Abstract
The correlation dimension D 2 is used to develop a method of classification for phase space orbits. D 2 depends only on the mutual distances of the orbit’s points, therefore the time development of the orbit is reflected in the time development of the correlation dimension approximants. It is shown, that this technique allows to investigate the dynamical properties of a phase space orbit, a classification of chaotic and regular orbits and a detection of sticky orbits.
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Benettin, G., Casati, D., Galgani, L., Giorgilli, A. and Sironi, L.: 1986, ‘Apparent fractal dimensions in conservative dynamical systems', Phys. Letters A 118(7), 325.
Dvorak, R., Contopoulos, G. and Efthymiopoulos, C.: 1997, ‘The Stickiness in Mappings and Dynamical Systems', in: Benest and Froeschlé (eds), Chaos dans les Systèmes gravitationelles p. 55.
Dvorak, R. and Sun, Y. S.: 1997, ‘The phase space of the extended Sitnikov problem', Celest. Mech. & Dyn. Astr. 67, 87–106.
Freistetter, F.: 2000, Fractal Dimensions in Mappings and Dynamical Systems with Applications in Celestial Mechanics, Diplomarbeit, Universität Wien.
Grassberger, P. and Procaccia, I.: 1983, ‘Characterization of strange attractors', Phys. Rev. Letters 50, 346–349.
Hentschl, H. and Procaccia, I.: 1983, ‘The infinite number of generalized dimensions of fractals and strange attractors', Physica 8D, 435.
Lewis, M. and Levison, H.: 1995, ‘Using the correlation exponent as a measure of chaos in celestial mechanics', Am. Astr. Soc. Meet. 187, 42.19.
Mandelbrot, B.: 1977, The Fractal Geometry of Nature, Freemann, New York.
Sitnikov, K.: 1960, ‘Existence of oscillatory motion for the three-body problem', Dokl. Akad. Nauk, USSR 133(2), 303.
Wodnar, K.: 1993, ‘The Original Sitnikov Article-New Insights’, in: R. Dvorak and J. Henrard (eds), Qualitative and Quantitative Behavior of Planetary Systems, Kluwer Academic Publishers, Dordrecht, pp. 99–101.
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Freistetter, F. Fractal Dimensions as Chaos Indicators. Celestial Mechanics and Dynamical Astronomy 78, 211–225 (2000). https://doi.org/10.1023/A:1011157505026
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DOI: https://doi.org/10.1023/A:1011157505026