Research Article

Exotic states in a simple network of nanoelectromechanical oscillators

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Science  08 Mar 2019:
Vol. 363, Issue 6431, eaav7932
DOI: 10.1126/science.aav7932

Quickly exploring weakly coupled oscillators

Synchronizing oscillators have been useful models for exploring coupling in dynamic systems. However, many macroscopic platforms such as pendula evolve on slow time scales, which can limit the observation of states that emerge after many cycles. Matheny et al. fabricated a ring of eight nanoelectromechanical oscillators resonating at ∼2.2 megahertz with quality factors of ∼4000 that could be rapidly controlled and read out. Analysis of these large datasets revealed exotic synchronization states with complex dynamics and broken symmetries. Theoretical modeling showed that emergent higher-order interactions (such as biharmonic and next-nearest neighbor) stabilized complex dynamics, despite the network having weak nearest-neighbor coupling.

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Structured Abstract


A paramount contemporary scientific challenge is to understand and control networks. General studies of networks are essential to a variety of disciplines, including materials science, neuroscience, electrical engineering, and microbiology. To date, most studies are observational or “top-down,” relying on phenomenological models of nodal behavior deduced from data extracted from observations of the entire network. On the other hand, oscillator synchronization provides a popular “bottom-up” experimental paradigm for studies of network behavior. Synchronization occurs when a large number of networked oscillators tend to phase-lock and reach global consensus, despite the presence of internal disorder (such as differences in oscillator frequencies). However, the state of global consensus is not the only dynamical state manifested within networks of coupled oscillators. Recent work has discovered long-lived states that spontaneously break the underlying symmetries of the network, even when its constituent nodes are identical. Understanding the mechanisms that underlie these exotic symmetry-breaking dynamics will be of general benefit to network science and engineering. To enable experimental studies with unprecedented control and resolution, we developed an oscillator network based on nonlinear nanoelectromechanical systems (NEMS).


Complex oscillator dynamics emerge in a variety of settings. Previous experimental studies on symmetry breaking in oscillator networks have manifested such complex behavior only by implementing complicated coupling mechanisms designed for that purpose. By contrast, real-world networks are often dominated by simple coupling, so these previous experimental results cannot readily be generalized. Here, exotic states are seen to emerge within NEMS oscillator networks, just beyond the weak coupling limit in simple settings.


Using an array of coupled nonlinear NEMS oscillators, we observed spontaneous symmetry breaking in a simple and general network setting. These NEMS oscillator nodes were made to be nearly identical, and their fastest dynamical time scales were short enough to generate large datasets, permitting observation and statistical analyses of exotic, slowly emerging network phenomena. In addition, NEMS are very stable, so that transient effects within the network were able to relax within experimental time scales. We examined the network dynamics in detail throughout parameter space and showed experimentally that a simple network of oscillators can reproduce the predictions of theoretical models with explicit complex interactions. We fully explained these phenomena by applying a higher-order phase approximation to the full oscillator model. As strong evidence of the symmetry-breaking argument, we delineated the symmetry subgroups associated with each state.


Our results show that simple real-world oscillator networks display complex and exotic system states without the need for complex interactions. Our findings can be applied to observational studies of the behavior of natural or engineered oscillator networks. This work clearly elucidates how a 16-dimensional system, comprising eight magnitude-phase oscillators, collapses into synchronized states that evolve within lower-dimensional subspaces. Real-world networks based on our nonlinear NEMS oscillator platform will enable further insight into the mechanisms by which such behavior emerges in more complex topologies at even larger network scales.

States stabilized by emergent interactions.

(A) Oscillator network showing the physical connections and the emergent phase interactions. NN, nearest neighbor; NNN, next-nearest neighbor. (B) Stable fixed points for states with no magnitude variations. Colors correspond to colors of interactions from (A) required for stability. (C to E) Experimental heat maps of time-domain data for different combinations of phase differences (using the same data). Exotic states appear as bands in the plots (2-precess, WC-I, 2-TW-I, 2-TW-II).


Synchronization of oscillators, a phenomenon found in a wide variety of natural and engineered systems, is typically understood through a reduction to a first-order phase model with simplified dynamics. Here, by exploiting the precision and flexibility of nanoelectromechanical systems, we examined the dynamics of a ring of quasi-sinusoidal oscillators at and beyond first order. Beyond first order, we found exotic states of synchronization with highly complex dynamics, including weak chimeras, decoupled states, traveling waves, and inhomogeneous synchronized states. Through theory and experiment, we show that these exotic states rely on complex interactions emerging out of networks with simple linear nearest-neighbor coupling. This work provides insight into the dynamical richness of complex systems with weak nonlinearities and local interactions.

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