2014/02/01 by Dario Bambusi, Antonio Giorgilli, Bambusi, Dario +5
Computer Science · Engineering · Mathematics · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #Nonlinear Dynamics and Pattern Formation #Quantum chaos and dynamical systems #Stability and Controllability of Differential Equations #math.DS
paper · pdf · doi:10.48550/arxiv.1402.0076
arxiv created 2014/02/01 · openalex publication_date 2014/02/01 · arxiv updated 2014/02/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider a system in which some high frequency harmonic oscillators are coupled with a slow system. We prove that up to very long times the energy of the high frequency system changes only by a small amount. The result we obtain is completely independent of the resonance relations among the frequencies of the fast system. More in detail, denote by ε-1 the smallest high frequency. In the first part of the paper we apply the main result of [BG93] to prove almost conservation of the energy of the high frequency system over times exponentially long with ε-1/n (n being the number of fast oscillators). In the second part of the paper we give e new self-contained proof of a similar result which however is valid only over times of order ε-N with an arbitrary N. Such a second result is very similar to the main result of the paper [GHL13], which actually was the paper which stimulated our work.