Periodic orbits and escapes in dynamical systems

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• 作者：George Contopoulos (1) gcontop@academyofathens.gr
  Mirella Harsoula (1) mharsoul@academyofathens.gr
  Georgios Lukes-Gerakopoulos (12) gglukes@gmail.com
• 关键词：Hamiltonian systems &#8211 ; Periodic orbits &#8211 ; Manko ; Novikov metric
• 刊名：Celestial Mechanics and Dynamical Astronomy
• 出版年：2012
• 出版时间：July 2012
• 年：2012
• 卷：113
• 期：3
• 页码：255-278
• 全文大小：1.6 MB
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• 作者单位：1. Research Center for Astronomy and Applied Mathematics, Academy of Athens, Soranou Efesiou 4, 11527 Athens, Greece2. Theoretical Physics Institute, University of Jena, 07743 Jena, Germany
• ISSN：1572-9478

文摘

We study the periodic orbits and the escapes in two different dynamical systems, namely (1) a classical system of two coupled oscillators, and (2) the Manko-Novikov metric which is a perturbation of the Kerr metric (a general relativistic system). We find their simple periodic orbits, their characteristics and their stability. Then we find their ordered and chaotic domains. As the energy goes beyond the escape energy, most chaotic orbits escape. In the first case we consider escapes to infinity, while in the second case we emphasize escapes to the central “bumpy” black hole. When the energy reaches its escape value, a particular family of periodic orbits reaches an infinite period and then the family disappears (the orbit escapes). As this family approaches termination it undergoes an infinity of equal period and double period bifurcations at transitions from stability to instability and vice versa. The bifurcating families continue to exist beyond the escape energy. We study the forms of the phase space for various energies, and the statistics of the chaotic and escaping orbits. The proportion of these orbits increases abruptly as the energy goes beyond the escape energy.