周期轨道的李雅普诺夫指数与邻域
The Lyapunov exponents and the neighbourhood of periodic orbits
摘要：
我们证明了周期轨道的李雅普诺夫指数可以很容易地从monodromy矩阵的特征值得到。结果表明：简单稳定周期轨道的李雅普诺夫指数均为零，简单不稳定周期轨道只有一个正的李雅普诺夫指数，双不稳定周期轨道有两个不同的正的李雅普诺夫指数，复不稳定周期轨道的两个正的李雅普诺夫指数相等.我们提出了一个数值例子周期轨道在一个现实的银河势。此外，中心流形定理使我们能够表明，稳定的，简单的不稳定和双不稳定的周期轨道是家庭的母亲，分别定期，部分和完全混乱的轨道在其附近。
英文摘要：
We show that the Lyapunov exponents of a periodic orbit can be easily obtained from the eigenvalues of the monodromy matrix. It turns out that the Lyapunov exponents of simply stable periodic orbits are all zero, simply unstable periodic orbits have only one positive Lyapunov exponent, doubly unstable periodic orbits have two different positive Lyapunov exponents and the two positive Lyapunov exponents of complex unstable periodic orbits are equal. We present a numerical example for periodic orbits in a realistic galactic potential. Moreover, the center manifold theorem allowed us to show that stable, simply unstable and doubly unstable periodic orbits are the mothers of families of, respectively, regular, partially and fully chaotic orbits in their neighbourhood.
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