Hopf bifurcation in a three-dimensional system (Q1178003)

This paper deals with the following third-order system of a nuclear spin generator \[ \dot x=-\beta x+y,\quad \dot y=-x-\beta y(1-kz),\quad \dot z=\beta[\alpha(1-x)-ky^2], \] with parameters \(\alpha\in (0,1]\), \(\beta\in [0,1]\) and \(k>0\). \textit{S. Sherman} [Contrib. Differ. Equ. 2, 197--227 (1963; Zbl 0134.30901)] proved that there is a unique (stable) periodic solution of the above system whenever \(\beta\) is sufficiently small with \(0<k-2<\beta\) and \(2\beta < \alpha\). Using the unfolding theory, the authors of the present paper demonstrate that if the above Sherman's condition does not hold, then the uniqueness of the periodic solution may be violated, and the jumping phenomena and the hysteresis may appear.