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Spectrum of the Koopman Operator, Spectral Expansions in Functional Spaces, and State-Space Geometry

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Abstract

We examine spectral operator-theoretic properties of linear and nonlinear dynamical systems with globally stable attractors. Using the Kato decomposition, we develop a spectral expansion for general linear autonomous dynamical systems with analytic observables and define the notion of generalized eigenfunctions of the associated Koopman operator. We interpret stable, unstable and center subspaces in terms of zero-level sets of generalized eigenfunctions. We then utilize conjugacy properties of Koopman eigenfunctions and the new notion of open eigenfunctions—defined on subsets of state space—to extend these results to nonlinear dynamical systems with an equilibrium. We provide a characterization of (global) center manifolds, center-stable, and center-unstable manifolds in terms of joint zero-level sets of families of Koopman operator eigenfunctions associated with the nonlinear system. After defining a new class of Hilbert spaces, that capture the on- and off-attractor properties of dissipative dynamics, and introducing the concept of modulated Fock spaces, we develop spectral expansions for a class of dynamical systems possessing globally stable limit cycles and limit tori, with observables that are square-integrable in on-attractor variables and analytic in off-attractor variables. We discuss definitions of stable, unstable, and global center manifolds in such nonlinear systems with (quasi)-periodic attractors in terms of zero-level sets of Koopman operator eigenfunctions. We define the notion of isostables for a general class of nonlinear systems. In contrast with the systems that have discrete Koopman operator spectrum, we provide a simple example of a measure-preserving system that is not chaotic but has continuous spectrum, and discuss experimental observations of spectrum on such systems. We also provide a brief characterization of the data types corresponding to the obtained theoretical results and define the coherent principal dimension for a class of datasets based on the lattice-type principal spectrum of the associated Koopman operator.

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Notes

  1. Koopman modes are defined up to a constant, the same as eigenvectors. However, here we have defined projection with respect to a specific basis with an orthonormal dual.

  2. This is not the standard notion of conjugacy, since the dimensions of spaces that \(\mathbf{h}\) maps between is not necessarily the same, i.e., \(m\ne n\) necessarily.

  3. The definition of factor conjugacy can be generalized to include dynamical systems on spaces M and N, where \(\mathbf{h}:M\rightarrow N\) and \(N\ne M\) (see Mezić and Banaszuk 2004, where such concept was defined for the case \(N=S^1,\) indicating conjugacy to a rotation).

  4. Note that the change of variables in the Poincaré-Siegel linearization is an analytic diffeomorphism (Arnold 2012), and thus \(\mathbf{s}^{-1}\) is analytic.

  5. Specifically, if an equilibrium is asymptotically stable from an open set U, then there is a Lyapunov function L such that, sufficiently close to the origin (but not at the origin) \(\dot{L}<0\) and thus the vector field “points inwards” on level sets of L sufficiently close to the origin. We can choose \(\Sigma \) to be one of those level sets. Then clearly \(t(\mathbf{x})\) and \(i(\mathbf{x})\) are unique, for every trajectory as if not, the trajectory would need to “enter” and then “exit” the interior of L.

  6. Kelley in fact presented a similar example, with a stable direction instead of the unstable one in \(x_2\).

  7. I am thankful to Professor Mihai Putinar for directing my attention to the Fock space.

  8. The author proved the result a while ago and presented it in an early draft manuscript (UCSB preprint, 2011) with Dr. Yuehang Lan. That manuscript never got completed or published. The announcement of the result can be found in Mohr and Mezić (2014), where function spaces were built based on the intuition from this example. Since only the statement and no proof was provided in Mohr and Mezić (2014), the author felt it appropriate to provide the complete result with the proof here.

  9. Note the increase of dimension of state space by 1, to \(n+1,\) to accommodate easier notation since it makes the off-attractor vector \(\mathbf{y}\) n-dimensional.

  10. Note on the terminology: these types of systems were classically called linear skew-product systems.

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Acknowledgements

I am thankful to John Guckenheimer, Dimitris Giannakis, Mihai Putinar, Yueheng Lan, Alex Mauroy, Ryan Mohr and Mathias Wanner for their useful comments. This work was supported in part by the DARPA contract HR0011-16-C-0116 and ARO Grants W911NF-11-1-0511 and W911NF-14-1-0359.

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  1. Mechanical Engineering and Mathematics, University of California Santa Barbara, Santa Barbara, CA, 93106, USA

    Igor Mezić

  2. University of Rijeka, Rijeka, Croatia

    Igor Mezić

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  1. Igor Mezić
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Correspondence to Igor Mezić.

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Communicated by Dr. Paul Newton.

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Mezić, I. Spectrum of the Koopman Operator, Spectral Expansions in Functional Spaces, and State-Space Geometry. J Nonlinear Sci 30, 2091–2145 (2020). https://doi.org/10.1007/s00332-019-09598-5

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