Transient amplification of broken symmetry in elastic snap-through

Abstract: A snap-through bifurcation occurs when a bistable structure loses one of its stable states and moves rapidly to the remaining state. For example, a buckled arch with symmetrically clamped ends can snap between an inverted and a natural state as the ends are released. A standard linear stability analysis suggests that the arch becomes unstable to asymmetric perturbations. Surprisingly, our experiments show that this is not always the case: symmetric transitions are also observed. Using experiments, numerics, and a toy model, we show that the symmetry of the transition depends on the rate at which the ends are released with sufficiently fast loading leading to symmetric snap-through. Our toy model reveals that this behavior is caused by a region of the system's state space in which any initial asymmetry is amplified. The system may not enter this region when loaded fast (hence remaining symmetric) but will traverse it for some interval of time when loaded slowly, causing a transient amplification of asymmetry. Our toy model suggests that this behavior is not unique to snapping arches, but rather can be observed in dynamical systems where both a saddle-node and a pitchfork bifurcation occur in close proximity.