arXiv 1806.075292018-06-20math.DSnlin.CD

具有固定观测函数的延迟嵌入的普遍性

Prevalence of Delay Embeddings with a Fixed Observation Function

Raymundo Navarrete, Divakar Viswanath

摘要：

设$x_{j+1}=ϕ(x_{j})$，$x_{j}\in\mathbb{R}^{d}$是一个动力系统，其中$ϕ$是一个自同构.虽然状态向量$x_{j}$通常是不可观测的，但动态可以从延迟向量$\left(o(x_{1}),\ldots,o(x_{D})\right)$恢复，其中$o$是标量值观测函数，$D$是嵌入维数。延迟映射是通用$o$的嵌入，并且更强烈地，嵌入属性是普遍的。我们考虑观测函数固定在$o=π_{1}$的情况，其中$π_{1}$是到第一坐标的投影。然而，我们允许多项式扰动直接应用于非同构$ϕ$，从而模仿动力系统的参数化方式。我们证明了延迟映射是关于扰动的概率为1的嵌入。我们的证明引入了一种新的技术，证明流行使用勒贝格点的概念。

英文摘要：

Let $x_{j+1}=ϕ(x_{j})$, $x_{j}\in\mathbb{R}^{d}$, be a dynamical system with $ϕ$ being a diffeomorphism. Although the state vector $x_{j}$ is often unobservable, the dynamics can be recovered from the delay vector $\left(o(x_{1}),\ldots,o(x_{D})\right)$, where $o$ is the scalar-valued observation function and $D$ is the embedding dimension. The delay map is an embedding for generic $o$, and more strongly, the embedding property is prevalent. We consider the situation where the observation function is fixed at $o=π_{1}$, with $π_{1}$ being the projection to the first coordinate. However, we allow polynomial perturbations to be applied directly to the diffeomorphism $ϕ$, thus mimicking the way dynamical systems are parametrized. We prove that the delay map is an embedding with probability one with respect to the perturbations. Our proof introduces a new technique for proving prevalence using the concept of Lebesgue points.

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