Bifurcations emerging from a double Hopf bifurcation for a BWR

This work illustrates several scenarios of nonlinear patterns for torus generating from a Double Hopf (HH) bifurcation in a dynamical system of boiling water reactor (BWR). Co-dimension one bifurcations of fixed points and limit cycles were explored for the system of BWR. A few explanations of generalized Hopf (GH) bifurcation can also be found in literature. However, other co-dimension two bifurcations of fixed points and limit cycles are unexplored. Therefore, although bifurcations of limit cycles which are originating from Hopf bifurcation are also investigated, the major focus in this work is bifurcations of higher co-dimensions. The system considered shows a rich and interesting dynamics because of the existence of double Hopf bifurcation. Neimark-Sacker (NS) curves start from double Hopf and limit cycles from Hopf become torus near to NS points. Resonances and other bifurcations on NS curve are discussed as well. The dynamics of torus near resonance and Chenciner bifurcations are also noteworthy. The bursting of the torus is shown for the system. In short, co-dimension one and two bifurcations of limit cycles in bifurcation diagrams have been investigated with a discussion of origin of limit point, period doubling and Neimark-Sacker bifurcations of limit cycles.