Eur. Phys. J. Plus (2013) 128: 21
https://doi.org/10.1140/epjp/i2013-13021-5

Regular Article

The symmetric 1:2 resonance

¹ Dipartimento di Scienze di Base e Applicate per l’Ingegneria, Università di Roma “la Sapienza”, Via Antonio Scarpa, 16, 00161, Roma, Italy
² Dipartimento di Fisica, Università di Roma “Tor Vergata” and INFN, Sezione di Roma Tor Vergata, Via della Ricerca Scientifica, 1, 00133, Rome, Italy

^* e-mail: pucacco@roma2.infn.it

Received: 1 August 2012
Accepted: 22 January 2013
Published online: 21 February 2013

Abstract

This paper illustrates the application of the Lie transform normal-form theory to the construction of the 1:2 resonant normal form corresponding to a wide class of natural Hamiltonian systems. We show how to compute the bifurcations of the main periodic orbits in a potential with double-reflection symmetries. The stability analysis of the normal modes and of the periodic orbits in general position allows us to get overall information on the phase-space structure of systems in which this resonance is dominating. As an example we apply these results to a class of models useful as galactic potentials.