解析动力学的分段解析子单元

解析动力学的遍历迁移理论和分段解析子单元

Ergodic Transport Theory and Piecewise Analytic Subactions for Analytic Dynamics

本文考虑一个保方向的分段解析实扩张映射$f: [0,1]\to [0,1]$，其度为$d$，以及一个实解析正势$g: [0,1] \to \mathbb{R}$ .我们假设映射和势函数在复平面上的区间邻域内有一个复解析延拓。我们还假设$\log g$是为这个扩展定义的。在复动力学中已知，在上述假设下，对于给定的势$β\,\log g$，其中$β$是实常数，存在定义在$[0,1]$上的实解析本征函数$ϕ_β$（具有复解析扩展），用于$β\,\log g$的Ruelle算子。在某些假设下，我们证明了$\frac{1}β\, \log ϕ_β$收敛，是一个分段解析校准子作用。我们的理论可以应用于$\log g(x)=-\log f'(x)$ .在这种情况下，我们将对合核与所谓的标度函数联系起来。

Length Spectrum Rigidity for piecewise analytic Bunimovich Billiards

In the paper, we establish Squash Rigidity Theorem - the dynamical spectral rigidity for piecewise analytic Bunimovich squash-type stadia. We also establish Stadium Rigidity Theorem - the dynamical spectral rigidity for piecewise analytic Bunimovich stadia whose flat boundaries are a priori fixed. In addition, for smooth Bunimovich squash-type stadia we compute the Lyapunov exponents along the maximal period two orbit, as well as the value of the Peierls' Barrier function from the maximal marked length spectrum associated to the rotation number $\frac{2n}{4n+1}$.

在非光滑系统中检测极限环面：一种分析方法及其在三维分段线性系统中的应用

Detecting Limit Tori in Non-Smooth Systems: An Analytic Approach with Applications to 3D Piecewise Linear Systems

本文提出了一种分析方法，用于在非光滑系统中检测孤立不变环面，并在三维分段线性系统中应用，首次实现了对极限环面的分析检测。

分形分析中的伪焦点在分段解析系统中的螺旋轨迹

Fractal Analysis of Pseudo Foci in Piecewise Analytic Systems

本文通过分形方法研究分段解析系统中伪焦点附近螺旋轨迹的Minkowski维度和非退化性，为分析此类伪焦点的分支现象提供理论支持。

Piecewise closed form expression of Rössler-like trajectories

The use of iteration and piecewise functions allows analytic expression of the trajectories of an Rössler-like attractor, avoiding infinite series solution. It seems possible to extend this approach to other attractors, even if the differential equations are not known, but the experimental data have been collected.

Shilnikov-type Dynamics in Three-Dimensional Piecewise Smooth Maps

We show the existence of Shilnikov-type dynamics and bifurcation behaviour in general discrete three-dimensional piecewise smooth maps and give analytical results for the occurence of such dynamical behaviour. Our main example in fact shows a `two-sided' Shilnikov dynamics, i.e. simultaneous looping and homoclinic intersection of the one-dimensional eigenmanfolds of fixed points on both sides of the border. We also present two complementary methods to analyse the return time of an orbit to the border: one based on recursion and another based on complex interpolation.