Linear structure of nonlinear dynamic systems via Koopman decomposition

Abstract

Linear structure and invariant subspaces of nonlinear dynamics are revealed, extending the superposition principle and invariant subspaces from linear dynamics. They are achieved by considering dynamics in its dual space and the local spectral Koopman theory. The Koopman eigenfunctions constitute invariant subspaces under the given dynamic system, providing convenient bases for the linear structure. On the other hand, the locality and infinite dimensionality are identified as two unique properties of nonlinear dynamics, where the former refers to the spectral problem is locally defined, and the latter refers to Koopman spectrums are recursively proliferated by nonlinear interaction. Koopman spectral theory is studied. For a linear time-invariant (LTI) system, its linear spectrum is a subset of Koopman spectrums. High order Koopman spectrum can be obtained for nonlinear observables using the proliferation rule. For a linear time-variant system (LTV), Koopman decomposition is obtained by the eigenvalue problem of its fundamental matrix. Besides the general LTV, the periodic LTV system is studied using the Floquet theory. The Floquet spectrums are found to be Koopman spectrums. For a nonlinear system, a local Koopman spectrum problem is defined for a parameterized semigroup Koopman operator, and the simple local spectra are found to be conditionally continuous from the operator perturbation theory. The proliferation is found to recursively applicable to nonlinear dynamics. Moreover, the hierarchy structure of the Koopman decomposition of nonlinear systems is discovered, by decomposing dynamics into base and perturbation on top of it. The numerical algorithm, dynamic mode decomposition (DMD), is examined for its applicability to capture the spectra and modes for a variety of dynamic systems. A more robust and efficient framework based on generalized eigenvalue problem (GEV) is proposed, which is then solved by a least-square solution (LS) or a total least square solution (TLS). Therefore, two algorithms, DMD-LS and DMD-TLS algorithm, are developed. DMD-LS algorithm is mathematically equivalent to the standard DMD algorithm first proposed by Schmid (2010) but more robust. DMD-TLS is more accurate for noise data. A residue-based criterion is developed to choose dynamically important or true DMD modes from trivial or spurious modes that often appear in DMD computations. Linear structure via Koopman decomposition is first applied to a linear dynamic system and an asymptotic nonlinear system, for example. Then flow past fixed cylinder of a Hopf bifurcation process is numerically studied via DMD technique. The equivalence of Koopman decomposition to the GSA is verified at the primary instability stage. The Fourier modes, the least stable Floquet modes, and their high-order derived Koopman modes are found to be the superposition of countable infinite Koopman modes when the flow reaches periodic by considering continuity of Koopman spectrum and the invariance of Koopman modes to the nonlinear transition process. The nonlinear modulation effects, namely, the modulation of the mean flow and the resonance phenomena is explained similarly. The coherent structures are also found to be related to the decomposition. A DMD based model order reduction method is implemented based on Galerkin projection. The model reduction approach is applied to both the transitional and the periodic stages of flow passing a fixed cylinder. Accurate dynamics and frequencies are rebuilt.