Abstract
Coherence resonance (CR) is a phenomenon in which the response of a stable nonlinear system to external noise exhibits a peak in coherence at an intermediate noise amplitude. We report the first experimental evidence of CR in a hydrodynamic system, a low-density jet capable of undergoing both supercritical and subcritical Hopf bifurcations. By applying noise to the jet in its unconditionally stable regime, we find that, for both types of bifurcation, the coherence factor peaks at an intermediate noise amplitude and increases as the stability boundary is approached. We also find that the autocorrelation function decays differently between the two types of bifurcation, indicating that CR can reveal information about the nonlinearity of a system even before it bifurcates to a limit cycle. We then model the CR dynamics with a stochastically forced van der Pol oscillator calibrated in two different ways: (i) via the conventional method of measuring the amplitude evolution in transient experiments and (ii) via the system-identification method of Lee et al. (J. Fluid Mech., vol. 862, 2019, pp. 200–215) based on the Fokker–Planck equation. We find better experimental agreement with the latter method, demonstrating the deficiency of the former method in identifying the correct form of system nonlinearity. The fact that CR occurs in the unconditionally stable regime, prior to both the Hopf and saddle-node points, implies that it can be used to forecast the onset of global instability. Although demonstrated here on a low-density jet, CR is expected to arise in almost all nonlinear dynamical systems near a Hopf bifurcation, opening up new possibilities for the development of global-instability precursors in a variety of hydrodynamic systems.
Highlights
Low-density jets are central to many technological processes, such as thermal plasma spraying and fuel injection
By applying noise to the jet in its unconditionally stable regime, we find that, for both types of bifurcation, the coherence factor peaks at an intermediate noise amplitude and increases as the stability boundary is approached
We find that the autocorrelation function decays differently between the two types of bifurcation, indicating that Coherence resonance (CR) can reveal information about the nonlinearity of a system even before it bifurcates to a limit cycle
Summary
Low-density jets are central to many technological processes, such as thermal plasma spraying and fuel injection. Lee et al (2019) used a high-order Landau equation and its corresponding Fokker–Planck equation to perform system identification (SI) of a low-density jet, with input data coming from measurements of the noise-induced dynamics in the unconditionally stable regime (USR, i.e. the regime in which the system is stable to infinitesimal as well as finite-amplitude perturbations), prior to both the Hopf and SN points These researchers showed that this approach can enable accurate prediction of (i) the order of nonlinearity, (ii) the types and locations of the bifurcation points, and the stability boundaries, and (iii) the limit-cycle dynamics beyond such boundaries. This is a universal feature of nonlinear dynamical systems near a Hopf bifurcation, and provides a suitable framework within which to understand and unify the stochastic dynamics of such systems (Ushakov et al 2005)
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