Unsteady aerodynamics of flapping wings and airfoils are in the center of research for quite some time in order to understand the flight of natural locomotion systems as well as the design of futuristic bio-mimetic devices. The present study focuses on dynamical aperiodicity, chaos and the corresponding bifurcation route in the wake and near-field of flapping airfoils. Numerical simulations have been performed at different parametric regimes associated with the different dynamical states, using an in-house flow solver developed based on the discrete forcing type Immersed Boundary Method (IBM). Results obtained using the IBM solver are compared with those from a well-validated body-fitted Arbitrary Lagrangian-Eulerian (ALE) approach. This explores the scope of body non-conformal mesh methods in comparison to body fitted approaches in capturing complex flow topologies, especially, during aperiodic flow regimes. The unsteady flow-field is seen to undergo a transition from periodicity to chaos through a quasi-periodic route as non-dimensional plunge velocity (kh) is increased from 1.0 to 1.9. In the transitional regime (kh = 1.5), the quasi-periodic movement of the wake vortices plays the key role in bringing out jet-switching. Furthermore, the effect of stochastic inflow fluctuations on the jet-switching characteristics has been analysed in the present study. The fluctuating inflows alter the organized arrangement of the vortex street even at a lower kh (kh = 1.0), giving way to an advanced jet-switching onset. More frequent switching with larger deflection angle is also observed at kh = 1.5 as compared to the no fluctuation case. The underlying flow physics are investigated through a qualitative study of the near-field interactions as well as various quantitative measures derived from the unsteady flow-field. Finally, the role of phase-offset between pitch and plunge motions on the transitional flow dynamics is studied. A novel scaling relation considering all the relevant kinematic parameters is proposed in order to differentiate the distinct dynamical wake regimes in a robust manner.