Prevalence of Delay Embeddings with a Fixed Observation Function
Let $x_{j+1}=\phi(x_{j})$, $x_{j}\in\mathbb{R}^{d}$, be a dynamical system with $\phi$ 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 observat...
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| Main Authors | , |
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| Format | Journal Article |
| Language | English |
| Published |
19.06.2018
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| Subjects | |
| Online Access | Get full text |
| DOI | 10.48550/arxiv.1806.07529 |
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| Summary: | Let $x_{j+1}=\phi(x_{j})$, $x_{j}\in\mathbb{R}^{d}$, be a dynamical system
with $\phi$ 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
$o=\pi_{1}$, with $\pi_{1}$ being the projection to the first coordinate.
However, we allow polynomial perturbations to be applied directly to the
diffeomorphism $\phi$, 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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| DOI: | 10.48550/arxiv.1806.07529 |