Mathematics – Dynamical Systems
Scientific paper
Sep 1991
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1991cemda..51..227d&link_type=abstract
Celestial Mechanics and Dynamical Astronomy (ISSN 0923-2958), vol. 51, no. 3, 1991, p. 227-250. Research supported by DARPA.
Mathematics
Dynamical Systems
17
Dynamical Systems, Elliptical Orbits, Hamiltonian Functions, Harmonic Oscillation, Lissajous Figures, Nonlinear Systems, Normalizing, Orbital Mechanics, Perturbation Theory, Transformations (Mathematics)
Scientific paper
Normalization of a perturbed elliptic oscillator, when executed in Lissajous variables, amounts to averaging over the elliptic anomaly. The reduced Lissajous variables constitute a system of cylindrical coordinates over the orbital spheres of constant energy, but the pole-like singularities are removed by reverting to the subjacent Hopf coordinates. The two-parameter coupling that is a polynomial of degree for admitting the symmetries of the square is studied in detail. It is shown that the normalized elliptic oscillator in that case behaves everywhere in the parameter plane like a rigid body in free rotation about a fixed point, and that it passes through butterfly bifurcations wherever its phase flow admits non-isolated equilibria.
Deprit Andre
Elipe Antonio
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