Fig. 4: High sensitivity near the PhT cusp singularity.
From: Higher-order singularities in phase-tracked electromechanical oscillators
a The PhT frequency \({\omega }_{{{{{{{{\rm{d}}}}}}}}}^{*}\) as a function of angular velocity Ω and degeneracy condition Δω. The contours of Ω = Ω0 (dark red curve) and Δω = 0 (green curves) portray the sharp variation of \({\omega }_{{{{{{{{\rm{d}}}}}}}}}^{*}\) near the singularity nexus (blue point). b The PhT frequency at the critical angular velocity \({\omega }_{{{{{{{{\rm{d}}}}}}}}}^{*}({\Omega }_{0})\) and its shift from ω1, \(\delta {\omega }_{{{{{{{{\rm{X}}}}}}}}}={\omega }_{{{{{{{{\rm{d}}}}}}}}}^{*}({\Omega }_{0})-{\omega }_{1}\) as functions of Δω. Here, ω0 represents ω1 at Δω = 0. In the range of −0.25γ ≤ Δω ≤ 0.25γ, Frequency output δωX decreases monotonically with Δω. c Frequency output δωX near the singularity nexus versus the natural-frequency perturbation ϵ = Δω from both simulation (red solid curve) and experiment (red circles). The eigenfrequency splits near an EP (blue dashed curve) and a DP (black dot-dashed curve) are also simulated. Error bars are the standard deviation. d Logarithmic plot of the absolute data in c. The PhT cusp singularity has a cubic-root output, providing higher sensitivity compared to the EP and DP.
