In mathematics, catastrophe theory is a branch
of bifurcation theory in the study of dynamical
systems; it is also a particular special case
of more general singularity theory in geometry.
Bifurcation theory studies and classifies
phenomena characterized by sudden shifts in
behavior arising from small changes in circumstances,
analysing how the qualitative nature of equation
solutions depends on the parameters that appear
in the equation. This may lead to sudden and
dramatic changes, for example the unpredictable
timing and magnitude of a landslide.
Catastrophe theory originated with the work
of the French mathematician René Thom in
the 1960s, and became very popular due to
the efforts of Christopher Zeeman in the 1970s.
It considers the special case where the long-run
stable equilibrium can be identified as the
minimum of a smooth, well-defined potential
function (Lyapunov function).
Small changes in certain parameters of a nonlinear
system can cause equilibria to appear or disappear,
or to change from attracting to repelling
and vice versa, leading to large and sudden
changes of the behaviour of the system. However,
examined in a larger parameter space, catastrophe
theory reveals that such bifurcation points
tend to occur as part of well-defined qualitative
geometrical structures.
== Elementary catastrophes ==
Catastrophe theory analyzes degenerate critical
points of the potential function — points
where not just the first derivative, but one
or more higher derivatives of the potential
function are also zero. These are called the
germs of the catastrophe geometries. The degeneracy
of these critical points can be unfolded by
expanding the potential function as a Taylor
series in small perturbations of the parameters.
When the degenerate points are not merely
accidental, but are structurally stable, the
degenerate points exist as organising centres
for particular geometric structures of lower
degeneracy, with critical features in the
parameter space around them. If the potential
function depends on two or fewer active variables,
and four or fewer active parameters, then
there are only seven generic structures for
these bifurcation geometries, with corresponding
standard forms into which the Taylor series
around the catastrophe germs can be transformed
by diffeomorphism (a smooth transformation
whose inverse is also smooth). These seven
fundamental types are now presented, with
the names that Thom gave them.
== Potential functions of one active variable
==
=== Fold catastrophe ===
V
=
x
3
+
a
x
{\displaystyle V=x^{3}+ax\,}
At negative values of a, the potential has
two extrema - one stable, and one unstable.
If the parameter a is slowly increased, the
system can follow the stable minimum point.
But at a = 0 the stable and unstable extrema
meet, and annihilate. This is the bifurcation
point. At a > 0 there is no longer a stable
solution. If a physical system is followed
through a fold bifurcation, one therefore
finds that as a reaches 0, the stability of
the a < 0 solution is suddenly lost, and the
system will make a sudden transition to a
new, very different behaviour. This bifurcation
value of the parameter a is sometimes called
the "tipping point".
=== Cusp catastrophe ===
V
=
x
4
+
a
x
2
+
b
x
{\displaystyle V=x^{4}+ax^{2}+bx\,}
The cusp geometry is very common, when one
explores what happens to a fold bifurcation
if a second parameter, b, is added to the
control space. Varying the parameters, one
finds that there is now a curve (blue) of
points in (a,b) space where stability is lost,
where the stable solution will suddenly jump
to an alternate outcome.
But in a cusp geometry the bifurcation curve
loops back on itself, giving a second branch
where this alternate solution itself loses
stability, and will make a jump back to the
original solution set. By repeatedly increasing
b and then decreasing it, one can therefore
observe hysteresis loops, as the system alternately
follows one solution, jumps to the other,
follows the other back, then jumps back to
the first.
However, this is only possible in the region
of parameter space a < 0. As a is increased,
the hysteresis loops become smaller and smaller,
until above a = 0 they disappear altogether
(the cusp catastrophe), and there is only
one stable solution.
One can also consider what happens if one
holds b constant and varies a. In the symmetrical
case b = 0, one observes a pitchfork bifurcation
as a is reduced, with one stable solution
suddenly splitting into two stable solutions
and one unstable solution as the physical
system passes to a < 0 through the cusp point
(0,0) (an example of spontaneous symmetry
breaking). Away from the cusp point, there
is no sudden change in a physical solution
being followed: when passing through the curve
of fold bifurcations, all that happens is
an alternate second solution becomes available.
A famous suggestion is that the cusp catastrophe
can be used to model the behaviour of a stressed
dog, which may respond by becoming cowed or
becoming angry. The suggestion is that at
moderate stress (a > 0), the dog will exhibit
a smooth transition of response from cowed
to angry, depending on how it is provoked.
But higher stress levels correspond to moving
to the region (a < 0). Then, if the dog starts
cowed, it will remain cowed as it is irritated
more and more, until it reaches the 'fold'
point, when it will suddenly, discontinuously
snap through to angry mode. Once in 'angry'
mode, it will remain angry, even if the direct
irritation parameter is considerably reduced.
A simple mechanical system, the "Zeeman Catastrophe
Machine", nicely illustrates a cusp catastrophe.
