Mathematics
Scientific paper
Jan 1986
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1986cemec..38....1c&link_type=abstract
Celestial Mechanics (ISSN 0008-8714), vol. 38, Jan. 1986, p. 1-22.
Mathematics
26
Astrodynamics, Branching (Mathematics), Degrees Of Freedom, Galactic Evolution, Orbital Mechanics, Elliptical Galaxies, Many Body Problem, Periodic Variations, Stability, Topology
Scientific paper
The bifurcations of families of double and quadruple period orbits in a simple Hamiltonian system of three degrees of freedom are studied. The bifurcations are either 'simple' or 'double', depending on whether a stability curve crosses or is tangent to the axis b = -2. The double period families bifurcate from simple families of periodic orbits. 'Existence diagrams' are constructed to show where any given family exists in the control space, where it is stable, simply unstable, doubly unstable, or complex unstable. 'Stability diagrams' that give the stability parameters b(1) and b(2) as functions of epsilon (for constant eta), or of eta (for constant epsilon) are also constructed. The quadruple period orbits are generated either from double period orbits, or directly from simple period orbits (at double bifurcations). Several rules about the various types of bifurcations are derived. The most important phenomenon is the 'collision of bifurcations'. At any such collision of bifurcations the interconnections between the various families change as well as the general character of the dynamical system.
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