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Bifurcations of Periodic Orbits in Axisymmetric Scalefree Potentials a
Author(s) -
HUNTER C.,
TERZIĆ BALŠA,
BURNS AMY M.,
PORCHIA DONALD,
ZINK CHRIS
Publication year - 1998
Publication title -
annals of the new york academy of sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.712
H-Index - 248
eISSN - 1749-6632
pISSN - 0077-8923
DOI - 10.1111/j.1749-6632.1998.tb11250.x
Subject(s) - bifurcation , instability , physics , plane (geometry) , pitchfork bifurcation , orbit (dynamics) , classical mechanics , perpendicular , periodic orbits , rotational symmetry , period (music) , mechanics , geometry , bifurcation theory , mathematics , quantum mechanics , nonlinear system , aerospace engineering , acoustics , engineering
We study orbits in potentials with central cusps, emphasizing the spheriodal equidensity (SED) potentials generated by mass distributions with spheroidal equidensity surfaces. The most prominent bifurcations are those related to 1:1 and 4:3 resonances between radial motions and motions perpendicular to the central plane. We find that 1:1 resonances can cause the thin tube orbit, as well as the equatorial plane orbit, to become unstable. We concentrate on period‐tripling bifurcations because they appear to be the least understood. We study them via a class of analytic maps. This study suggests that stable period‐three orbits generally arise de novo in stable and unstable pairs via a turning‐point bifurcation, and not through a bifurcation from the thin tube at a 120° rotation angle. The stable period‐three orbits typically have only a short span of existence before becoming unstable to a period‐doubling instability through a supercritical pitchfork bifurcation.