A continuous one-dimensional map with period three includes all periods. This raises the following question: Can we obtain any types of periodic orbits solely by learning three data points? We consider learning period three with random neural networks and report the universal property associated with it. We first show that the trained networks have a thermodynamic limit that depends on the choice of target data and network settings. Our analysis reveals that almost all learned periods are unstable and each network has its characteristic attractors (which can even be untrained ones). Here, we propose the concept of characteristic bifurcation expressing embeddable attractors intrinsic to the network, in which the target data points and the scale of the network weights function as bifurcation parameters. In conclusion, learning period three generates various attractors through characteristic bifurcation due to the stability change in latently existing numerous unstable periods of the system.

翻译：具有周期三的一维连续映射包含所有周期。这引出了以下问题：我们能否仅通过学习三个数据点就能获得任意类型的周期轨道？我们考虑用随机神经网络学习周期三，并报告与之相关的普适性质。我们首先证明，训练后的网络存在一个依赖于目标数据选择和网络设置的热力学极限。我们的分析表明，几乎所有学习到的周期都是不稳定的，每个网络都有其特征吸引子（甚至可以是未经训练的吸引子）。在此，我们提出特征分岔的概念，用以表达网络内禀的可嵌入吸引子，其中目标数据点和网络权重的尺度充当分岔参数。结论是，通过学习周期三，系统因潜在存在的众多不稳定周期的稳定性变化，通过特征分岔生成了各种吸引子。