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Catastrophe Theory

texpower math example · Catastrophe Theory studies and classi es phenomena characterized by sudden shifts in behavior arising from small changes in ... Gilmore, R. Catastrophe Theory

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Page 1: texpower math example · Catastrophe Theory studies and classi es phenomena characterized by sudden shifts in behavior arising from small changes in ... Gilmore, R. Catastrophe Theory

Catastrophe Theory

Page 2: texpower math example · Catastrophe Theory studies and classi es phenomena characterized by sudden shifts in behavior arising from small changes in ... Gilmore, R. Catastrophe Theory

Catastrophe Theory

Vladimir I. Rene Christopher

ARNOLD THOM ZEEMANN

Page 3: texpower math example · Catastrophe Theory studies and classi es phenomena characterized by sudden shifts in behavior arising from small changes in ... Gilmore, R. Catastrophe Theory

Originated by Vladimir I. Arnold, Rene Thom, and

Christopher Zeemann in the 1960-70s,

Catastrophe Theory studies and classifies

phenomena characterized by sudden shifts in

behavior arising from small changes in

circumstances.

Page 4: texpower math example · Catastrophe Theory studies and classi es phenomena characterized by sudden shifts in behavior arising from small changes in ... Gilmore, R. Catastrophe Theory

Originated by Vladimir I. Arnold, Rene Thom, and

Christopher Zeemann in the 1960-70s,

Catastrophe Theory studies and classifies

phenomena characterized by sudden shifts in

behavior arising from small changes in

circumstances.

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. Catastrophe theory has been applied to

a number of different physical, chemical, biological,

and even social phenomena.

Page 5: texpower math example · Catastrophe Theory studies and classi es phenomena characterized by sudden shifts in behavior arising from small changes in ... Gilmore, R. Catastrophe Theory

First Examples

� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �

� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �

� �� �� �� �� �� �

� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �

� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �� � � � � � � �C

AT

AST

RO

PHE

POIN

T

88 92 96 100 104 108

� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �

� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �

Page 6: texpower math example · Catastrophe Theory studies and classi es phenomena characterized by sudden shifts in behavior arising from small changes in ... Gilmore, R. Catastrophe Theory

Zeemann machine

This device, invented by the mathematician

Christopher Zeeman consists of a wheel which is

tethered by an elastic to a fixed point B in its plane.

The control input to the system is another elastic

attached to the same point A as the first and roughly

of the same length. The other end C of the elastic

can be moved about an area diametrically opposite to

the fixed point.��

�� �� ��

�� ��O

A

B

C

Page 7: texpower math example · Catastrophe Theory studies and classi es phenomena characterized by sudden shifts in behavior arising from small changes in ... Gilmore, R. Catastrophe Theory

The experiment shows that:

• To each position of C there correspond 1 or 3

equilibrium points of the system.

• If we slowly move C then usually the wheel rotates

slowly (this means that the equilibrium point

slowly changes), however sometimes the wheel

(the equilibrium point) suddenly jumps!

Page 8: texpower math example · Catastrophe Theory studies and classi es phenomena characterized by sudden shifts in behavior arising from small changes in ... Gilmore, R. Catastrophe Theory

• The positions of C where the equilibrium point

jumps lie on a cusp.

Wheel jumps when C crosses the red curve

��� ���

��� ���

��� ���

��� ���

O

A

B

C

Page 9: texpower math example · Catastrophe Theory studies and classi es phenomena characterized by sudden shifts in behavior arising from small changes in ... Gilmore, R. Catastrophe Theory

• The cusp is the common boundary of two regions:

I) the set of C such that the system has one

equilibrium point; II) the set of C such that the

system has three equilibrium points;

• Hysteresis behavior. If the equilibrium point

jumps while C crosses the boundary, then while C

is moving in the opposite direction the equilibrium

point does not jump!

Page 10: texpower math example · Catastrophe Theory studies and classi es phenomena characterized by sudden shifts in behavior arising from small changes in ... Gilmore, R. Catastrophe Theory

wheel jumps!

wheel does not jump!

Threeequilibriumpoints

One equlibriumpoint

Page 11: texpower math example · Catastrophe Theory studies and classi es phenomena characterized by sudden shifts in behavior arising from small changes in ... Gilmore, R. Catastrophe Theory

Let us consider another machine: “parabolic rock”.

