Decidability of chaos for some families of dynamical systems

One-Parameter Families of Smooth Interval Maps: Density of Hyperbolicity and Robust Chaos

In this note we will discuss the notion of robust chaos, and show that (i) there are natural one-parameter families with robust chaos and (ii) hyperbolicity is dense within generic one-parameter families (and so these families are not robustly chaotic).

Asymptotic likelihood of chaos for smooth families of circle maps

We consider a smooth two-parameter family $f_{a,L}\colonθ\mapsto θ+a+LΦ(θ)$ of circle maps with a finite number of critical points. For sufficiently large $L$ we construct a set $A_L^{(\infty)}$ of $a$-values of positive Lebesgue measure for which the corresponding $f_{a,L}$ exhibits an exponential growth of derivatives along the orbits of the critical points. Our construction considerably improves the previous one of Wang and Young for the same class of families, in that the following asymptotic estimate holds: the Lebesgue measure of $A_L^{(\infty)}$ tends to full measure in $a$-space as $L$…

The boundary of chaos for interval mappings

A goal in the study of dynamics on the interval is to understand the transition to positive topological entropy. There is a conjecture from the 1980's that the only route to positive topological entropy is through a cascade of period doubling bifurcations. We prove this conjecture in natural families of smooth interval maps, and use it to study the structure of the boundary of mappings with positive entropy. In particular, we show that in families of mappings with a fixed number of critical points the boundary is locally connected, and for analytic mappings that it is a cellular set.

关于 s-极限跟踪与 Li-Yorke 型混沌的若干结果

Some results on s-limit shadowing and Li-Yorke type chaos

本文研究紧度量空间连续自映射的 s-极限跟踪性质，通过相空间的 $G_δ$-划分将 Li-Yorke 型混沌的描述推广到一般 Furstenberg 族，补充了已有结果并给出例证。

分形插值曲面的有限八面体族覆盖

Covers of fractal interpolation surfaces with finite families of octahedrons

本文针对分形插值曲面，提出了一种基于寻找集合最大值的改进方法，用于构造有限八面体族覆盖吸引子，较先前基于排序算法的方法计算效率更高。

Elements of contemporary mathematical theory of dynamical chaos. Part 1. Pseudohyperbolic attractors

The paper deals with topical issues of modern mathematical theory of dynamical chaos and its applications. At present, it is customary to assume that dynamical chaos in finitedimensional smooth systems can exist in three different forms. This is dissipative chaos, the mathematical image of which is a strange attractor; conservative chaos, for which the entire phase space is a large "chaotic sea" with randomly spaced elliptical islands inside it; and mixed dynamics, characterized by the principal inseparability in the phase space of attractors, repellers and conservative elements of dynamics. I…