Linearization in the large of nonlinear systems and Koopman operator spectrum

According to the Hartman–Grobman Theorem, a nonlinear system can be linearized in a neighborhood of a hyperbolic stationary point. Here, we extend this linearization around stable (unstable) equilibria or periodic orbits to the whole basin of attraction, for both discrete diffeomorphisms and flows. We discuss the connection of the linearizing transformation to the spectrum of Koopman operator.

► Linearization around stable equilibria in the attraction basin.
► The linearization transformation is continuously differentiable.
► It works for both maps and flows, autonomous or non-autonomous.
► Applicable to the (un)stable manifold of periodic orbits.
► Linearization and Koopman operator spectrum.