Catastrophic Change
Catastrophic change involves four stages:
Stage 1:
If a system is ‘at rest’ (that is, not undergoing change), then it will
remain at rest in its standard resting state within a defined range of states. It
tends to occupy a preferred stable state, or at least a defined range of states.
Stage 2:
The system is subjected to forces.
Stage 3:
The system initially responds by trying to absorb the forces. If it
is possible, the system will attempt to return to its preferable initial state.
Stage 4:
It is possible that the system cannot absorb the forces because
they are so strong. In this case the system gives way under the force. Two
consequences follow. (i) There is no direct or continuous way back to the
former stable state. (ii) Catastrophic change
occurs and establishes a new
preferred stable state or range of states. These changes can take any of several
forms.
Small changes in certain parameters of a nonlinear system can cause equilibria
to appear or disappear, and the system can change from attracting to repelling
and vice versa. The
common characteristics is that a small change in one
paramter on the margin between stability leads to large and sudden changes of
the behaviour of the system. In the illustration below of tilting the box, past a
point (called the degenerate critical point), a small additional force was a
enough to tip the box over the edge, both literally and metaphorically.
Catastrophe theory identifies seven elementary types of catastrophes. These
are labelled
fold, cusp, swallowtail,
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butterfly, hyperbolic umbilic, elliptic
umbilic, and parabolic umbilic.
Illustration
Catastrophic change can be illustrated by a comparison. Assume there is a
wooden box on the floor with four sides. This box is resting on one side, (the
down side). Some force is applied to the top of the box on one side (the
adjacent side). The box lifts a little by leaning to the other side, then falls back
to its original position. It may then bounce a little on the floor and shift the
whole position of the box slightly.
Assume now that this force is increased. Initially the same reaction occurs
except the box probably bounces a little more as the side that was uplifted
returns to the floor. However, past a point the box will not return. Instead if it
will keep falling so that it now rests on the side opposite the adjacent side.
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last painting, The Swallow's Tail, was based on this
catastrophe.
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