Analytical approach to identifying a bifurcation point in reduced-nonlinear dynamical systems formed by shift mode and oscillation modes

The reduced-order models for fluid flow provide a simple tool for analyzing nonlinear aspects of the Navier-Stokes equations. Bifurcation is a classical nonlinear phenomenon, and analyzing dynamical systems of reduced-order models that capture it facilitates our understanding of bifurcation. In this study, we analyze a reduced-order model based on the Galerkin projection for the Hopf bifurcation of the flow around a cylinder, which consists of a steady-state solution referred to as the shift mode and the oscillatory modes of the wake vortex street. The dynamical system of the reduced-order model is linearized at the equilibrium points. Eigenvalue analysis for a linearized dynamical system yields eigenvalues as a function of Reynolds number, and the bifurcation point is identified analytically. In the ROM constructed by changing the Reynolds number of the shift mode and the oscillation modes, the bifurcation Reynolds number is investigated using our analytical approach. Our understanding of the Hopf bifurcation is advanced by examining the Reynolds number of the modes used in the dynamical system and its bifurcation point. Furthermore, it provides valuable insights into constructing practical ROMs that accurately capture the Hopf bifurcation.