Faculty of Economics, Rissho University, 4-2-16 Osaki, Shinagawa-ku, Tokyo 141-8602, Japan.
Graduate School of Commerce and Management, Hitotsubashi University, 2-1 Naka, Kunitachi, Tokyo 186-8601, Japan.
Chaos. 2017 Aug;27(8):081103. doi: 10.1063/1.4995043.
A chaotic motion can be considered an irregular transition process near unstable periodic orbits embedded densely in a chaotic set. Therefore, unstable periodic orbits have been used to characterize properties of chaos. Statistical quantities of chaos such as natural measures and fractal dimensions can be determined in terms of unstable periodic orbits. Unstable periodic orbits that can provide good approximations to averaged quantities of chaos or turbulence are also known to exist. However, it is not clear what type of unstable periodic orbits can capture them. In this paper, a model for an irregular transition process of a chaotic motion among unstable periodic orbits as nodes is constructed by using a network. We show that unstable periodic orbits which have lots of links in the network tend to capture time averaged properties of chaos. A scale-free property of the degree distribution is also observed.
混沌运动可以被认为是在密集嵌入混沌集中的不稳定周期轨道附近的不规则过渡过程。因此，不稳定周期轨道被用于描述混沌的特性。可以根据不稳定周期轨道来确定混沌的统计量，如自然测度和分形维数。也存在能够提供混沌或湍流的平均量的良好逼近的不稳定周期轨道。然而，尚不清楚哪种类型的不稳定周期轨道可以捕获它们。在本文中，通过使用网络构建了一个在不稳定周期轨道作为节点的混沌运动的不规则过渡过程的模型。我们表明，在网络中具有大量链接的不稳定周期轨道往往会捕获混沌的时间平均特性。还观察到了度分布的无标度性质。
