  Under a steady DC electric field of sufficient strength, a weakly conducting
dielectric sphere in a dielectric solvent with higher conductivity can undergo
spontaneous spinning (Quincke rotation) through a pitchfork bifurcation. We
design an object composed of a dielectric sphere and an elastic filament. By
solving an elasto-electro-hydrodynamic (EEH) problem numerically, we uncover an
EEH instability exhibiting diverse dynamic responses. Varying the bending
stiffness of the filament, the composite object displays three behaviours: a
stationary state, undulatory swimming and steady spinning, where the swimming
results from a self-oscillatory instability through a Hopf bifurcation. By
conducting a linear stability analysis incorporating an elastohydrodynamic
model, we theoretically predict the growth rates and critical conditions, which
agree well with the numerical counterparts. We also propose a reduced model
system consisting of a minimal elastic structure which reproduces the EEH
instability. The elasto-viscous response of the composite structure is able to
transform the pitchfork bifurcation into a Hopf bifurcation, leading to
self-oscillation. Our results imply a new way of harnessing elastic media to
engineer self-oscillations, and more generally, to manipulate and diversify the
bifurcations and the corresponding instabilities. These ideas will be useful in
designing soft, environmentally adaptive machines.
