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We propose a novel operator-theoretic framework to study global stability ofnonlinear systems. Based on the spectral properties of the so-called Koopmanoperator, our approach can be regarded as a natural extension of classic linearstability analysis to nonlinear systems. The main results establish the(necessary and sufficient) relationship between the existence of specificeigenfunctions of the Koopman operator and the global stability property ofhyperbolic fixed points and limit cycles. These results are complemented withnumerical methods which are used to estimate the region of attraction of thefixed point or to prove in a systematic way global stability of the attractorwithin a given region of the state space.

Global stability analysis using the eigenfunctions of the Koopman operator