A formal classification of Hopf-Bogdanov-Takens singularities of codimension three

[摘要] This paper is concerned with Hopf-Bogdanov-Takens singularities, that is, singularities of 4-dimensional vector fields whose 1-jet is linearly conjugate to y partial derivative/partial derivative x - v partial derivative/partial derivative u + u partial derivative/partial derivative v. They first occur in generic 3-parameter families. A normal form for these singularities and their unfoldings is given up to an arbitrary order. Working with a second order truncation of the normal form for the singularity, we provide a formal classification at the level of codimension three. The proof relies on blowing-up techniques and estimation of Abelian integrals. Previous results in the literature about the classification of 3-dimensional nilpotent singularities of codimension four are essential for our purposes. (C) 2019 Elsevier Inc. All rights reserved.

[发布日期] 2019-12-15

[关键词] Singularities;Normal forms;Blow-up;Hopf-Bogdanov-Takens