The present study attempts to capture the fluid-structure interaction dynamics of a chord-wise flexible flapping wing system using a limited mode structural model coupled with a high-fidelity Navier-Stokes (N-S) solver. The wing is modeled as two elliptic rigid foils connected by a non-linear torsional spring that incorporates the chord-wise bending stiffness. The front link is subjected to an active pitching-plunging motion while the rear link undergoes flow-induced passive oscillation. The structural governing equation for the rear link takes the form of a Duffing equation subjected to base excitation and external aerodynamic forcing. The aerodynamic loads on the foil are computed using a discrete forcing Immersed Boundary Method based in-house N-S solver which is coupled with the structural solver by a staggered weak coupling strategy. A bifurcation study is performed considering the free-stream velocity as the control parameter, in the presence of both structural and aerodynamic non-linearities. A dynamical transition in the unsteady flow-field from a periodic reverse-Kármán wake to an aperiodic wake is observed as the flow parameters are varied. The same transition is also reflected in the passive oscillation of the rear foil when analyzed with tools from the dynamical systems theory.