Perturbed normalizers and Melnikov functions

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:464

Article

Buica, Adriana¹ 

[1] Univ Babes Bolyai, Dept Matemat, Str Kogalniceanu 1, Cluj Napoca 400084, Romania

关键词: Melnikov functions;    Perturbed normalizers;    Planar vector field;    Lie bracket;   

DOI  :  10.1016/j.jmaa.2018.04.004

来源: Elsevier

【 摘 要 】

Given m >= 1 and a smooth family of planar vector fields (X-epsilon)(epsilon) that is a perturbation of a period annulus, we provide a characterization, in terms of Lie brackets, of the property that the first (m - 1) Melnikov functions of (X-epsilon)(epsilon) vanish identically. The equivalent condition is the existence of a smooth family of planar vector fields (U-epsilon)(epsilon), called here perturbed normalizers of order m. We also provide an effective procedure for computing U-epsilon when the first (m - 1) Melnikov functions of (X-epsilon)(epsilon) vanish identically. A formula for the derivative of the m-th order Melnikov function is given. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free