Celestial mechanics

, Volume 38, Issue 1, pp 1–22 | Cite as

Bifurcations in systems of three degrees of freedom

  • G. Contopoulos


We study the bifurcations of families of double and quadruple period orbits in a simple Hamiltonian system of three degrees of freedom. The bifurcations are either ‘simple’ or ‘double’, depending on whether a stability curve crosses or is tangent to the axis b=−2. We have also generation of a new family whenever a given family has a maximum or minimum ε or ∌.

The double period families bifurcate from simple families of periodic orbits. We construct ‘existence diagrams’ to show where any given family exists in the control space (ε, η) and where it is stable (S), simply unstable (U), doubly unstable (DU), or complex unstable (Δ), We construct also ‘stability diagrams’ that give the stability parameters b1 and b2 as functions of ε (for constant η), or of η (for constant ε).

The quadruple period orbits are generated either from double period orbits, or directly from simple period orbits (at double bifurcations). We derive several rules about the various types of bifurcations. The most important phenomenon is the ‘collision of bifurcations’. At any such collision of bifurcations the interconnections between the various families change and the general character of the dynamical system changes.


Period Orbit Hamiltonian System General Character Stability Parameter Double Period 
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Copyright information

© D. Reidel Publishing Company 1986

Authors and Affiliations

  • G. Contopoulos
    • 1
  1. 1.European Southern ObservatoryGarchingGermany

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