On the use of POD-based ROMs to analyze bifurcations in some dissipative systems

This paper deals with the use of POD-based reduced order models to construct bifurcation diagrams (which requires calculating steady and time-dependent attractors) in complex bifurcation problems involving dissipative systems. The method proposed in the paper relies on the observation that POD manifolds resulting from snapshots calculated in time-dependent runs for specific values of the parameters of the problem also contain the attractors for other values of the parameters. The reason for this property is explained for a general class of dissipative systems, which includes many problems of scientific/industrial interest. The consequence is that appropriate POD manifolds can be calculated in a quite computationally efficient way. The method is illustrated considering both a simple bifurcation problem for a Fisher-like equation and a fairly complex bifurcation problem for the complex Ginzburg–Landau equation.

► Calculation of bifurcation diagrams may require huge computational resources.
► Complex bifurcation diagrams can be calculated using POD plus Galerkin projection.
► POD manifolds are fairly independent of the parameters in dissipative systems.
► Some basic ideas to efficiently compute POD manifolds are provided.
► A Fisher-like equation and the complex Ginzburg–Landau equation are considered.