辛向量场椭圆不动点的有效通有超指数稳定性

一般四维辛映射中正则环面的整体结构

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

对于混合相空间的一般四维辛映射的情况下，我们研究了正则环面的全局组织。为此，我们计算椭圆1-环面的两个耦合的标准地图，并显示它们在一个3D相空间切片。这可视化了在4D相空间中，所有规则的2-环面是如何围绕椭圆1-环面的骨架组织的。1-环面出现在两种类型的单参数族中：（a）从椭圆-椭圆周期轨道产生的李雅普诺夫族，观察到甚至存在于远离它们和超越主要共振间隙的地方，以及（b）从秩-1共振产生的族。在两种类型的族的共振间隙处，（i）存在周期轨道，类似于2D映射的庞加莱-伯克霍夫定理，或者（ii）族可以形成大弯曲。结合这些结果允许描述的4D相空间中的定期环面的层次结构类似于在2D地图中的岛屿周围的岛屿层次。

Linear stability of the elliptic relative equilibria for the restricted $4$-body problem: the Euler case

In this paper, we consider the elliptic relative equilibria of the restricted $4$-body problems, where the three primaries form an Euler collinear configuration and the four bodies span $\mathbf{R}^2$. We obtain the symplectic reduction to the general restricted $N$-body problem. By analyzing the relationship between this restricted $4$-body problems and the elliptic Lagrangian solutions, we obtain the linear stability of the restricted $4$-body problem by the $ω$-Maslov index. Via numerical computations, we also obtain conditions of the stability on the mass parameters for the symmetric cases…

$C1$-Genericity of Symplectic Diffeomorphisms and Lower Bounds for Topological Entropy

There is a $C^1$-residual (Baire second class) subset $\mathcal{R}$ of symplectic diffeomorphisms on $2d$-dimensional manifold, $d\geq 1$, such that for every non-Anosov $f$ in $\mathcal{R}$ its topological entropy is lower bounded by the supremum of the Lyapunov exponents of their hyperbolic periodic points in the \emph{unbreakable central subbundle} (i.e., central direction with no dominated splitting) of $f$. The previous result deals with the fact that for $f$ in a residual set $\tilde{\mathcal{R}}$ of symplectic diffeomorphisms (containing $\mathcal{R}$) satisfies a trichotomy: or $f$ is …

关于辛动力学的Kolmogorov典型性质

On Kolmogorov-typical properties of symplectic dynamics

本文提出一个通用框架，证明了具有椭圆点的有限参数辛微分同胚一般族在所有参数值下均具有周期点快速增长、普适性和高涌现性等典型性质。

Maslov-type indices and linear stability of elliptic Euler solutions of the three-body problem

In this paper, we use the central configuration coordinate decomposition to study the linearized Hamiltonian system near the elliptic Euler solutions. Then using the Maslov-type ω-index theory of symplectic paths and the theory of linear operators we compute the ω-indices and obtain certain properties of linear stability of the Euler elliptic solutions of the classical three-body problem.

Elliptic and non-hyperbolic closed characteristics on compact convex P-cyclic symmetric hypersurfaces in ${\bf R}^{2n}$

Let $Σ$ be a compact convex hypersurface in ${\bf R}^{2n}$ which is P-cyclic symmetric, i.e., $x\in Σ$ implies $Px\inΣ$ with P being a $2n\times2n$ symplectic orthogonal matrix and $P^k=I_{2n}$, where $n, k\geq2$, $ker(P-I_{2n})=0$. In this paper, we first generalize Ekeland index theory for periodic solutions of convex Hamiltonian system to a index theory with P boundary value condition and study its relationship with Maslov P-index theory, then we use index theory to prove the existence of elliptic and non-hyperbolic closed characteristics on compact convex P-cyclic symmetric hypersurfaces i…

Bifurcations of families of 1D-tori in 4D symplectic maps

The regular structures of a generic 4D symplectic map with a mixed phase space are organized by one-parameter families of elliptic 1D-tori. Such families show prominent bends, gaps, and new branches. We explain these features in terms of bifurcations of the families when crossing a resonance. For these bifurcations no external parameter has to be varied. Instead, the longitudinal frequency, which varies along the family, plays the role of the bifurcation parameter. As an example we study two coupled standard maps by visualizing the elliptic and hyperbolic 1D-tori in a 3D phase-space slice, loc…

Moser's quadratic, symplectic map

In 1994, Jürgen Moser generalized Hénon's area-preserving quadratic map to obtain a normal form for the family of four-dimensional, quadratic, symplectic maps. This map has at most four isolated fixed points. We show that the bounded dynamics of Moser's six parameter family is organized by a codimension-three bifurcation, which we call a quadfurcation, that can create all four fixed points from none. The bounded dynamics is typically associated with Cantor families of invariant tori around fixed points that are doubly elliptic. For Moser's map there can be two such fixed points: this structure…

平面辛台球的一般性质

Generic properties of planar symplectic billiards

研究平面辛台球的一般性质，证明对于\(C^\infty\)强凸域剩余集，周期轨迹相关性质，还证弗兰克斯引理等，得出\(C^\infty\)强凸辛台球一般有正拓扑熵的结论。

椭圆不动点处平面映射的规范形与首次积分

Normal forms and first integrals of planar maps at elliptic fixed points

本文研究椭圆不动点邻域内的实解析平面映射，在存在首次积分时构造规范形，并证明第二个首次积分的存在迫使映射线性化，涵盖共振与非共振及辛形式保持与否的情形。

High-order control for symplectic maps

We revisit the problem of introducing an a priori control for devices that can be modeled via a symplectic map in a neighborhood of an elliptic equilibrium. Using a technique based on Lie transform methods we produce a normal form algorithm that avoids the usual step of interpolating the map with a flow. The formal algorithm is completed with quantitative estimates that bring into evidence the asymptotic character of the normal form transformation. Then we perform an heuristic analysis of the dynamical behavior of the map using the invariant function for the normalized map. Finally, we discuss…

辛动力学中鲁棒的遍历行为

Robustly transitive behavior in symplectic dynamics

研究辛映射的直接积在满足支配条件时，实解析变形展现出大范围鲁棒遍历集，同时得到非均匀超几何的实解析鲁棒遍历辛映射新例子。

球摆的半全局辛不变量

Semi-global symplectic invariants of the spherical pendulum

我们显式地计算了球摆的焦点-焦点附近的半全局辛不变量。本文提出了一种改进的Birkhoff规范形方法，用于计算Eliasson变元中不稳定平衡点附近哈密顿量的展开式。结合这与明确的公式的行动，我们发现半全球辛不变量附近的焦点，焦点介绍武Ngoc 2003年。我们还证明了Birkhoff规范形是零圈上完全椭圆积分的逆。据我们所知，这是第一次，半全球辛不变量附近的焦点焦点已明确计算。最后，我们对钟摆作一些评论，它的不变量可以用一种漂亮的方式与θ函数联系起来。

Double exponential stability for generic real-analytic elliptic equilibrium points

We consider the dynamics in a neighborhood of an elliptic equilibrium point with a Diophantine frequency of a symplectic real analytic vector field and we prove the following result of effective stability. Generically, both in a topological and measure-theoretical sense, any solution starting sufficiently close to the equilibrium point remains close to it for an interval of time which is doubly exponentially large with respect to the inverse of the distance to the equilibrium point. We actually prove a more general statement assuming the frequency is only non-resonant. This improves previous r…