三维系统共振周期轨道的延拓与稳定性推导
Continuation and stability deduction of resonant periodic orbits in three dimensional systems
摘要：
在少自由度动力系统中，周期解构成相空间的主干，其稳定性的确定和计算对于理解全局动力学是至关重要的。在本文中，我们研究了经典的三维三体问题，并利用其动力学来评估太阳系外系统的长期演化。我们计算周期轨道，对应于精确的共振运动，并确定其线性稳定性。通过计算动力学稳定性的映射，我们表明，稳定的周期轨道包围在相空间中有规律的运动，即使在两个以上的自由度的系统，而混沌是明显接近不稳定的。因此，稳定周期轨道族实际上是相空间中稳定域的主干。
英文摘要：
In dynamical systems of few degrees of freedom, periodic solutions consist the backbone of the phase space and the determination and computation of their stability is crucial for understanding the global dynamics. In this paper we study the classical three body problem in three dimensions and use its dynamics to assess the long-term evolution of extrasolar systems. We compute periodic orbits, which correspond to exact resonant motion, and determine their linear stability. By computing maps of dynamical stability we show that stable periodic orbits are surrounded in phase space with regular motion even in systems with more than two degrees of freedom, while chaos is apparent close to unstable ones. Therefore, families of stable periodic orbits, indeed, consist backbones of the stability domains in phase space.
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