VISUALIZING RELATIVE EQUILIBRIA AND BIFURCATIONS BY PAINTING HAMILTONIANS ON PERSONAL COMPUTERS
P. PASCUA, Jesús Rubio, A. Viartola, S. Ferrer · International Journal of Bifurcation and Chaos · 1996
In this paper, we present a graphical tool running on a PC for studies of integrable systems of one degree of freedom with several parameters, in particular for perturbed Hamiltonian systems with symmetries whose orbit space is the two-dimensional sphere S2 after they are reduced by normalization. The pioneering work was performed by Coffey et al. [1990], who developed a graphical method on a Connection Machine in order to paint the Hamiltonian function ℋ* over phase space, instead of numerical integrations of the corresponding differential equations. We have pursued their idea and the procedure proposed here renders pictures at the rate of one picture per minute. For research or teaching activities, the above mentioned method may be used in two directions: (a) in the stability analysis of equilibria and their bifurcations in parameter space; (b) for the specification of initial conditions and the corresponding qualitative long term global dynamics, in order to make proper choices for numerical studies of the full problem. As examples of how to interact with the program, and of the different role played by physical and dynamical parameters, we consider the dynamics of the Laplace–Runge–Lenz vector in two perturbed Keplerian systems of recent interest: the generalized Van der Waals effect in the hydrogen atom, and Harrington’s model for triple stellar systems.