Control of the global parameters of complex networks has been explored experimentally in a variety of contexts. Yet, the more difficult prospect of realizing arbitrary network architectures, especially analog physical networks that provide dynamical control of individual nodes and edges, has remained elusive. Given the vast hierarchy of time scales involved, it also proves challenging to measure a complex network’s full internal dynamics. These span from the fastest nodal dynamics to very slow epochs over which emergent global phenomena, including network synchronization and the manifestation of exotic steady states, eventually emerge. Here, we demonstrate an experimental system that satisfies these requirements. It is based upon modular, fully controllable, nonlinear radio frequency nanomechanical oscillators, designed to form the nodes of complex dynamical networks with edges of arbitrary topology. The dynamics of these oscillators and their surrounding network are analog and continuous-valued and can be fully interrogated in real time. They comprise a piezoelectric nanomechanical membrane resonator, which serves as the frequency-determining element within an electrical feedback circuit. This embodiment permits network interconnections entirely within the electrical domain and provides unprecedented node and edge control over a vast region of parameter space. Continuous measurement of the instantaneous amplitudes and phases of every constituent oscillator node are enabled, yielding full and detailed network data without reliance upon statistical quantities. We demonstrate the operation of this platform through the real-time capture of the dynamics of a three-node ring network as it evolves from the uncoupled state to full synchronization.

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完全可控的非线性纳米机械振荡器构成的复杂动力学网络

在各种情况下，已经通过实验探索了复杂网络全局参数的控制。然而，实现任意网络体系结构，尤其是提供对单个节点和边缘的动态控制的模拟物理网络的更困难的前景仍然难以捉摸。考虑到所涉及的时间尺度是巨大的层次结构，因此测量复杂网络的全部内部动态也具有挑战性。从跨越最快的节点动力学到非常缓慢的时期，最终出现的全局现象（包括网络同步和奇异稳态的表现）最终出现了。在这里，我们演示了满足这些要求的实验系统。它基于模块化的，完全可控的非线性射频纳米机械振荡器，设计以形成具有任意拓扑结构边缘的复杂动态网络的节点。这些振荡器及其周围网络的动力学是模拟值和连续值，可以实时进行全面查询。它们包括压电纳米机械膜谐振器，其用作电反馈电路中的频率确定元件。该实施例允许完全在电域内的网络互连，并在广阔的参数空间区域上提供空前的节点和边缘控制。连续测量瞬时振幅和相位 它们包括压电纳米机械膜谐振器，其用作电反馈电路中的频率确定元件。该实施例允许完全在电域内的网络互连，并在广阔的参数空间区域上提供空前的节点和边缘控制。连续测量瞬时振幅和相位 它们包括压电纳米机械膜谐振器，其用作电反馈电路中的频率确定元件。该实施例允许完全在电域内的网络互连，并在广阔的参数空间区域上提供空前的节点和边缘控制。连续测量瞬时振幅和相位每个组成的振荡器节点都被启用，无需依赖统计量即可产生完整而详细的网络数据。我们通过实时捕获三节点环形网络从未耦合状态到完全同步的动态过程来演示该平台的操作。