Bifurcation-catastrophe theorists roughly define a catastrophe as a sudden transition resulting from a continuous parameter change. Here are some basic definitions for understanding the fold catastrophe.

1. A critical point

of a differentiable function

of one variable

satisfies

.

2. A nondegenerate critical point

of a differentiable function

of one variable

satisfies

and

; if

and

,

is called a degenerate critical point.

For

and

, there are two nondegenerate critical points; for

, there is one degenerate critical point; and for

, there are no critical points.

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