Normal Forms for Manifolds of Normally Hyperbolic Singularities and Asymptotic Properties of Nearby Transitions

三维多项式向量场中单值奇点附近的法向双曲极限环面

Normally hyperbolic limit tori near monodromic singularities in 3D polynomial vector fields

本文利用平均理论与受 Christopher-Lloyd 方法启发的递归构造，研究三维多项式向量场在单值奇点附近的法向双曲极限环面最大数，并给出改进的 Hilbert 数下界。

微分方程中法向双曲不变环面检测机制

A mechanism for detecting normally hyperbolic invariant tori in differential equations

本文提出一种检测微分方程中法向双曲不变环面的机制，通过引导系统的双曲极限环条件，证明非自治周期方程存在此类环面，并应用于 jerk 方程族。

Bifurcations of Normally Hyperbolic Invariant Manifolds and Consequences for Reaction Dynamics

In this paper we study the breakdown of normal hyperbolicity and its consequences for reaction dynamics; in particular, the dividing surface, the flux through the dividing surface (DS), and the gap time distribution. Our approach is to study these questions using simple, two degree-of-freedom Hamiltonian models where calculations for the different geometrical and dynamical quantities can be carried out exactly. For our examples, we show that resonances within the normally hyperbolic invariant manifold may, or may not, lead to a `loss of normal hyperbolicity'. Moreover, we show that the onset o…

将离散几何奇异摄动理论扩展到非双曲点

Extending Discrete Geometric Singular Perturbation Theory to Non-Hyperbolic Points

本文利用Takens嵌入定理将离散几何奇异摄动理论扩展到非正规双曲区域，证明快慢映射在非双曲奇点附近的动力学可由相应快慢向量场的时间1映射局部逼近，并应用于折叠、跨临界、叉形等奇点及中心流形分析。

Slow-fast normal forms arising from piecewise smooth vector fields

We studied piecewise smooth differential systems of the form $$\dot{z} = Z(z) = \dfrac{1 + \operatorname{sgn}(F)}{2}X(z) + \dfrac{1 - \operatorname{sgn}(F)}{2}Y(z),$$ where $F: \mathbb{R}^{n}\rightarrow \mathbb{R}$ is a smooth map having 0 as a regular value. We consider linear regularizations of the vector field $Z$ given by $$\dot{z}= Z_{\varepsilon}(z) = \dfrac{1 + φ(F/\varepsilon)}{2}X(z) +\dfrac{1 - φ(F /\varepsilon)}{2}Y(z),$$where $φ$ is a transition function (not necessarily monotonic) and nonlinear regularizations of the vector field $Z$ whose transition function is monotonic. It is a…