In this paper, we study the dynamics of a two-parameter unfolding system χ` = y, y` = βy+αf (χ,α)±χy+yg(χ), where f (χ, α) is a second order polynomial in x and g(χ) is strictly nonlinear in χ. We show that the higher order term yg(χ) in the system does not change qulitative structure of the Hopf bifurcations near the fixed points for small α and β if the nontrivial fixed point approaches to the origin as α approaches zero.