2013/03/29 by Antonio Giorgilli, Giorgilli, Antonio
Computer Science · Mathematics · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #Nonlinear Dynamics and Pattern Formation #Numerical methods for differential equations #Quantum chaos and dynamical systems #math.DS
paper · pdf · doi:10.48550/arxiv.1303.7322
arxiv created 2013/03/29 · openalex publication_date 2013/03/29 · arxiv updated 2013/04/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
It is shown that a Hamiltonian system in the neighbourhood of an equilibrium may be given a special normal form in case the eigenvalues of the linearized system satisfy non--resonance conditions of Melnikov's type. The normal form possesses a two dimensional (local) invariant manifold on which the solutions are known. If the eigenvalue is pure imaginary then these solutions are the natural continuation of a normal mode of the linear system. The latter result was first proved by Lyapounov. The present paper completes Lyapounov's result in that the convergence of the transformation of the Hamiltonian to a normal form is proven and the condition that the eigenvalues be pure imaginary is removed.