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 codimension-two point. We adopt a dynamic 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. DOI: 10.1103/PhysRevLett.119.064502 Droplets, capsules, and vesicles in flow often exhibit interestingly rich dynamics even in the linear shear flow [1][2][3][4][5][6][7][8][9]. Despite substantial work on the dynamics of these soft systems enclosing homogeneous fluids, limited effort has been directed to studying their behavior when they include an internal structure. However, such a configuration is common in nature and engineering applications: cells like leukocytes and megakaryocytes contain a nucleus of up to 50%-80% of themselves in volume [10]; double emulsions playing an important role in chemical and pharmaceutical engineering are featured with a coreshell geometry [11][12][13]; droplet-based encapsulation for high-throughput biological assays utilizes droplets as microchambers to compartment cells for analysis at the single-cell level, where the cell size can be comparable to the droplet size in certain applications [14][15][16].These fluid particles are characterized by complex hydrodynamic interactions between the internal structures and the external interface. Few works conducted for nucleated model cells in shear [17][18][19][20] all assumed their compound structures to be concentric, preserving the rotational symmetry of order 2 (C2) about the y axis and reflection symmetry about the y ¼ 0 shear plane [see Fig. 1(a)]. The symmetries do hold for a single shear-driven deformable particle that attains a steady ellipsoidal shape undergoing tank-treading motion [21][22][23]]. Yet, they are not guaranteed in the presence of an internal structure.In this Letter we focus on the stability of the concentricity of composite fluid particles. By considering a droplet encaging a spherical particle as a model system, we formulate the following questions: will the composite structures remain concentric? How does the dynamics depend on interfacial tension and particle size? What is the role of the hydrodynamic interaction?We begin our discussion by presenting 3D hydrodynamic simulations of a compound particle droplet subjected to unbounded shear U ∞ ¼ G · x, in the creeping flow regime, where the only nonzero component G xz ¼ _ γ represents the shear rate [ Fig. 1(a)]. The incompressible Stokes equations are solved by a boundary integral meth...