Predicting Saddle-Node Bifurcations Using Transient Dynamics: A Model-Free Approach

Abstract This paper proposes a novel method for predicting the presence of saddle-node bifurcations in dynamical systems. The method exploits the effect that saddle-node bifurcations have on transient dynamics in the surrounding phase space and parameter space, and does not require any information about the steady-state solutions associated with the bifurcation. Specifically, trajectories of a system obtained for parameters close to the saddle-node bifurcation present local minima of the logarithmic decrement trend in the vicinity of the bifurcation. By tracking the logarithmic decrement for these trajectories, the saddle-node bifurcation can be accurately predicted. The method does not strictly require any mathematical model of the system, but only a few time series, making it directly implementable for gray- and black-box models and experimental apparatus. The proposed algorithm is tested on various systems of different natures, including a single-degree-of-freedom system with nonlinear damping, the mass-on-moving-belt, a time-delayed inverted pendulum, and a pitch-and-plunge wing profile. Benefits, limitations, and future perspectives of the method are also discussed. The proposed method has potential applications in various fields, such as engineering, physics, and biology, where the identification of saddle-node bifurcations is crucial for understanding and controlling complex systems.