Forecasting Bifurcation of Parametrically Excited Systems: Theory & Experiments

Abstract

A system is parametrically excited when one or some of its coefficients vary with time. Parametric excitation can be observed in various engineered and physical systems. Many systems subject to parametric excitation exhibit critical transitions from one state to another as one or several of the system parameters change. Such critical transitions can either be caused by a change in the topological structure of the unforced system, or by synchronization between a natural mode of the system and the parameter variation. Forecasting bifurcations of parametrically excited systems before they occur is an active area of research both for engineered and natural systems. In particular, anticipating the distance to critical transitions, and predicting the state of the system after such transitions, remains a challenge, especially when there is an explicit time input to the system. In this work, a new model-less method is presented to address these problems based on monitoring transient recoveries from large perturbations in the pre-bifurcation regime. Recoveries are studied in a Poincare section to address the challenge caused by explicit time input. Numerical simulations and experimental results are provided to demonstrate the proposed method. In numerical simulation, a parametrically excited logistic equation and a parametrically excited Duffing oscillator are used to generate simulation data. These two types of systems show that the method can predict transitions induced by either bifurcation of the unforced system, or by parametric resonance. We further examine the robustness of the method to measurement and process noise by collecting recovery data from an electrical circuit system which exhibits parametric resonance as one of its parameters varies.