Analytic expressions for the unstable manifold at equilibrium points in dynamical systems of differential equations

Unstable, or stable, manifolds are important invariant surfaces for nonlinear dynamical systems. For instance, they characterize "pieces" on the boundary of the region of attraction for a given stable point In this paper, we prove the analyticity of the unstable (stable) manifold for a class of dynamical systems described by differential equations: an explicit parametrization of the unstable (stable) manifold at saddle points is introduced. The parametrization is analytic, converging on the whole domain of definition (i.e., entire). The coefficients of the Taylor expansion of this parametrization are explicitly given in a recursive formulae form, thus the manifold can be characterized computationally.