@article{Żołądek2023, 
author = {Henryk Żołądek},
title = {Normal forms, invariant manifolds and Lyapunov theorems},
year = {2023},
journal = {Communications in Analysis and Mechanics},
volume = {15},
number = {2},
pages = {300-341},
keywords = {Lyapunov theorems, Poincaré–Dulac normal form, invariant manifolds},
url = {https://www.sciopen.com/article/10.3934/cam.2023016},
doi = {10.3934/cam.2023016},
abstract = {We present an approach to Lyapunov theorems about a center for germs of analytic vector fields based on the Poincaré–Dulac and Birkhoff normal forms. Besides new proofs of three Lyapunov theorems, we prove their generalization: if the Poincaré–Dulac normal form indicates the existence of a family of periodic solutions, then such a family really exists. We also present new proofs of Weinstein and Moser theorems about lower bounds for the number of families of periodic solutions; here, besides the normal forms, some topological tools are used, i.e., the Poincaré–Hopf formula and the Lusternik–Schnirelmann category on weighted projective spaces.}
}