End-to-end delay is one of the most important characteristics of Internet end-to-end packet dynamics, which can be applied to quality of services (QoS) management, service level agreement (SLA) management, congestion control algorithm development, etc. Nonstationarity and nonlinearity are found by the analysis of various delay series measured from different links. The fact that different types of links have different degree of Self-Similarity is also obtained. By constructing appropriate network architecture and neural functions, functional networks can be used to model the Internet end-to-end nonlinear delay time series. Furthermore, by using adaptive parameter studying algorithm, the nonstationarity can also be well modeled. The numerical results show that the provided functional network architecture and adaptive algorithm can precisely characterize the Internet end-to-end delay dynamics.