# Koopman representation

Abstract: The Koopman operator has recently garnered much attention for its value in dynamical systems analysis and data-driven model discovery. However, its application has been hindered by the computational complexity of extended dynamic mode decomposition; this requires a combinatorially large basis set to adequately describe many nonlinear systems of interest, e.g. cyber-physical infrastructure systems, biological networks, social systems, and fluid dynamics.
The Koopman operator is also known as the composition operator, which is formally the pull-back operator on the space of scalar observable functions . The Koopman operator is the dual, or left-adjoint, of the Perron-Frobenius operator, or transfer operator, which is the push-forward operator on the space of probability density functions. Oct 22, 2015 · Koopman observable subspaces and finite linear representations of nonlinear dynamical systems for control by Steven L. Brunton, Bingni W. Brunton, Joshua L. Proctor, and J. Nathan Kutz For more ...

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Koopman, Shephard and Doornik for technical details on the algorithms used in the S+FinMetrics/SsfPack functions. This chapter is organized as follows. Section 14.2 describes the gen-eral state space model and state space representation required for the S+FinMetrics/SsfPack state space functions. Subsections describe the
In this work, we explore finite-dimensional linear representations of nonlinear dynamical systems by restricting the Koopman operator to an invariant subspace. The Koopman operator is an infinite-dimensional linear operator that evolves observable functions of the state-space of a dynamical system [Koopman 1931, PNAS]. 2.2 The Koopman representation De nition 2.2.1. Let y(X;B;˛) be an action on a measure space which preserves the measure class ˛. The Koopman representation of associated to this action is the unitary representation ˇ: !U(L2(X;˛)) given by (ˇ()f)(x) = f(1x)(d ˛ d˛)1=2(x): Note that for all 1; 2 2, and f2L2(X;˛) we have ˇ(1 2)f= ˙ 1 2 (f)(d(1 2) ˛ d˛)1=2

Let G be a locally compact second countable Abelian group. Given a measure preserving action T of G on a standard probability space (X,μ), let M(T) denote the set of essential values of the spectral multiplicity function of the Koopman representation associated to T.
PDF | The Koopman operator has recently garnered much attention for its value in dynamical systems analysis and data-driven model discovery. However,... | Find, read and cite all the research you ... portions of the Koopman representation that embed into the left-regular rep-resentation. By combining this result with the generalized factor theorem of the previous paper, we conclude that for actions having completely positive outer entropy, the Koopman representation must be isomorphic to the count-

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Abstract In a prior paper, the author generalized the classical factor theorem of Sinai to actions of arbitrary countably infinite groups. In the present paper, we use this theorem and the techniques of its proof in order to study connections between positive entropy phenomena and the Koopman representation.
of the Koopman eigenfunctions. Overall, the “tuples” of Koopman eigenfunctions, eigenvalues, and modes enable us to: (a) transform state space so that the dynamics appear to be linear, (b) determine the temporal dynamics of the linear system, and (c) reconstruct the state of the original system from our new linear representation. In