Exploiting noise-induced dynamics for system identification near a Hopf bifurcation

We propose a system identification framework that exploits the noise-induced dynamics inherent to nonlinear systems near a supercritical or subcritical Hopf bifurcation. The key assumption made is that the system response can be modeled with a Stuart--Landau equation and its corresponding Fokker--Planck equation. We demonstrate the framework on two different flow systems: a hydrodynamic system (a low-density jet) undergoing a subcritical Hopf bifurcation, and a thermoacoustic system (a Rijke tube) undergoing a supercritical Hopf bifurcation. For both systems, we extract the model coefficients using experimental measurements of the noise-induced dynamics in only the unconditionally stable regime, prior to both the Hopf and saddle-node points. We show that the framework can accurately predict (i) the order of nonlinearity, (ii) the types and locations of the bifurcation points, and (iii) the limit-cycle characteristics beyond such points. As the noise-induced dynamics of nonlinear systems are expected to be universal in the vicinity of a Hopf bifurcation (Ushakov et al. 2005, Phys. Rev. Lett., vol. 95, 123903), the proposed framework should be applicable to a variety of flow systems in nature and engineering.