  To understand the behavior of composite fluid particles such as nucleated
cells and double-emulsions in flow, we study a finite-size particle
encapsulated in a deforming droplet under shear flow as a model system. In
addition to its concentric particle-droplet configuration, we numerically
explore other eccentric and time-periodic equilibrium solutions, which emerge
spontaneously via supercritical pitchfork and Hopf bifurcations. We present the
loci of these solutions around the codimenstion-two point. We adopt a dynamical
system approach to model and characterize the coupled behavior of the two
bifurcations. By exploring the flow fields and hydrodynamic forces in detail,
we identify the role of hydrodynamic particle-droplet interaction which gives
rise to these bifurcations.
