Hopf bifurcation analysis of scalar implicit neutral delay differential equation

摘要

Hopf bifurcation analysis is conducted on a scalar implicit Neutral Delay Differential Equation (NDDE) by means of the extension of two analytical methods: 1) center manifold reduction combined with normal form theory; 2) method of multiple scales. The modifications of the classical algorithms originally developed for explicit differential equations lead to the same algebraic results, which are further confirmed by numerical simulations. It is shown that the generalizations of these regular normal form calculation methods are useful for the local nonlinear analysis of implicit NDDEs where the explicit formalism is typically not accessible and the existence and uniqueness of solutions around the equilibrium are only assumed together with the existence of a smooth local center manifold.