A spectral operator theoretic perspective was taken for modeling nonlinear dynamics subjected to external inputs, due to prescribed forcing or an environment. Under this Director's Research Initiative (DRI), we focused on spectral properties (eigenfunctions, eigenvalues) of the Koopman operator (i.e., an infinite dimensional linear operator that can describe the full dynamics of a nonlinear system). We showed how external forcing could be treated in a Koopman decomposition-based framework. Furthermore, during the course of our study using Koopman decompositions of periodically excited Hopf bifurcation systems, we characterized a new attractor that had never been described in the literature or texts. We called this attractor the quasi-periodic intermittency attractor and showed that it may be a fundamental building block of a variety of multi-scale dynamics systems. Perhaps the most far reaching impact of this DRI will be a contribution that was not planned in the original proposal. This contribution has to do with the generalization of Koopman decompositions using a fractional calculus perspective on complexity. By using a combination of Koopman operator theory and fractional calculus, we showed that our generalized spectral decompositions are better suited for complex systems influenced by an external environment in which long-term memory is introduced.
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