Abstract
We consider the problem of finding a normal form for differential equations in the neighbourhood of an equilibrium point, and produce general explicit estimates for both the normal form at a finite order and the remainder, using the method of Lie transforms. With such technique, the classical Poincaré-Dulac theorems are recovered, and the problem of the stability of a reversible system of coupled harmonic oscillators up to exponentially large times is discussed.
Riassunto
Si considera il problema di porre in forma normale un sistema di equazioni differenziali nell'intomo di un punto di equilibrio, e si danno in generale stime esplicite sia per la forma normale troncata ad un ordine finito che per i resti. Si fa uso dell'algoritmo della trasformata di Lie. Con questo metodo si riottengono i teoremi classici di Poincaré-Dulac, e si discute il problema della stabilità per tempi esponenzialmente lunghi di un sistema reversibile di oscillatori armonici accoppiati.
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Giorgilli, A., Posilicano, A. Estimates for normal forms of differential equations near an equilibrium point. Z. angew. Math. Phys. 39, 713–732 (1988). https://doi.org/10.1007/BF00948732
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DOI: https://doi.org/10.1007/BF00948732