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Order and Chaos in Dynamical Systems

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Abstract

We consider the characteristics of order and chaos in dynamical systems, with emphasis on the orbits in astronomical systems. Celestial mechanics deals with orbits in the solar system, which are mainly ordered. On the other hand the orbits of stars in galaxies were considered to be chaotic. However numerical experiments have shown that in general a system contains both ordered and chaotic orbits. Thus a new classification of dynamical systems has been established. We describe ordered and chaotic orbits in galaxies and in mappings. Some ordered orbits appear even in strongly perturbed systems. The transition from order to chaos is due to resonance overlapping. Then we describe some recent developments concerning order and chaos in the solar system and in galaxies. The outer spiral arms in strong barred galaxies are composed mainly of sticky chaotic orbits. Ordered and chaotic orbits appear also in Bohmian quantum mechanics. If the initial probability p is not equal to the square of the wave function |ψ|2, then in the case of ordered orbits p never approaches |ψ|2, while in the case of chaotic orbits p → |ψ|2 after a time interval called “quantum Nekhoroshev time”.

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  1. Research Center for Astronomy of the Academy of Athens, Soranou Efesiou 4, 115 27, Athens, Greece

    George Contopoulos

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  1. George Contopoulos
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Correspondence to George Contopoulos.

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Lecture held in the Seminario Matematico e Fisico on March 3, 2008

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Contopoulos, G. Order and Chaos in Dynamical Systems. Milan J. Math. 77, 101–126 (2009). https://doi.org/10.1007/s00032-009-0102-y

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  • DOI: https://doi.org/10.1007/s00032-009-0102-y

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