# Complex vs. rigid behavior of chains of identical harmonic oscillators

Università degli Studi Trento

38123 Povo (TN)

Tel +39 04 61/281508-1625-1701-3786

dept.math [at] unitn.it

**Luogo**: Povo Zero, via Sommarive 14 (Povo) - Sala Seminari "-1"**Ore**: 15:00

**Relatore**:

- Massimo Villarini (Università di Modena e Reggio Emilia)

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**Abstract**:

We will be interested in dynamical systems which are isoenergetic, periodic perturbation of n identical uncoupled harmonic oscillators: isoenergetic means that all the vector fields of the perturbation share the same energy level 1, a closed manifold, and are studied there. Periodic means that all the orbits of any perturbation are closed. Systems all of whose orbits are closed are named oscillators. We will consider a dichotomy between complex, or truly non-linear, behaviour, and rigid behaviour of isoenergetic periodic perturbations of oscillators. The prototype of rigid behaviour is Poincare'-Lyapunov Centre Theorem: a 1-parameter family of non-degenerate analytic centers in the plane are orbitally conjugated through a smooth isotopy to the linear center. We will show that the natural generalization of this theorem is false (M.V. Ergodic th. dyn. syst. 2019), by giving examples of non-linearizable non-degenerate multi-centers in dimension 8 of the phase space. On the other hand, we will show that such examples are nearly optimal in the sense that perturbations keeping fixed all but one harmonic oscillators must be rigid (M.V. submitted). If possible, we will give applications to the classification of small Seifert manifolds and to the study of discontinuous changing of integer-valued invariants of oscillators.

**Referente**: Marco Sabatini