An Optimal Model Identification For Oscillatory Dynamics With a Stable Limit Cycle

TL;DR

This paper introduces a parameter-free, variational data assimilation framework for identifying dynamical systems with oscillatory behavior, demonstrating improved modeling of transient vortex shedding over traditional mean-field models.

Contribution

It presents a novel, flexible identification method that does not rely on polynomial expansions, enhancing accuracy in modeling oscillatory dynamics with stable limit cycles.

Findings

Improved reconstruction of vortex shedding dynamics

Effective identification without precise initial guesses

Validation shows robustness and accuracy of the method

Abstract

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.