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The cusp catastrophe is exemplified by the minimum of the function f(x) = x4 at x = 0 . This critical point is unstable: perturbing the function to f(x) = x4 + ax2 gives a function with three critical points (two minima) near zero if a is negative, no matter how small. The minimum of f(x) = x4 has an additional instability: the perturbation to f(x) = x4 + ax2 + bx will have two minima near zero if b < (4/3)(-a3/6)1/2 and one otherwise. (For a mechanical instantiation of this catastrophe see The Catastrophe Machine.)
| The (a,b) parameter space for the cusp catastrophe. For parameter values (a,b) in the region between the red curves, the energy function f(x) = x4 + ax2 + bx has two minima. Otherwise it has one. |
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Folllowing from right to left the one-dimensional section given by the blue line, one of the local minima disappears and is captured by the other. The verb class corresponding to this morphology is capture: The fish swallows the bug. |
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Following the blue section from left to right, a new local minimum appears beside the old one. The verb class corresponding to this morphology is emission: The bird lays an egg. |
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Following the green line in either direction, one local minimum is substituted by another. The verb class corresponding to this morphology is transformation: The caterpillar becomes a butterfly. |
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