In this device, smooth variations in the position
of the end of a spring can cause sudden changes
in the rotational position of an attached
wheel.Catastrophic failure of a complex system
with parallel redundancy can be evaluated
based on relationship between local and external
stresses. The model of the structural fracture
mechanics is similar to the cusp catastrophe
behavior. The model predicts reserve ability
of a complex system.
Other applications include the outer sphere
electron transfer frequently encountered in
chemical and biological systems and modelling
Real Estate Prices.Fold bifurcations and the
cusp geometry are by far the most important
practical consequences of catastrophe theory.
They are patterns which reoccur again and
again in physics, engineering and mathematical
modelling.
They produce the strong gravitational lensing
events and provide astronomers with one of
the methods used for detecting black holes
and the dark matter of the universe, via the
phenomenon of gravitational lensing producing
multiple images of distant quasars.
The remaining simple catastrophe geometries
are very specialised in comparison, and presented
here only for curiosity value.
=== Swallowtail catastrophe ===
V
=
x
5
+
a
x
3
+
b
x
2
+
c
x
{\displaystyle V=x^{5}+ax^{3}+bx^{2}+cx\,}
The control parameter space is three-dimensional.
The bifurcation set in parameter space is
made up of three surfaces of fold bifurcations,
which meet in two lines of cusp bifurcations,
which in turn meet at a single swallowtail
bifurcation point.
As the parameters go through the surface of
fold bifurcations, one minimum and one maximum
of the potential function disappear. At the
cusp bifurcations, two minima and one maximum
are replaced by one minimum; beyond them the
fold bifurcations disappear. At the swallowtail
point, two minima and two maxima all meet
at a single value of x. For values of a>0,
beyond the swallowtail, there is either one
maximum-minimum pair, or none at all, depending
on the values of b and c. Two of the surfaces
of fold bifurcations, and the two lines of
cusp bifurcations where they meet for a<0,
therefore disappear at the swallowtail point,
to be replaced with only a single surface
of fold bifurcations remaining. Salvador Dalí's
last painting, The Swallow's Tail, was based
on this catastrophe.
=== Butterfly catastrophe ===
V
=
x
6
+
a
x
4
+
b
x
3
+
c
x
2
+
d
x
{\displaystyle V=x^{6}+ax^{4}+bx^{3}+cx^{2}+dx\,}
Depending on the parameter values, the potential
function may have three, two, or one different
local minima, separated by the loci of fold
bifurcations. At the butterfly point, the
different 3-surfaces of fold bifurcations,
the 2-surfaces of cusp bifurcations, and the
lines of swallowtail bifurcations all meet
up and disappear, leaving a single cusp structure
remaining when a>0.
== Potential functions of two active variables
==
Umbilic catastrophes are examples of corank
2 catastrophes. They can be observed in optics
in the focal surfaces created by light reflecting
off a surface in three dimensions and are
intimately connected with the geometry of
nearly spherical surfaces.
Thom proposed that the Hyperbolic umbilic
catastrophe modeled the breaking of a wave
and the elliptical umbilic modeled the creation
of hair like structures.
=== Hyperbolic umbilic catastrophe ===
V
=
x
3
+
y
3
+
a
x
y
+
b
x
+
c
y
{\displaystyle V=x^{3}+y^{3}+axy+bx+cy\,}
=== Elliptic umbilic catastrophe ===
V
=
x
3
3
−
x
y
2
+
a
(
x
2
+
y
2
)
+
b
x
+
c
y
{\displaystyle V={\frac {x^{3}}{3}}-xy^{2}+a(x^{2}+y^{2})+bx+cy\,}
=== Parabolic umbilic catastrophe ===
V
=
x
2
y
+
y
4
+
a
x
2
+
b
y
2
+
c
x
+
d
y
{\displaystyle V=x^{2}y+y^{4}+ax^{2}+by^{2}+cx+dy\,}
== Arnold's notation ==
Vladimir Arnold gave the catastrophes the
ADE classification, due to a deep connection
with simple Lie groups.
A0 - a non-singular point:
V
=
x
{\displaystyle V=x}
.
A1 - a local extremum, either a stable minimum
or unstable maximum
V
=
±
x
2
+
a
x
{\displaystyle V=\pm x^{2}+ax}
.
A2 - the fold
A3 - the cusp
A4 - the swallowtail
A5 - the butterfly
Ak - a representative of an infinite sequence
of one variable forms
V
=
x
k
+
1
+
⋯
{\displaystyle V=x^{k+1}+\cdots }
D4− - the elliptical umbilic
D4+ - the hyperbolic umbilic
D5 - the parabolic umbilic
Dk - a representative of an infinite sequence
of further umbilic forms
E6 - the symbolic umbilic
V
=
x
3
+
y
4
+
a
x
y
2
+
b
x
y
+
c
x
+
d
y
+
e
y
2
{\displaystyle V=x^{3}+y^{4}+axy^{2}+bxy+cx+dy+ey^{2}}
E7
E8There are objects in singularity theory
which correspond to most of the other simple
Lie groups.
== See also
