论文标题
检测不变的扩展锥以生成单词集以识别分段线性地图中的混乱
Detecting invariant expanding cones for generating word sets to identify chaos in piecewise-linear maps
论文作者
论文摘要
我们展示了三个对象的存在，$ω_ {\ rm trap} $，$ {\ bf w} $和$ c $，对于连续的piewise-linear map $ f $ in $ \ mathbb {r}^n $上的连续piewise-linear map $ f $，这意味着$ f $ $ f $具有阳性lyapunov exporment。首先，$ω_ {\ rm陷阱} \ subset \ mathbb {r}^n $是$ f $的捕获区域。第二，$ {\ bf w} $是一组有限的单词，它以$ω_ {\ rm trap} $编码所有点的前轨道。最后，$ c \ subset t \ mathbb {r}^n $是一个不变的扩展锥，用于$ {\ bf w} $中单词形成的$ f $构图的衍生物。我们开发了一种算法，该算法可以识别这些对象的二维同构形态，由两个仿射片组成。主要的工作是在$ω_ {\ rm陷阱} $和$ c $的明确构造中。它们的存在等同于以一般方式的一组可计算条件。这导致在相对较大的参数空间方面产生了计算机辅助的混乱证明。我们还观察到$ c $扩展的失败如何与$ f $的分叉相吻合。使用单方方向衍生物对Lyapunov指数进行评估，以便分析中可以包含与开关歧管相交的正向轨道（其中$ f $不可分测）。
We show how the existence of three objects, $Ω_{\rm trap}$, ${\bf W}$, and $C$, for a continuous piecewise-linear map $f$ on $\mathbb{R}^N$, implies that $f$ has a topological attractor with a positive Lyapunov exponent. First, $Ω_{\rm trap} \subset \mathbb{R}^N$ is trapping region for $f$. Second, ${\bf W}$ is a finite set of words that encodes the forward orbits of all points in $Ω_{\rm trap}$. Finally, $C \subset T \mathbb{R}^N$ is an invariant expanding cone for derivatives of compositions of $f$ formed by the words in ${\bf W}$. We develop an algorithm that identifies these objects for two-dimensional homeomorphisms comprised of two affine pieces. The main effort is in the explicit construction of $Ω_{\rm trap}$ and $C$. Their existence is equated to a set of computable conditions in a general way. This results in a computer-assisted proof of chaos throughout a relatively large regime of parameter space. We also observe how the failure of $C$ to be expanding can coincide with a bifurcation of $f$. Lyapunov exponents are evaluated using one-sided directional derivatives so that forward orbits that intersect a switching manifold (where $f$ is not differentiable) can be included in the analysis.
