Singular perturbation of nonlinear systems with regular singularity

TL;DR

This paper extends the Balser-Kostov method to analyze the summability of solutions in nonlinear singular perturbation problems with regular singularities, demonstrating 1-summability under specific eigenvalue conditions.

Contribution

It introduces a novel extension of the Balser-Kostov method to nonlinear systems, establishing conditions for 1-summability of formal solutions with new estimates similar to KAM theory.

Findings

Formal solutions are 1-summable under eigenvalue conditions.

Develops a lemma to handle convolutions in nonlinear power series.

Provides estimates akin to those in KAM theorem.

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.