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This paper explores the interactions of both phase and amplitude in a network of N MEMS-Colpitts oscillators that are resistively coupled. The numerical simulations of the extensive networks of oscillators, required for emerging applications such as clock synchronization and neuromorphic computing, become computationally prohibitive as the number of oscillators increases. This complicates the design and evaluation of such systems, as understanding the effects of changes in coupling and design parameters can require many simulations. This study employs the method of multiple scales (MS) in combination with the harmonic balance method to convert the coupled differential equations governing the system of oscillators into a set of nonlinear evolution equations for the amplitude and phase of the oscillators. The amplitude and phase evolve on a timescale that is slow, commensurate with the damping in the system, compared with the fast timescale of the oscillation frequencies. The approach used in this study to describe the amplitude and phase dynamics offers significant computational efficiency (gains of 10× to 50× are shown) compared to direct numerical integration while maintaining an accurate representation of the response. The results of the presented simulations demonstrate the effect of coupling strength on the dynamics of the network, accounting for both phase and amplitude dynamics.

Complex structural systems deployed for aerospace, civil, or mechanical applications must operate reliably under varying operational conditions. Structural health monitoring (SHM) systems help ensure the reliability of these systems by providing continuous monitoring of the state of the structure. SHM relies on synthesizing measured data with a predictive model to make informed decisions about structural states. However, these models—which may be thought of as a form of a digital twin—need to be updated continuously as structural changes (e.g., due to damage) arise. We propose an uncertainty-aware machine learning model that enforces distance preservation of the original input state space and then encodes a distance-aware mechanism via a Gaussian process (GP) kernel. The proposed approach leverages the spectral-normalized neural GP algorithm to combine the flexibility of neural networks with the advantages of GP, subjected to structure-preserving constraints, to produce an uncertainty-aware model. This model is used to detect domain shift due to structural changes that cannot be observed directly because they may be spatially isolated (e.g., inside a joint or localized damage). This work leverages detection theory to detect domain shift systematically given statistical features of the prediction variance produced by the model. The proposed approach is demonstrated on a nonlinear structure being subjected to damage conditions. It is shown that the proposed approach is able to rely on distances of the transformed input state space to predict increased variance in shifted domains while being robust to normative changes.