Page 12: texpower math example · Catastrophe Theory studies and classi es phenomena characterized by sudden shifts in behavior arising from small changes in ... Gilmore, R. Catastrophe Theory

Consider the mathematical model:

(X,Y)

x

y

� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � �

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Page 13: texpower math example · Catastrophe Theory studies and classi es phenomena characterized by sudden shifts in behavior arising from small changes in ... Gilmore, R. Catastrophe Theory

Consider the mathematical model:

(X,Y)

x

y

" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "" " " " " " " " " " " " " " " " "

# # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # ## # # # # # # # # # # # # # # # #

$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %

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B

A

M

N

O

(�( )�)

y = x2 ⇒

x = t, y = t2 ⇒

MN : (x−t)+2t(y−t2) = 0

Page 14: texpower math example · Catastrophe Theory studies and classi es phenomena characterized by sudden shifts in behavior arising from small changes in ... Gilmore, R. Catastrophe Theory

Consider the mathematical model:

(X,Y)

x

y

* * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * *

+ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + + + + +

, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

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B

A

M

N

O

0�0 1�1

y = x2 ⇒

x = t, y = t2 ⇒

MN : (x−t)+2t(y−t2) = 0

Then, setting X = − 12x, Y = 1

2(1 − 2y), we get

the equation t3 + Y t + X = 0

Page 15: texpower math example · Catastrophe Theory studies and classi es phenomena characterized by sudden shifts in behavior arising from small changes in ... Gilmore, R. Catastrophe Theory

The set of normals to parabola.

Page 16: texpower math example · Catastrophe Theory studies and classi es phenomena characterized by sudden shifts in behavior arising from small changes in ... Gilmore, R. Catastrophe Theory

Let us construct the graph of the surface

Σ : t3 + Y t + X = 0 in the three-dimensional space with

coordinates (X, Y, t).

Consider the graph of Σ as the family of graphs of the

function X = −t3 − Y t while Y varies.

Y<0 Y>0Y=0

t

X X X

t t

Page 17: texpower math example · Catastrophe Theory studies and classi es phenomena characterized by sudden shifts in behavior arising from small changes in ... Gilmore, R. Catastrophe Theory

The graph of Σ : t3 + Y t + X = 0.

Y

X

t

Page 18: texpower math example · Catastrophe Theory studies and classi es phenomena characterized by sudden shifts in behavior arising from small changes in ... Gilmore, R. Catastrophe Theory

Explanation of experimental phenomena.

1. The number of equilibrium points depends of the

position of C.

2�2 3�3

4�4 5�56�6 7�7

8�89�9:�: ;�;

<�< =�=C

C1

3

Page 19: texpower math example · Catastrophe Theory studies and classi es phenomena characterized by sudden shifts in behavior arising from small changes in ... Gilmore, R. Catastrophe Theory

2. The equilibrium point jumps when C crosses the

cusp.

C

C1

jump downjump up

Page 20: texpower math example · Catastrophe Theory studies and classi es phenomena characterized by sudden shifts in behavior arising from small changes in ... Gilmore, R. Catastrophe Theory

3. One can avoid the jump (the catastrophe)

Page 21: texpower math example · Catastrophe Theory studies and classi es phenomena characterized by sudden shifts in behavior arising from small changes in ... Gilmore, R. Catastrophe Theory

The mathematical formulation.

We are given a system with one degree of freedom

(one-dimensional configuration space) and

m-dimensional control space.

t is a coordinate in the configuration space (a

parameter of the system);

u1, u2, . . . , um are coordinates in the control space

(control parameters);

U(t, u1, . . . , um) is the “potential energy” of the system;

The main problem: How does an equilibrium

point of the system behave when we vary control

parameters?

