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 = ( , , ) with a C ] -valued function, holomorphic in aWe show that its unique formal solution in power series of , whose coefficients are holomorphic functions of , is 1-summable under a Siegel-type condition on the eigenvalues of (0, 0, 0). The estimates employed resemble the ones used in KAM theorem. A simple lemma is applied to tame convolutions that appear in the power series expansion of nonlinear equations. Applications to spherical Bessel functions and probability theory are indicated. The proposed summability method has certain advantages as it may be applied as well to (singularly perturbed) nonlinear partial differential equations of evolution type.

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“…Definition 1. Reference ( [52,53]) a system is said to be in singularity perturbed form if its dynamics can be represented as:…”

“…References ([ 52 , 53 ]) a system is said to be in singularity perturbed form if its dynamics can be represented as: where and are continuously differentiable vector fields, is singular perturbation parameter and satisfies . The state vectors are defined by and , while denotes the input vector.…”