Probability of rate-induced tipping in two slow–fast stochastic mechanical systems

par Baptiste Bergeot (INSA CVL)

jeudi 8 janv. 2026, 14:00 → 15:00 Europe/Paris

Salle de Séminaires (Orléans)

Rate-induced tipping refers to a large and sudden shift, or critical transition, in the behavior of a dynamical system when a parameter changes at a speed exceeding a critical rate. Primarily studied in conceptual climate models, it remains largely unexplored in engineering, despite possible dramatic effects which are not predicted (and even unsuspected) with a constant-parameter bifurcation analysis.

This presentation explores the impact of noise on rate-induced tipping in two types of fast–slow mechanical systems. The first system is a self-sustained reed musical instrument (such as the clarinet) in which one of the bifurcation parameters – specifically, the pressure inside the musician’s mouth – varies slowly over time accounting for attack transients performed by the musician. The second system is a nonlinear passive vibration absorber, referred to as a Nonlinear Energy Sink (NES), which is coupled to a self-sustained oscillator (SSO) requiring vibration attenuation. 

Although these two systems differ greatly in terms of their applications, they indeed both exhibit rate-induced tipping: when a critical tipping point is crossed, the reed instrument can abruptly transition from silence to a musical note, while the SSO–NES system can switch from mitigated to unmitigated responses. Moreover, both systems are modeled by differential equations containing a small parameter that highlights their singularly perturbed, fast–slow nature. As a result, their dynamics (i) cannot be fully explained by the concepts traditionally used in engineering such as bifurcation diagrams and basins of attraction and (ii) can be significantly influenced by the presence of noise. For each of these systems, the deterministic dynamics will first be described. Subsequently, the influence of noise will be analyzed using both numerical simulations and analytical methods. The aim is, for each system, to estimate the probability of tipping, a quantity that, in the deterministic case, abruptly changes from zero to one when the tipping point is crossed.