Border Singularities as Solutions of an Ordinary Differential Equation

The border singularities of a sixth-degree smooth function will be examined in this article by using real analysis and catastrophe theory. Next that, we provide an application of an ordinary differential equation (ODE) together with its boundary conditions. Using the local Lyapunov-Schmidt approach, we demonstrate that this function is identical to the key function that corresponds to the functional of the ODE. The bifurcation analysis of the function has been investigated by border singularities. The parametric equation for the bifurcation set (caustic) and its geometric description together with the critical points’ bifurcation spreading has been found.