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arXiv · 1207.4212

Singular perturbation of nonlinear systems with regular singularity

William R. P. Conti·Domingos H. U. Marchetti
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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}^ν-valued function, holomorphic in a polydisc \bar{D}_ρ\times \bar{D}_ρ\times \bar{D}_ρ^ν. 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.

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William R. P. Conti, Domingos H. U. Marchetti. 2012-07-17. Singular perturbation of nonlinear systems with regular singularity. https://arxiv.org/abs/1207.4212

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