When an evolving system loses stability it will be at a generic codimension-one bifurcation of nonlinear dynamics, which can be safe, explosive or dangerous. The safe and explosive forms give a new determinate response with no hysteresis. Dangerous bifurcations give a dynamic jump to a remote attractor, with subsequent hysteresis. The jump can be always to the same attractor: or it can be indeterminate, with jumps to one of two or more distinct solutions; here simulations or experiments might give acceptable jumps, while the manufactured system might jump to failure. This generic indeterminacy gives the greatest unpredictability of nonlinear dynamics, because it sweeps trajectories precisely onto a basin boundary; from which they are dispersed more widely than by chaos. Invariant manifolds and basin boundaries govern the phenomenon. Illustrations with a smooth boundary in R~2 are the cyclic fold and sub-critical Hopf. Examples in R~3, with the complexity of a fractal boundary, include the tangled saddle node, sub-critical flip, and chaotic crisis. The tangled saddle-node is locally a regular fold, but its fractal manifolds give unpredictable jumps to resonance in driven mechanical and electrical oscillators.
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