A Rigidity Property of Perturbations of n Identical Harmonic Oscillators
- PDF / 400,710 Bytes
- 28 Pages / 439.37 x 666.142 pts Page_size
- 9 Downloads / 181 Views
A Rigidity Property of Perturbations of n Identical Harmonic Oscillators Massimo Villarini1 Received: 26 June 2020 / Accepted: 9 September 2020 © Springer Nature Switzerland AG 2020
Abstract Let X ε : S 2n−1 → T S 2n−1 be a smooth perturbation of X 0 , 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 X ε is closed: while in general is false that the vector fields of the perturbation are orbitally equivalent to the unperturbed X 0 (Villarini in Ergod Theory Dyn Syst 39:1–32, 2019), we prove that this rigidity behaviour is indeed true if each X ε restricted to a codimension 2 sphere in S 2n−1 is orbitally conjugated to a subsystem of X 0 made by n − 1 harmonic oscillators. In other words: to have a non-rigid, or truly non-linear, perturbation of X 0 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 X 0 showing discontinuous changing of integer invariants of the vector fields of the perturbation.
1 Introduction Dynamical systems whose orbits are all closed, for short oscillators, are often interesting examples of special mechanical systems, whose pertubation theory is fundamental both for theoretical and applied studies. Examples range from centres in planar dynamics to coupled harmonic oscillators in Lagrange’s discrete model of an oscillating string. Perturbing an oscillator within the class of oscillators, i.e. keeping the condition that the perturbed system’s orbits are all closed, results in a fundamental dichotomy. • Rigid behaviour: no dynamical change is possible perturbing an oscillator, i.e. all the perturbed oscillators are orbitally equivalent to the unperturbed one
B 1
Massimo Villarini [email protected] Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Universitá di Modena e Reggio Emilia, via Campi 213/b, 41100 Modena, Italy 0123456789().: V,-vol
86
Page 2 of 28
M. Villarini
• Complex or truly non-linear behaviour: some analytic characteristics of the oscillators in the perturbations change suddenly , e.g. the period function of the oscillators is locally unbounded near the unperturbed oscillator [13], or an integer invariant of the oscillators (for instance, the Euler number of the circle bundle associated to it) changes discontinuously. Of course, in these cases orbital equivalence of the vector fields of the perturbation with the unperturbed model is impossible. The prototype of rigid behaviour is the Poincaré Centre Theorem, [9], which, in a slightly generalized form, states that a 1-parameter family of real analytic nondegenerate centres in the plane are orbitally equivalent, through a smooth isotopy, to one element of such family. On the other hand, the search for complex behaviour of perturbations of oscillators dates back at least to the Periodic
Data Loading...
