Research Article

Exotic states in a simple network of nanoelectromechanical oscillators

See allHide authors and affiliations

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

  • Fig. 1 Experimental setup and representative data.

    (A) Each oscillator node rotates on a limit cycle (orange on black circle), with coupling edge to nearest neighbors (springs). (B) An example of a synchronization transition. The real-time network dynamics shows the emergence of synchronization as the coupling is initialized. It depicts the oscillator magnitudes aj and the phase differences between nearest neighbors Δj = (ϕj+1 – ϕj) mod(2π). (C) Phases of the eight oscillators on four 2D tori before and after synchronization.

  • Fig. 2 Experimental data on splay states.

    (A) Frequencies Ω(k) of the unity-magnitude synchronized states k ∈ {0, 1, … 7}, for which the neighboring phase differences are Δ(k) = 2πk/8. At α = 0.074 and β = 1, the data (green spheres) agree well with the fit from the theoretical predictions (purple line). (B) Probabilities of finding the antiphase state from 300 experimental trials for each point with random initial phases, as a function of the phase-lag parameter γ from Eq. 4 (green error bars). We simulated the full set of equations (Eq. 2) with a stochastic white noise term of strength D added to both magnitude and phase. (C) Experimental data on a perturbation of the antiphase state. We perturbed the natural frequency of a single oscillator (oscillator 1) by an amount δω1 and examined its magnitude a1 and the deviation of the mean frequency of the ring δω1/8, hence δΩ(k=4) – δω1/8 is shown. We numerically determined the effect of the perturbation; the plot shows the result from the full model (solid lines) and the Kuramoto-Sakaguchi model (dashed lines). Horizontal error bars are given by digital step resolution of the circuit used for frequency control. Vertical error for the oscillator frequencies is given by the average resolvable frequency over eight oscillators within a ~100-ms time window.

  • Fig. 3 Inhomogeneous synchronization.

    (A) Oscillator nearest-neighbor phase differences showing pattern formation. At t ≈ 0.53 s, we suddenly shifted the coupling from a value of β = 1 to β = 0.28. (B) Phase differences of the inhomogeneous state as the system is quenched from the data in (A). The color corresponds to the time axis, with more blue representing later times. (C) Numerical data on the state space of the inhomogeneous synchronization. The plot shows the magnitude of the Kuramoto order parameter Zip characterizing the steady state found in these simulations. (D) Data showing the overlap of the uniform and inhomogeneous synchronized states as a function of coupling parameter β at α = 0.09. The plot shows the mean (in trial number) of the absolute value of the order parameter with a threshold for each trial given by Δ¯j < 0.02 and Mn < 0.94 (blue stars) or Mn > 0.94 (maroon triangles). A clear separation of the states appeared when this threshold was applied. The experimental results are compared with the numerical simulations of Eq. 2 (black dots) and Eq. 3 (green circles).

  • Fig. 4 Decoupled states.

    (A) Phase differences, frequencies, mutual information matrix, and a snapshot of the 2-precess state (see text). Note that the next-nearest neighbors (e.g., oscillators 1 and 3) are in the same group and are antiphase-locked. (B) Phase differences, frequencies, mutual information matrix, and a snapshot of the 2-TW-I state. (C) Mean frequency difference of the clusters (within the decoupled state) as a function of the difference of the mean natural frequencies of the clusters. The data show that for two different values of coupling, the intercept is near zero. Thus, in a ring without disorder, no frequency difference between clusters exists in the decoupled state. (D) The in-phase synchronized state, the 2-precess state, and the 2-TW-I state are plotted in a reduced phase space of Δ8, Δ1, a1.

  • Fig. 5 Weak chimeras.

    (A) Phase differences, frequencies, mutual information matrix, and a snapshot of the WC-I state. (B) Phase differences, frequencies, mutual information matrix, and a snapshot of the WC-II state. (C) Simulations of the full magnitude-phase model (Eq. 2) and the approximate phase mode (Eq. 3) for α = 0.075 and β = 0.5. (D) Data for the frequency difference of the oscillator clusters (grouped as {2,3,6,7} and {1,4,5,8}) as a function of the coupling β. Blue, experimental data; red, data from numerical simulations of the magnitude-phase model (Eq. 2); green, data from numerical simulations of the phase-only model (Eq. 3). (E) Data for the magnitudes from simulation and experiment for β = 0.2 and β = 0.55. The magnitude deviation changes from ~0.4 to ~0.8.

  • Table 1 Isotropy subgroups of D8 × T.
    SubgroupSubspace dimensionGeneratorsPhase pattern
    D81σ, κ{a, a, a, a, a, a, a, a}
    D8(+, –)1(κ, 1), (κσ, –1){a, –a, a, –a, a, –a, a, –a}
    Z8(p), p ∈ 1, 2, 31σωp{a, ωpa, ω2pa, ω3pa, ω4pa,
    ω5pa, ω6pa, ω7pa}
    D4(+, –)1(σκ, 1), (κσ, –1){a, a, –a, –a, a, a, –a, –a}
    D4(κ)2σ2κ, κ{a, b, a, b, a, b, a, b}
    Z4(p), p ∈ 1, 22σ2ω2p{a, b, ipa, ipb, i2pa, i2pb, i3pa, i3pb}
    D2(κ)3σ4κ, κ{a, b, c, b, a, b, c, b}
    D1(κ)5κ{a, b, c, d, e, d, c, b}
    D2(κσ)2σ3κ, κσ{a, b, b, a, a, b, b, a}
    D1(κσ)4σ7κ, κσ{a, b, c, d, d, c, b, a}
    D2(–, –)23κ, –1), (κσ, –1){a, b, –b, –a, a, b, –b, –a}
    D1(–, –)47κ, –1), (κσ, –1){a, b, c, d, –d, –c, –b, –a}
    D2(+, –)23κ, 1), (κσ, –1){a, b, b, a, –a, –b, –b, –a}
    Z24σ4{a, b, c, d, a, b, c, d}
    Z2(p = 1)4σ4ω4{a, b, c, d, –a, –b, –c, –d}
  • Movie 1. The transition from an unsynchronized state to an antiphase synchronized state after the inter-oscillator coupling is induced (composed of experimental data).
  • Movie 2. Highlighting the spatiotemporal symmetry of a synchronized splay state, k = 7 (composed of experimental data).
  • Movie 3. Formation of the spatial pattern of phases in the inhomogeneous synchronized state through a quench of the coupling parameter β (composed of experimental data).
  • Movie 4. The 2-precess state with two sets of decoupled oscillators (composed of experimental data).
  • Movie 5. The WC-I state and synchronization between clusters across the ring (composed of experimental data).
  • Movie 6. The WC-II state demonstrating an additional symmetry not described by D8 × T (composed of experimental data).
  • Exotic states in a simple network of nanoelectromechanical oscillators

    Matthew H. Matheny, Jeffrey Emenheiser, Warren Fon, Airlie Chapman, Anastasiya Salova, Martin Rohden, Jarvis Li, Mathias Hudoba de Badyn, Márton Pósfai, Leonardo Duenas-Osorio, Mehran Mesbahi, James P. Crutchfield, M. C. Cross, Raissa M. D'Souza, Michael L. Roukes

    Materials/Methods, Supplementary Text, Tables, Figures, and/or References

    Download Supplement
    • Materials and Methods
    • Supplementary Text
    • Figs. S1 to S6
    • Tables S1 to S3
    • References

    Images, Video, and Other Media

    Movies S1 to S32

Stay Connected to Science

Navigate This Article