Symbolic computation of normal form for Hopf bifurcation in a neutral delay differential equation and an application to a controlled crane

A symbolic computation scheme and its corresponding Maple program are developed to compute the normal form for the Hopf bifurcation in a neutral delay differential equation. In the symbolic computation scheme, the neutral delay differential equation is considered as an ordinary differential equation in an appropriate infinite-dimensional phase space so that both center manifold reduction and normal form computation can be simultaneously conducted without computing center manifold beforehand. The Maple program is proved to provide an easy way to compute the normal form of the neutral delay differential equation automatically by only inputting some basic information of the equation. As an illustrative example, the application of the Maple program to a container crane with a delayed position feedback control is given. The results reveal that the normal form obtained by using the center manifold reduction and the normal form computation is in a full agreement with the result derived by applying the method of multiple scales. Moreover, numerical analysis is presented to validate the analytical results.