Bifurcations in systems of three degrees of freedom

Details Bifurcations in systems of three degrees of freedom Bifurcations in systems of three degrees of freedom

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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