On the Angular Singularities of a Smooth Function

The paper studies angular singularities of a real smooth function of the 4th degree using real analysis and catastrophe theory. After that, we apply an ordinary differential equation (ODE) with its boundary conditions. We show that the real smooth function equivalent to the key function associated with the ODE's function by applying the Lyapunov-Schmidt local technique. The angular singularities have been used to study the bifurcation analysis of the real smooth function. We have discovered the (caustic) bifurcation set's parametric equation and geometric interpretation. Moreover, the critical spots' bifurcated spread has been identified.