Page 22: texpower math example · Catastrophe Theory studies and classi es phenomena characterized by sudden shifts in behavior arising from small changes in ... Gilmore, R. Catastrophe Theory

The surface of “equilibrium points”:

F (t, u1, . . . , um) =∂

∂tU(t, u1, . . . , um) = 0

t

u2

u1

Equilibrium points surface: F(t,u)=0

Control space

>�> ?�?@�@�

@A�A�

A

Equilibrium point

Control point

Page 23: texpower math example · Catastrophe Theory studies and classi es phenomena characterized by sudden shifts in behavior arising from small changes in ... Gilmore, R. Catastrophe Theory

Given F (t, u1, . . . , un) we define the bifurcation set:

BF = {u | ∃t0 :∂

∂tF (t0, u) = 0}

BCB DCD

ECE FCF

GCG HCH

ICI JCJ

Equilibrium surface

Tangent planeis vertical

Tangent plane isnot vertical

Bifurcation set

1−1 correspondence

Page 24: texpower math example · Catastrophe Theory studies and classi es phenomena characterized by sudden shifts in behavior arising from small changes in ... Gilmore, R. Catastrophe Theory

Theorem. Assume that ∂j

∂tjU(t, u1, . . . , um) = 0, for all

j = 1, k, and ∂k+1

∂tk+1 U(t, u1, . . . , um) 6= 0.

Then, “in the general situation”, one can change

coordinates:

t = t(s, v) u = u(v)

in such a way that

U(t(s, v), u(v)) = sk+1 + vk−1sk−1 + · · · + v1s

Note that k < m!

This means that, in order to solve our problem, we

can study concrete surfaces!

Page 25: texpower math example · Catastrophe Theory studies and classi es phenomena characterized by sudden shifts in behavior arising from small changes in ... Gilmore, R. Catastrophe Theory

Case k = 2;

Potential energy: U(s, v1) = s3 + v1s

Equilibrium surface: U(s, v1) = 3s2 + v1

Bifurcation set: v1 = 0

K KK KK KL LL LL L

M MM MM MN NN NN N

s

v1

v1

Page 26: texpower math example · Catastrophe Theory studies and classi es phenomena characterized by sudden shifts in behavior arising from small changes in ... Gilmore, R. Catastrophe Theory

Case k = 3;

Potential energy: U(s, v1) = s4 + v2s2 + v1s

Equilibrium surface: F (s, v1) = 4s3 + 2v2s + v1

Bifurcation set:v21

8+

v32

27= 0

Page 27: texpower math example · Catastrophe Theory studies and classi es phenomena characterized by sudden shifts in behavior arising from small changes in ... Gilmore, R. Catastrophe Theory

v

v

s

2

1

v

v1

2

Bifurcation set

Equilibrium surface

Page 28: texpower math example · Catastrophe Theory studies and classi es phenomena characterized by sudden shifts in behavior arising from small changes in ... Gilmore, R. Catastrophe Theory

Case k = 4;

Potential energy: U(s, v1) = s5 + v3s3 + v2s

2 + v1s

Equilibrium surface: F (s, v1) = 5s4 + 3v3s2 + 2v2s + v1

Bifurcation set:

Page 29: texpower math example · Catastrophe Theory studies and classi es phenomena characterized by sudden shifts in behavior arising from small changes in ... Gilmore, R. Catastrophe Theory

Optics

O�O P�P

Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q QR R R R R R R R R R R R R R R R

S S S S S S S S S S S S S S S ST T T T T T T T T T T T T T T T

U U U U U U U U U U U U U U U UV V V V V V V V V V V V V V V V

W�W�W�W�

W�W�W�W�

W�W�W�W�

W�W�W�W

X�X�X�X�

X�X�X�X�

X�X�X�X�

X�X�X�X

Y Y Y Y Y Y Y YY Y Y Y Y Y Y YY Y Y Y Y Y Y YY Y Y Y Y Y Y YY Y Y Y Y Y Y YY Y Y Y Y Y Y YY Y Y Y Y Y Y YY Y Y Y Y Y Y YY Y Y Y Y Y Y YY Y Y Y Y Y Y YY Y Y Y Y Y Y YY Y Y Y Y Y Y YY Y Y Y Y Y Y YY Y Y Y Y Y Y YY Y Y Y Y Y Y YY Y Y Y Y Y Y YY Y Y Y Y Y Y YY Y Y Y Y Y Y YY Y Y Y Y Y Y YY Y Y Y Y Y Y YY Y Y Y Y Y Y YY Y Y Y Y Y Y Y

