<p>Fold bifurcations—also called saddle-node bifurcations—mark a point where two branches of stable and unstable steady-state solutions merge and annihilate each other. As fold bifurcations do not affect the linear stability of steady-state solutions that are not directly involved with them, they are often undetected or overlooked in real engineering systems. However, they can have an important impact on a system’s global dynamics and on its operational safety, as they often delimit regions of the parameter space where a solution is only locally stable but not globally, i.e., it is not robust against external perturbations. In this study, we present a new model-free method for identifying fold bifurcations experimentally, uniquely based on analyzing transient time series obtained for parameter values where the two branches of solutions merged at a fold do not even exist. The method leverages the so-called critical slowing down and exploits the effect that fold bifurcations have on the surrounding phase portrait. As a sequel to previous exhaustive numerical testing, the technique is experimentally validated on a towed wheel undergoing shimmy. Extensive investigation illustrates how the system under study loses stability through a subcritical Hopf bifurcation, generating a region of bistability delimited by a fold bifurcation. The position of the fold is accurately predicted by the proposed technique, which validates its effectiveness.</p>

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Model-free fold bifurcation prediction from pre-bifurcation scenario: experimental validation through wheel shimmy vibrations

  • Fanni Kadar,
  • Gabor Stepan,
  • Giuseppe Habib

摘要

Fold bifurcations—also called saddle-node bifurcations—mark a point where two branches of stable and unstable steady-state solutions merge and annihilate each other. As fold bifurcations do not affect the linear stability of steady-state solutions that are not directly involved with them, they are often undetected or overlooked in real engineering systems. However, they can have an important impact on a system’s global dynamics and on its operational safety, as they often delimit regions of the parameter space where a solution is only locally stable but not globally, i.e., it is not robust against external perturbations. In this study, we present a new model-free method for identifying fold bifurcations experimentally, uniquely based on analyzing transient time series obtained for parameter values where the two branches of solutions merged at a fold do not even exist. The method leverages the so-called critical slowing down and exploits the effect that fold bifurcations have on the surrounding phase portrait. As a sequel to previous exhaustive numerical testing, the technique is experimentally validated on a towed wheel undergoing shimmy. Extensive investigation illustrates how the system under study loses stability through a subcritical Hopf bifurcation, generating a region of bistability delimited by a fold bifurcation. The position of the fold is accurately predicted by the proposed technique, which validates its effectiveness.