Carleman linearization and normal forms for differential systems with quasi-periodic coefficients.

Sergey V Chermnykh

doi:10.1186/s40064-016-3015-6

Abstract

We study the matrix representation of Poincaré normalization using the Carleman linearization technique for non-autonomous differential systems with quasi-periodic coefficients. We provide a rigorous proof of the validity of the matrix representation of the normalization and obtain a recursive algorithm for computing the normalizing transformation and the normal form of the differential systems. The algorithm provides explicit formulas for the coefficients of the normal form and the corresponding transformation.

Highlights

The main aim of this paper is to develop a recursive algorithm for constructing transformations to the Poincaré normal form for non-autonomous differential systems with quasi-periodic coefficients (Arnold 1983), suitable for performing on a computer
The algorithm presented here for constructing the normal form and normalizing transformation is based on Carleman linearization of differential equations (4) to the form (6) by (5)
This algorithm permits us to write the normalizing transformation in matrix form (7) by (9), (3) and the normal form of the differential system by (10), (5) in the sequence (8)

Summary

Introduction

The main aim of this paper is to develop a recursive algorithm for constructing transformations to the Poincaré normal form for non-autonomous differential systems with quasi-periodic coefficients (Arnold 1983), suitable for performing on a computer. The algorithm is based on the Carleman linearization technique (Carleman 1932). Several applications of Carleman linearization have been presented up until now. Bellman (1961) used Carleman linearization to obtain approximate solutions to nonlinear systems. Babadzanjanz (1978) studied the existence of continuations and representation of the solutions in celestial mechanics. Steeb and Wilhelm (1980) and Kowalski and Steeb (1991) studied nonlinear dynamical systems and generalizations of Carleman linearization. Carleman linearization technique has been used in a series of applications in the field of control theory, for example, to controlability and observability of infinitedimentional linear dynamical systems (Mozyrska and Bartosiewicz 2006, 2008) and to stochastic systems (Germani et al 2007)

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