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Nonlinear convection due to Thermal and Compositional buoyancy in Earth's outer core has been investigated. We have obtained the values of Takens-Bogdanov bifurcation points and codimension-two bifurcation points by plotting graphs of neutral curves corresponding to stationary and oscillatory convection for different values of physical parameters. We have studied Nusselt number from Landau-Ginzburg equation at the onset of stationary convection for the steady case. We have also derived a nonlinear two dimensional Landau-Ginzburg equation near the onset of stationary convection at a supercritical pitchfork bifurcation and nonlinear one-dimensional coupled Landau-Gingburg type equations near the onset of oscillatory convection at a supercritical Hopf bifurcation. We have discussed the stability regions of standing and travelling waves. We have also discussed the occurrence of secondary instabilities such as Eckhaus, Zigzag and Benjamin-Feir instabilities.