Degenerate Critical Points and Catastrophes: Fold Catastrophe

​
time, t
1.5
perturbation, ϵ
0
θ
1
= 2π ×
0
1
8
1
4
θ
2
= 2π ×
-
1
2
-
7
16
-
3
8
-
5
16
-
1
4
-
3
16
-
1
8
-
1
16
0
ϕ
1
= π ×
1
8
1
4
3
8
1
2
ϕ
2
= π ×
1
16
1
8
3
16
1
4
5
16
3
8
7
16
1
2
The simple algebraic curve
φ(t)=
3
t
+ϵt
is a good enough example to explain degeneracy and catastrophe in the extended phase space
(t,φ(t),

φ
(t))
. With the help of this Demonstration, students can easily understand the fold catastrophe.

Details

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
x
0
of a differentiable function
y=f(x)
of one variable
x
satisfies
′
f
(
x
0
)=0
.
2. A nondegenerate critical point
x
0
of a differentiable function
y=f(x)
of one variable
x
satisfies
′
f
(
x
0
)=0
and
″
f
(
x
0
)≠0
; if
′
f
(
x
0
)=0
and
″
f
(
x
0
)=0
,
x
0
is called a degenerate critical point.
For
φ(t)=
3
t
+ϵt
and
ϵ<0
, there are two nondegenerate critical points; for
ϵ=0
, there is one degenerate critical point; and for
ϵ>0
, there are no critical points.

References

[1] V. I. Arnold, Ordinary Differential Equations (R. A. Silverman, ed. and trans.), Cambridge, MA: MIT Press, 1973.
[2] D. V. Anosov et al, eds., Dynamical Systems I: Ordinary Differential Equations and Smooth Dynamical Systems (Encyclopaedia of Mathematical Sciences, Vol. 1), New York: Springer, 1997.
[3] V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations (J. Szücs, trans.; M. Levi, ed.), New York: Springer-Verlag, 1983.
[4] V. I. Arnold, Mathematical Methods of Classical Mechanics (K. Vogtmann and A. Weinstein, trans.), New York: Springer-Verlag, 1978.
[5] V. I. Arnold, ed., Dynamical Systems V: Bifurcation Theory and Catastrophe Theory (Encyclopaedia of Mathematical Sciences, Vol. 5), New York: Springer, 1994.
[6] V. I. Arnold, Catastrophe Theory, 3rd ed. (G. S. Wassermann, trans.), New York: Springer-Verlag, 1992.
[7] D. P. L. Castrigiano and S. A. Hayes, Catastrophe Theory, Reading, MA: Addison-Wesley, 1993.
[8] R. Thom, Structural Stability and Morphogenesis: An Outline of a General Theory of Models (D. H. Fowler, trans.), Reading MA: Addison-Wesley., 1989.
[9] J. Milnor, Morse Theory, Princeton, NJ: Princeton University Press, 1963.
[10] Y. Matsumoto, An Introduction to Morse Theory (Translations of Mathematical Monographs, Vol. 208) (K. Hudson and M. Saito, trans.), Providence, RI: American Mathematical Society, 2002.
[11] S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, Reading, MA: Addison-Wesley, 1994.
[12] S. Wolfram, A New Kind of Science, Champaign, IL: Wolfram Media, 2002.

External Links

Degenerate (Wolfram MathWorld)
Degeneracy (Wolfram MathWorld)
Critical Point (Wolfram MathWorld)
Catastrophe (Wolfram MathWorld)
Catastrophe Theory (Wolfram MathWorld)
Fold Catastrophe (Wolfram MathWorld)
Some Past Initiatives (NKS|Online)
Multidimensional generalizations[of intrinsically defined curves] (NKS|Online)
Discreteness in space (NKS|Online)
History[of theories of biological form] (NKS|Online)
Phenomenology of microscopic fracture (NKS|Online)

Permanent Citation

Ki-Jung Moon
​
​"Degenerate Critical Points and Catastrophes: Fold Catastrophe"​
​http://demonstrations.wolfram.com/DegenerateCriticalPointsAndCatastrophesFoldCatastrophe/​
​Wolfram Demonstrations Project​
​Published: December 9, 2013