Singular perturbation of nonlinear systems with regular singularity

Abstract

We extend Balser-Kostov method of studying summability properties of a singularly perturbed inhomogeneous linear system with regular singularity at origin to nonlinear systems of the form \varepsilon zf^{\prime} = F(\varepsilon,z,f) with F a \mathbb{C}^{\nu}-valued function, holomorphic in a polydisc \bar{D}_{\rho}\times \bar{D}_{\rho}\times \bar{D}_{\rho}^{\nu}. We show that its unique formal solution in power series of \varepsilon, whose coefficients are holomorphic functions of z, is 1-summable under a Siegal-type condition on the eigenvalues of F_{f}(0,0,0). The estimates employed resemble the ones used in KAM theorem. A simple Lemma is developed to tame convolutions that appears in the power series expansion of nonlinear equations.