Periodic solutions of approximated normal forms and simplification of normal transform by using Gröbner basis

Abstract

The paper proposes a methodology based on the normal form perturbation method and Gröbner basis approach. Spectrum rearrangement allows to exhibit resonant terms associated with periodic behaviors of the system. Normal forms generate compatibility equations when one looks for periodic solutions and define amplitude-frequency conditions. Gröbner generators are defined from these equations and associated ideal is generated in order to simplify results of final normal form using generalized Euclidean division. Normal transform are simplified, and in some cases can be linearized with nonlinearity remaining hidden in amplitude-frequency conditions.