  We propose a general framework for parameter-free identification of a class
of dynamical systems. Here, the propagator is approximated in terms of an
arbitrary function of the state, in contrast to a polynomial or Galerkin
expansion used in traditional approaches. The proposed formulation relies on
variational data assimilation using measurement data combined with assumptions
on the smoothness of the propagator. This approach is illustrated using a
generalized dynamic model describing oscillatory transients from an unstable
fixed point to a stable limit cycle and arising in nonlinear stability analysis
as an example. This 3-state model comprises an evolution equation for the
dominant oscillation and an algebraic manifold for the low- and high-frequency
components in an autonomous descriptor system. The proposed optimal model
identification technique employs mode amplitudes of the transient vortex
shedding in a cylinder wake flow as example measurements. The reconstruction
obtained with our technique features distinct and systematic improvements over
the well-known mean-field (Landau) model of the Hopf bifurcation. The
computational aspect of the identification method is thoroughly validated
showing that good reconstructions can also be obtained in the absence of of
accurate initial approximations.
