Revisiting the Ginzburg-Landau Equation: Model Reduction and Data-Driven Analysis of the Bénard-von Kármán Instability

The Stuart-Landau and Ginzburg-Landau equations model the temporal and spatial evolution of a small perturbation to a nonlinear dynamical system close to a Hopf bifurcation point in a dimensionless parameter, such as the Reynolds number Re. These equations are particularly relevant for describing the dynamics of the viscous wake behind a 2D cylinder, including the transition from a steady flow to time-periodic vortex shedding as Re increases through the critical bifurcation value Re_c ≈ 47. In this work, we investigate the use of data-driven sparse nonlinear modeling techniques to learn nonlinear ordinary and partial differential equations based purely on data from high-fidelity numerical simulations. In particular, we explore various coarse graining options to reduce the dimensionality and also compare the learned equations with the classical Stuart-Landau and Ginzburg-Landau models.