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Normal forms, invariant manifolds and Lyapunov theorems

Communications in Analysis and Mechanics, ISSN: 2836-3310, Vol: 15, Issue: 2, Page: 300-341
2023
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Article Description

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.

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