In the present study, the double-diffusive oscillatory convection of binary mixture, 3He-4He, in porous medium heated from below and cooled from above was investigated with stress-free boundary conditions. The Darcy model was employed in the governing system of perturbed equations. An attempt was made, for the first time, to solve these equations by using the nonlinear analysis-based truncated Fourier series. The influence of the Rayleigh number (R), the separation ratio (?) due to the Soret effect, the Lewis number (Le), and the porosity number (?) on the field variables were investigated using the finite amplitudes. From the linear stability analysis, expressions for the parameters, namely, R and wavenumbers, were obtained, corresponding to the bifurcations such as pitchfork bifurcation, Hopf bifurcation, Takens-Bogdnanov bifurcation and co-dimension two bifurcation. The results reveal that the local Nusselt number (NL) increases with R. The total energy is enhanced for all increasing values of R. The deformation in the basic cylindrical rolls and the flow rate are enhanced with R. The trajectory of heat flow was studied using the heatlines concept. The influence of R on the flow topology is depicted graphically. It is observed that the intensity of heat transfer and the local entropy generation are increased as R increases.