Singularities occur when the parameter space of a system folds, or comes together, to a single point. These occur in several dynamical physical systems and therefore are key to our understanding of many phenomena, such as gravitational singularities in black holes and optical catastrophes. Recently, high-order phase singularities have been observed within micro-electro-mechanical systems (MEMS)-based devices, as they offer a unique, dynamically rich platform to observe and study singularities. However, existing studies required complex structures or carefully designed geometries in order to facilitate the observation. Here, we show the observation and characterization of phase singularities within a far simpler, generic, silicon-based double-ended-tuning-fork-type device. This suggests that the presented approach, utilizing the nonlinear dynamics of parametric modulation-based operation, is universally applicable to resonant MEMS-based devices. The device used here has a smaller footprint and less complex electrode layout for actuation, sensing, and tuning compared to previous work. Characterization of the phase response of the device-under-test presents the emergent cusp and pitchfork singularities, with a cusp catastrophe observed in the parameter space as both singularities coalesce to a single point—known as the nexus. This work represents a step toward the widespread observation and consequently application of higher-order phase singularities, which is promising for a range of applications, for example, mechanical-based computing, non-reciprocal devices and inertial sensors, and improved sensor performance, in particular in terms of sensitivity.
Uka et al. (2026) studied this question.