Duke University, Department of Physics and Center for Nonlinear and Complex Systems, Durham, NC 27708, USA.
Philos Trans A Math Phys Eng Sci. 2010 Jan 28;368(1911):495-513. doi: 10.1098/rsta.2009.0235.
We undertake a systematic study of the dynamics of Boolean networks to determine the origin of chaos observed in recent experiments. Networks with nodes consisting of ideal logic gates are known to display either steady states, periodic behaviour or an ultraviolet catastrophe where the number of logic-transition events circulating in the network per unit time grows as a power law. In an experiment, the non-ideal behaviour of the logic gates prevents the ultraviolet catastrophe and may lead to deterministic chaos. We identify certain non-ideal features of real logic gates that enable chaos in experimental networks. We find that short-pulse rejection and asymmetry between the logic states tend to engender periodic behaviour, at least for the simplest networks. On the other hand, we find that a memory effect termed 'degradation' can generate chaos. Our results strongly suggest that deterministic chaos can be expected in a large class of experimental Boolean-like networks. Such devices may find application in a variety of technologies requiring fast complex waveforms or flat power spectra, and can be used as a test-bed for fundamental studies of real-world Boolean-like networks.
我们对布尔网络的动力学进行了系统的研究，以确定最近实验中观察到的混沌的起源。众所周知，由理想逻辑门组成的网络要么呈现稳态，要么呈现周期性行为，要么呈现紫外灾难，即网络中每单位时间循环的逻辑转换事件数量呈幂律增长。在实验中，逻辑门的非理想行为阻止了紫外灾难，并可能导致确定性混沌。我们确定了现实逻辑门的某些非理想特征，这些特征使实验网络中的混沌成为可能。我们发现，短脉冲抑制和逻辑状态之间的不对称性往往会产生周期性行为，至少对于最简单的网络是这样。另一方面，我们发现一种称为“降级”的记忆效应可以产生混沌。我们的结果强烈表明，在一大类实验性布尔型网络中可以预期到确定性混沌。这种器件可以在许多需要快速复杂波形或平坦功率谱的技术中找到应用，并可用作研究现实世界布尔型网络的基础的试验台。
