Hopf bifurcation in a three-dimensional system

In this paper, Liapunov-Schmidt reduction and singularity theory are employed to discuss Hopf and degenerate Hopf bifurcations in global parametric region in a three-dimensional system x=–+y, y=–x–y(1–kz), =(1–z)–ky²]. The conditions on existence and stability are given.Project Supported by the National Natural Science Foundation of China.Thanks for the instruction and help from Associate Professor Lu Qi-shao.