Sebastian FERRER

Departamento de Matemática Aplicada, Universidad de Murcia,
Avenida Alfonso XIII, 42
30203 Cartagena, Murcia, Espagne
ferrer@plc.um.es

Reduction of Harmonic Oscillators in 1--1--1 Resonance: Relative Equilibria and Bifurcations of Quartic Perturbations

Abstract: This communication deals with the dynamics of a Hamiltonian system with three equal modes around an elliptic equilibrium. The Taylor expansion is truncated at fourth order. The process of the reduction is performed in two steps. First, a regular reduction, due to the oscillator symmetry, brings the system to the orbit space $\mbox{\bf CP}^2$. In systems with axial symmetry a singular reduction can be carried out. This permits to make a global study of a one--degree--of--freedom system which lies on a double reduced space. As an application we analyze a parametric family modelling an elliptical galaxy, showing its relative equilibria, bifurcations and stability.