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Top: Equilibrium position Xe as a function of normalized input power M for X0 = 0, s0 = 0.5, s1 = 0.66, Rm = 0.81, ξ = 0.43μm−1, and δ = 0.05. Unstable fixed points (dotted blue) connect saddles (defined by SN) between points with negative curvature. Stable (solid red) and unstable (dashed blue) fixed points connect saddles with Hopf bifurcations (defined by H). M(H1) = 1.923 defines transition from region I and II. Bottom: zoom on first (left) and second (right) SN points. M(H2) = 6.334 and M(H3) = 3.597 define transitions from IIIc to IIIb and IIIb to IIIa, respectively.
Validation of the subcritical Hopf bifurcation in the Hane experiment2 for the parameter set in Fig. 2: stable limit-cycle (dashed red), stable fixed point (solid red), unstable limit-cycle (blue dots) vs. maximal limit-cycle measurement (green circles). Bottom: Periodic limit-cycle M = 2.0: time-series (left), power spectra (right).
Strange attractor M = 8.0 (for parameter set in Fig 2): X(t) time-series (upper left), power spectrum (upper right), Z(X,Y) state-space (lower left), Poincaré map projection (lower right).
Top: Bifurcation diagram (for parameter set in Fig. 2) of self-excited response depicted by Poincaré map magnitude (XPM ) for increasing values of power M. Bottom: time-series (left), and power spectra (right) of a periodic limit-cycle (M = 3.85) depicting an ultrasubharmonic of order m/n = 3/2, where limit-cycle frequency is f ∼ 0.16 (see Fig.3, bottom right) and dominant energy content is obtained at 2f/3 ∼ 0.105.
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