Critical slowing down and phase radius filtering for forecasting supercritical Hopf bifurcation

Abstract

In this paper, critical slowing down is considered as a precursory signal of supercritical Hopf bifurcation in a smooth system. Critical slowing down is one of warning features for the threshold of catastrophic nonlinear instability such as jump phenomenon (fold or subcritical Hopf bifurcation) as well as non-catastrophic transitions such as supercritical Hopf bifurcation discussed herein. In practical, when the local equilibrium point is a spiral sink, the stable spiral is along rather elliptic trajectory than circular one. It implies that the phase radius of the stable spiral fluctuates periodically while decreasing to zero. Those periodic fluctuations are undesirable for measuring the recovery rate of the system. To measure the recovery rate of the system, the decrement of the phase radius is required to be regulated removing its periodic components. For this purpose, herein, phase radius filtering is introduced and validated with a simple mathematical model. Then, a 2D airfoil model with pitch nonlinearity is demonstrated as an example, predicting the limit cycle flutter point (exhibiting supercritical Hopf bifurcation) by the corresponding approach combined with phase radius filtering.