texpower math example · Catastrophe Theory studies and classi es phenomena characterized by sudden...

Catastrophe Theory

Vladimir I. Rene Christopher

ARNOLD THOM ZEEMANN

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.

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.

First Examples

� � � � � � ��

� ��

� � � � � � � ��

� � � � � � � �� C

88 92 96 100 104 108

� � � � ��

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

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!

• The positions of C where the equilibrium point

jumps lie on a cusp.

Wheel jumps when C crosses the red curve

��

• 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!

wheel jumps!

wheel does not jump!

Threeequilibriumpoints

One equlibriumpoint

Let us consider another machine: “parabolic rock”.

Consider the mathematical model:

� � � � � � � � � � � � � � � � ��

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

��

� !�!

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

&�&�&�&�

&�&�&

'�'�'�'�

'�'�'

(�( )�)

y = x2 ⇒

x = t, y = t2 ⇒

MN : (x−t)+2t(y−t2) = 0

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

.�.�.�.�

.�.�.

/�/�/�/�

/�/�/

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

The set of normals to parabola.

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

The graph of Σ : t3 + Y t + X = 0.

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

2. The equilibrium point jumps when C crosses the

jump downjump up

3. One can avoid the jump (the catastrophe)

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?

The surface of “equilibrium points”:

F (t, u1, . . . , um) =∂

∂tU(t, u1, . . . , um) = 0

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

Control space

>�> ?�?@�@�

@A�A�

Equilibrium point

Control point

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

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!

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

Case k = 3;

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

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

Bifurcation set:v21

Bifurcation set

Equilibrium surface

Case k = 4;

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

2 + v1s

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

Bifurcation set:

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

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

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 +

2Y t3 + Xt2 − 2Y t

Fermat’s principle ⇒ ∂L∂t

(X, Y, t) = 0.

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

caustics

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

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

maximal time sheet

minimal time sheet

maximal time sheet

]�] ^�^

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 .

phase transition

P ,T plane

Psychology

Results

Abilities

Efforts

Finances

Ecology

Real surface

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.

London etc.: Cambridge University Press. VI, 179 p.

Dujardin, L. ”Catastrophe Teacher: An

Introduction for Experimentalists.”

http://perso.wanadoo.fr/l.d.v.dujardin/ct/eng index.html.

Gilmore, R. Catastrophe Theory for Scientists and

Engineers. New York: Dover, 1993.

Thom, R. Structural Stability and Morphogenesis:

An Outline of a General Theory of Models.

Reading, MA: Addison-Wesley, 1993.

Thompson, J. M. T. Instabilities and Catastrophes

in Science and Engineering. New York: Wiley, 1982.

Woodcock, A. E. R. and Davis, M. Catastrophe

Theory. New York: E. P. Dutton, 1978.

Zahler, R. and Sussman, H. J. ”Claims and

Accomplishments of Applied Catastrophe

Theory.” Nature 269, 759-763, 1977.

Zeeman, E. C. Catastrophe Theory–Selected

Papers 1972-1977. Reading, MA: Addison-Wesley,

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Ian Stewart - Bookshelf Collection€¦ · CATASTROPHE THEORY* Ian Stewart (received 26 July, 1976) Rene Thom's catastrophe theory [15] has attracted considerable attention recently,