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The Catastrophe Theory

Definition

The word catastrophe comes from Greek tragic drama and refers to the sudden twist of development in the plots. Catastrophe theory is a method for describing the evolution of forms in nature. It is particularly applicable where gradually changing forces produce sudden effects. Its interdisciplinary character was, for people with widely differing agendas, a cultural connector linking mathematics, biology, social sciences and philosophy.

It is represented by using topology. Topology is a generalization of geometry that studies spaces with the degree of generality appropriate to a specific problem. One central concern of topology is to study the properties of spaces that do not change under a continuous transformation, that is, translation, rotation and stretching without tearing.

catastrophe topo

Representation

A new method was needed that would focus on shapes, account for their stability, and explain their creation and destruction. The models are not suited for action or prediction but rather aimed at describing, and intelligibly understanding natural phenomena.
Reflectivity: a=a
Symmetry: a=b then b=a
Transitivity: a=b and b=c then a=c

“Reality presents itself to us as phenomena and shapes.” – Rene Thom

catastrophe multi equi

The Fold

There was only one kind of singularity that could occur in the catastrophe theory and that is called a fold.

The fold made possible the definition of flux equilibrium by combining a series of different surfaces to define multiple points of equilibrium.

catastrophe multi equi

Mathematical Definition

Catastrophe models come in both dynamic and static forms, the static forms being simply the equilibrium (stable and unstable) of the dynamic forms. Multiple stable equilibriums are inherent in catastrophe models.

  1. Stable Equilibrium
    i. A state which depends continuously on the parameters
    ii. An equilibrium such that nearby states remain close as the state evolves.
  2. Catastrophe
    A sudden change in state
  3. Structural Stability
    A model is structurally stable if its qualitative behaviour is unchanged by small perturbations of the parameters
  4. Generic
    A technical term meant to suggest what the property usually holds.

Internal Variables = u
Control Variables = a,b
Potential Function = V
The Potential function related the internal and the control variable = V (a,b,u)

 

Types of Catastrophes

The classification of seven singularities later led to the seven elementary catastrophes.
1.The fold
2.The Cusp
3.The Swallowtail
4.The butterfly
5. The three Umbilics

Types of catastrophes

 

epigenetic_landscape 02

epigenetic_landscape

Legacy

Chaos: When the present determines the future, but the approximate present does not approximately determine the future.

It is true in a sense, the ambitions of the theory failed, but in practice, the theory has succeeded. This is especially true of chaos theory, with which catastrophe theory had important interactions.

Thom never argued for the intrinsic superiority of his method but rather for its greater capacity to explain the world as it is perceived.

Chaos

Application

The catastrophe theory will be applied as moments of catastrophe by overlapping two opposing situations:

  1. A twist in the narrative
  2. The physical fold on the surface

The different moments will be unrolled and unified on a singular surface to create multiple moments of catastrophes. Both the stable and unstable conditions of the continuous surface will be addressed by negotiating between shifts in unstable narratives on stable physical folds and shifts in the stable narrative on the unstable physical folds.

Application

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