Bifurcations of straight-line oscillations

Details Bifurcations of straight-line oscillations Bifurcations of straight-line oscillations

Jul 1978

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1978a%26a....67..229m&link_type=abstract

Astronomy and Astrophysics, vol. 67, no. 2, July 1978, p. 229-240. Research supported by the Science Research Council.

Physics

4

Branching (Physics), Charged Particles, Geomagnetism, Oscillations, Pendulums, Perturbation Theory, Calculus Of Variations, Dynamic Stability, Equations Of Motion, Hill Method, Numerical Integration, Resonant Vibration

Scientific paper

The effect of varying the size of a perturbation that is inversely proportional to the square of distance on the structure of periodic solutions to the two-dimensional Stoermer and spring-pendulum problems is studied by considering straight-line oscillations on the symmetry axis. For many different sizes of the perturbation the stability of straight-line oscillations under 'transversal' perturbations which tend to take a moving particle out of the symmetry axis and make its motion two-dimensional is analyzed. The equations of motion and variation are simultaneously integrated numerically, and vertical stability parameters are calculated for each periodic oscillation. The results concerning bifurcations of families of periodic solutions are examined, one-dimensional motion is shown to bifurcate into two-dimensional motion, and qualitative predictions are made about the nature of the bifurcating two-dimensional oscillations. Attention is also given to the effect of perturbation on the stability of equilibrium, the case of resonant equilibrium, and the effect of perturbation on bifurcations.

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