Co-dimension two bifurcations of limit cycles starting from generalized Hopf bifurcation for the dynamical system of nuclear reactors

Abstract

Nonlinear stability becomes critical when periodic and aperiodic oscillations arise and are important for the safe operation of nuclear reactors. The linear stability analysis and Hopf bifurcation are well studied in the context of nuclear reactors and fail to detect higher-order nonlinear entities. A reduced-order model which couples neutron dynamics with thermal-hydraulics is used in this work. Hopf bifurcation and limit cycles were reported earlier; however, the bifurcation analysis of limit cycles with two free parameters was not provided in the past and is investigated here. These co-dimension two bifurcations of limit cycles, which define the origin of several bifurcations such as (limit point (LPC), period doubling (PD), and Neimark-Sacker (NS) bifurcation of limit cycles), are our focus. We analyze the bifurcation starting from the Generalized Hopf (GH) and observe R1 resonance bifurcations along with the cusp bifurcation of limit cycles (CPC). We reveal dynamics near the R1 and quasiperiodic behavior are present in the vicinity. We observe cascading of R1 bifurcation due to multiple LPCs occurring in the supercritical Hopf region. We also detect CPC, which changes the direction of the LPC curve, which we call the global stability boundary. We present aperiodic and uncertain oscillations near R1, and for a safer operation, we should understand the existence of higher-order bifurcations.