Z Z Z Z Z Z Z ZZ Z Z Z Z Z Z ZZ Z Z Z Z Z Z ZZ Z Z Z Z Z Z ZZ Z Z Z Z Z Z ZZ Z Z Z Z Z Z ZZ Z Z Z Z Z Z ZZ Z Z Z Z Z Z ZZ Z Z Z Z Z Z ZZ Z Z Z Z Z Z ZZ Z Z Z Z Z Z ZZ Z Z Z Z Z Z ZZ Z Z Z Z Z Z ZZ Z Z Z Z Z Z ZZ Z Z Z Z Z Z ZZ Z Z Z Z Z Z ZZ Z Z Z Z Z Z ZZ Z Z Z Z Z Z ZZ Z Z Z Z Z Z ZZ Z Z Z Z Z Z ZZ Z Z Z Z Z Z ZZ Z Z Z Z Z Z Z

t

M(X,Y)

Let L(X, Y, t) be the length of the path joining ∞ and

M(X; Y ).

L(X, Y, t) ≈ −1

4(1 + 5X)t4 +

1

2Y t3 + Xt2 − 2Y t

Fermat’s principle ⇒ ∂L∂t

(X, Y, t) = 0.

Page 30: texpower math example · Catastrophe Theory studies and classi es phenomena characterized by sudden shifts in behavior arising from small changes in ... Gilmore, R. Catastrophe Theory

Then L(X, Y, t) is equivalent to t4 + v2t2 + v1t.

caustics

[ [ [ [ [ [ [ [ [ [ [ [[ [ [ [ [ [ [ [ [ [ [ [[ [ [ [ [ [ [ [ [ [ [ [

\ \ \ \ \ \ \ \ \ \ \ \\ \ \ \ \ \ \ \ \ \ \ \\ \ \ \ \ \ \ \ \ \ \ \

maximal time sheet

minimal time sheet

maximal time sheet

]�] ^�^

C1

Page 31: texpower math example · Catastrophe Theory studies and classi es phenomena characterized by sudden shifts in behavior arising from small changes in ... Gilmore, R. Catastrophe Theory

Chemistry

Van der Waals Equation of State

(P +α

V 2)(V − β) = RT

Here P is the gas pressure, V is the gas volume, and T

is the gas temperature, and α, β are parameters

depending on the gas molecule properties.

Set X = 1V, then after a linear change of coordinates

we appear at the equation

X3 + aX + b = 0,

where a, b are linearly expressed in terms of P , T .

Page 32: texpower math example · Catastrophe Theory studies and classi es phenomena characterized by sudden shifts in behavior arising from small changes in ... Gilmore, R. Catastrophe Theory

dens

ity

gas

fluid

phase transition

X

P ,T plane

Page 33: texpower math example · Catastrophe Theory studies and classi es phenomena characterized by sudden shifts in behavior arising from small changes in ... Gilmore, R. Catastrophe Theory

Psychology

Results

Abilities

Efforts

Page 34: texpower math example · Catastrophe Theory studies and classi es phenomena characterized by sudden shifts in behavior arising from small changes in ... Gilmore, R. Catastrophe Theory

Finances

Page 35: texpower math example · Catastrophe Theory studies and classi es phenomena characterized by sudden shifts in behavior arising from small changes in ... Gilmore, R. Catastrophe Theory

Ecology

Page 36: texpower math example · Catastrophe Theory studies and classi es phenomena characterized by sudden shifts in behavior arising from small changes in ... Gilmore, R. Catastrophe Theory

Real surface

Page 37: texpower math example · Catastrophe Theory studies and classi es phenomena characterized by sudden shifts in behavior arising from small changes in ... Gilmore, R. Catastrophe Theory

REFERENCES:

Arnold, V. I. Catastrophe Theory, 3rd ed. Berlin:

Springer-Verlag, 1992.

Poston, Tim; Stewart, Ian Catastrophe theory and

its applications. (English) Surveys and Reference

Works in Mathematics. 2. London - San Francisco -

Melbourne: Pitman. XVIII, 491 p.

Brocker, Th. Differentiable germs and

catastrophes. Translated by L. Lander. (English)

London Mathematical Society Lecture Note Series. 17.

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