two-dimensional Riemannian closed manifold M, i.e. diffeomorphisms f such that Ro f=f^{-1}o R where R is an isometric involution on M. We obtain a C1-residual subset where any map inside it is Anosov or else has a dense set of elliptic periodic orbits. As a consequence we obtain the proof of the stability conjecture for this class of maps. Along the paper we also derive the C1-closing lemma for reversible maps and other perturbation toolboxes.

非单调环面映射中指数混合性的丧失

Loss of Exponential Mixing in a Non-Monotonic Toral Map

该研究针对正交帐篷形剪切复合的二维环面保测映射，借助混沌台球成果证明其强混合性与赫尔德可观测量的多项式关联衰减，指出其为遍历边界分段线性映射的典型，存在减慢混合的零测集且扰动下可生椭圆岛。

Birkhoff Averages and Rotational Invariant Circles for Area-Preserving Maps

Rotational invariant circles of area-preserving maps are an important and well-studied example of KAM tori. John Greene conjectured that the locally most robust rotational circles have rotation numbers that are noble, i.e., have continued fractions with a tail of ones, and that, of these circles, the most robust has golden mean rotation number. The accurate numerical confirmation of these conjectures relies on the map having a time reversal symmetry, and these methods cannot be applied to more general maps. In this paper, we develop a method based on a weighted Birkhoff average for identifying…

含噪声Harper映射中的回归与黏滞性

Recurrence and Stickiness in the Noisy Harper Map

该研究探究三种噪声对保面积Harper映射庞加莱回归统计的影响，发现噪声增强规则岛俘获效应使中间时间尺度幂律衰减更缓，长时间尺度则呈指数衰减，并用三状态马尔可夫链解释了相关机制。

建模哈布蛇与mongooses的捕食者-被捕食者动态导致日本阿武隈大岛生态灾难

Modeling the Prey-Predator Dynamics of Habu Snakes and Mongooses Leading to Ecological Disaster on Amami Oshima Island in Japan

研究通过数学模型分析引入mongooses对阿武隈大岛生态的影响，揭示早期干预可能避免物种灭绝和生态破坏。

含时激励器下准可积保面积映射绝热俘获现象中的双共振穿越分析

Analysis of double-resonance crossing in adiabatic trapping phenomena for quasi-integrable area-preserving maps with time-dependent exciters

针对含时激励的Hamilton系统与保面积映射，本文分析双共振条件下的绝热共振穿越及相空间结构，拓展周期扰动系统的俘获与输运研究，并服务于加速器束流分裂应用。

圆盘上的保面积微分同胚与正双曲轨道

Area-preserving diffeomorphisms on the disk and positive hyperbolic orbits

本文证明了开圆盘上保面积非退化微分同胚在具有至少2个内部周期点时必有无穷多个正双曲周期点，并将其应用于切透镜空间与切3维球面，细化了Hofer等人的结果。

Non-integrability of measure preserving maps via Lie symmetries

We consider the problem of characterizing, for certain natural number $m$, the local $\mathcal{C}^m$-non-integrability near elliptic fixed points of smooth planar measure preserving maps. Our criterion relates this non-integrability with the existence of some Lie Symmetries associated to the maps, together with the study of the finiteness of its periodic points. One of the steps in the proof uses the regularity of the period function on the whole period annulus for non-degenerate centers, question that we believe that is interesting by itself. The obtained criterion can be applied to prove the…

双椭圆曲面上的QRT映射

QRT Map on a Bielliptic Surface

本文将QRT映射中的垂直与水平平移分别替换为两族椭圆曲线上的群运算，从而在双椭圆曲面上构造出QRT映射的推广。