Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A theoretical framework for Koopman analyses of fluid flows, part 1:
  local Koopman spectrum and properties   Local Koopman spectral problem is studied to resolve all dynamics for a
nonlinear system. The proposed spectral problem is compatible with the linear
spectral theory for various linear systems, and several properties of local
Koopman spectrums are discovered. Firstly, proliferation rule is discovered for
nonlinear observables and it applies to nonlinear systems recursively.
Secondly, the hierarchy structure of Koopman eigenspace of nonlinear dynamics
is revealed since dynamics can be decomposed into the base and perturbation
parts, where the former can be analyzed analytically or numerically and the
latter is further divided into linear and nonlinear parts. The linear part can
be analyzed by the linear spectrums theory. They are then recursively
proliferated to infinite numbers for the nonlinear part. Thirdly, local Koopman
spectrums and eigenfunctions change continuously and analytically in the whole
manifold under suitable conditions, derived from operator perturbation theory.
Two cases of fluid dynamics are numerically studied. One is the two-dimensional
flow past cylinder at the Hopf-bifurcation near the critical Reynolds number.
Two asymptotic stages, flow systems around an unstable fixed point and a stable
limit cycle were studied separately by the DMD algorithm. The triad-chain and
the lattice distribution of Koopman spectrums confirmed the proliferation rule
and hierarchy structure of Koopman eigenspace. Another example is the
three-dimensional secondary instability of flow past a fixed cylinder, where
the Fourier modes, Floquet modes, and high-order Koopman modes characterizing
the main structure of the flow are discovered.

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The categories of this paper are: acc-phys
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The categories of this paper are: ao-sci
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The categories of this paper are: cmp-lg
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The categories of this paper are: nucl-ex
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The categories of this paper are: eess
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The categories of this paper are: quant-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-ex
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The categories of this paper are: nucl-th
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The categories of this paper are: q-bio
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The categories of this paper are: math
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The categories of this paper are: funct-an
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The categories of this paper are: math-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: astro-ph