A Rigidity Property of Perturbations of n Identical Harmonic Oscillators
| Autor: | Massimo Villarini |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
Applied Mathematics
Perturbation (astronomy) Codimension 01 natural sciences Hartman–Grobman theorem 010101 applied mathematics Rigidity (electromagnetism) 0103 physical sciences Discrete Mathematics and Combinatorics Vector field 0101 mathematics 010301 acoustics Harmonic oscillator Mathematics Mathematical physics |
| Popis: | Let Xe:S2n-1→TS2n-1 be a smooth perturbation of X0, the vector field associated to the dynamical system defined by n identical uncoupled harmonic oscillators constrained to their 1-energy level. We are dealing with the case when any orbit of every Xe is closed: while in general is false that the vector fields of the perturbation are orbitally equivalent to the unperturbed X0 (Villarini in Ergod Theory Dyn Syst 39:1–32, 2019), we prove that this rigidity behaviour is indeed true if each Xe restricted to a codimension 2 sphere in S2n-1 is orbitally conjugated to a subsystem of X0 made by n-1 harmonic oscillators. In other words: to have a non-rigid, or truly non-linear, perturbation of X0 at least two harmonic oscillators must be destroyed by the perturbation. We use this rigidity result to prove a linearization theorem for real analytic multicentres. Finally we give an example of a real analytic perturbation of X0 showing discontinuous changing of integer invariants of the vector fields of the perturbation. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|