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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