Abstract
Among the new phenomena that appear in Hamiltonian systems of three degrees of freedom (that are not found in systems of two degrees of freedom) are “complex instability” and “collisions of bifurcations”. We report some results from a study of simple perturbed three dimensional galactic systems. We find the stability types and bifurcations of the main families of periodic orbits in such systems. Complex instability is a general phenomenon, that introduces stochasticity without bifurcations. It seems that in general 3-D systems there are no infinite sequences of period doubling bifurcations. Collisions of bifurcations lead to a change of the interconnections between the various families of periodic orbits. Thus they produce qualitative differences between the various systems. The most important collisions of bifurcations occur in limiting cases, when a system is reduced to a two dimensional system plus on an independent oscillation. Thus the study of general 3-D systems is simplified considerably.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Benettin, G., Cercignani, C., Galgani, L. and Giorgilli, A.: 1980, Lett. Nuovo Cim. 28, 1.
Bountis, T.: 1981, Physica 3D, 577.
Broucke, R.: 1969, Amer. Inst. Aeronautics and Astronautics J. 7, 1003.
Contopoulos, G.: 1970, Astron. J. 75, 96, 108.
Contopoulos, G.: 1983a, Astron. Astrophys. 117, 89.
Contopoulos, G.: 1983b, Lett. Nuovo Cim. 37, 149.
Contopoulos, G.: 1983c, Astrophys. J. 275, 511.
Contopoulos, G.: 1983d, Lett. Nuovo Cim. 38, 257.
Contopoulos, G. and Barbanis, B.: 1985, in preparation.
Contopoulos, G. and Magnenat, P.: 1984, preprint.
Contopoulos, G. and Pinotsis, A.: 1984, Astron. Astrophysics, 133, 49.
Contopoulos, G. and Zikides, M.: 1980, Astron. Astrophysics, 90, 198.
Coullet, P. and Tresser, C.: 1978, J. Phys. 5C, 25.
Feigenbaum, M.J.: 1978, J. Stat. Phys. 19, 25.
Froeschlé, C: 1970, Astron. Astvophys. 4, 115.
Greene, J.M., MacKay, R.S., Vivaldi, F. and Feigenbaum, M. J.: 1981, Physica 3D, 468,
Hadjidemetriou, J.D.: 1975, Celes. Mech. 12, 255.
Heggie, D.S.: 1983, Celes. Mech. 29, 207.
Krein, M.G.: 1950, Doklady Acad. Nauk USSR 73, 445.
Moser, J.K.: 1958, Commun. Pure Appl. Math. 11, 81.
Widom, M. and Kadanoff, L.P.: 1982, Physica 5D, 287.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1985 D. Reidel Publishing Company
About this chapter
Cite this chapter
Contopoulos, G. (1985). Bifurcations and Stability in Three-Dimensional Systems. In: Szebehely, V.G. (eds) Stability of the Solar System and Its Minor Natural and Artificial Bodies. NATO ASI Series, vol 154. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-5398-7_8
Download citation
DOI: https://doi.org/10.1007/978-94-009-5398-7_8
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-8883-1
Online ISBN: 978-94-009-5398-7
eBook Packages: Springer Book Archive