Normal forms, invariant manifolds and Lyapunov theorems

Henryk Żołądek; Henryk Żołądek

doi:10.3934/cam.2023016

Communications in Analysis and Mechanics

2023, Volume 15, Issue 2: 300-341. doi: 10.3934/cam.2023016

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Research article

Normal forms, invariant manifolds and Lyapunov theorems

Henryk Żołądek ^,

Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warsaw, Poland

Received: 23 November 2022 Revised: 11 May 2023 Accepted: 22 May 2023 Published: 26 June 2023
Primary 05C38, 15A15; Secondary 05A15, 15A18

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.
- Lyapunov theorems,
- Poincaré–Dulac normal form,
- invariant manifolds
Citation: Henryk Żołądek. Normal forms, invariant manifolds and Lyapunov theorems[J]. Communications in Analysis and Mechanics, 2023, 15(2): 300-341. doi: 10.3934/cam.2023016

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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.

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