  We present novel sufficient conditions for the global stability of equilibria
in the case of nonlinear dynamics with analytic vector fields. These conditions
provide stability criteria that are directly expressed in terms of the Taylor
expansion coefficients of the vector field (e.g. in terms of first order
coefficients, maximal coefficient, sum of coefficients). Our main assumptions
is that the flow be holomorphic, and the linearized system be locally
exponentially stable and diagonalizable. These results are based on the
properties of the Koopman operator defined on the Hardy space on the polydisc.
