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swallowtail and butterfly buy back the smooth DE deterministic intuition lost with discontinuous changes in results.”
He grinned mischievously as he asked, “Can you see it?”
Thom’s catastrophes serve as accessible and powerful theoretical settings for the use of energy as a generalizable, one dimensional, dependent, resulting effect, influenced by one or several, sometimes conflicting, independent, causal, variables. For more examples: the weight of a ship (smaller to greater, left to right, along the x, normal dimension) and the position of center of gravity (smaller to greater, front to back, along the y splitting dimension) are causal with a jump in roof-height energy from stability to capsizing, a discontinuity emerging from initially smooth changes in stability. As above, gradually increasing tension (the left to right normal factor) and alienation (the back to front (splitting factor) in inmates generate a sudden increment in energy, from subtlety increasing tension in relative quiet to the sudden outbreak in a riot in the prison population. Embryological notochord somitogenesis, (that which become the vertebrate of the spinal column) has a smooth (left to right) causal influence that Chris Zeeman named a normal factor. It is the smooth growth of the material wave of mesodermal (to become muscle, connective tissue and bone) tissue. Zeeman called the front to back dimensional gradient of influence, the secondary wave of adhesiveness, the splitting factor. The value of this secondary wave co-determined a critical-valued interaction between these causal parameters leading to a discontinuous change in the “energy” equivalent continuity of developmental growth and vertebral column segmentation.
A little more technically: Thom’s basic mathematical contributions were in differential topology and analysis with particular emphasis on what is called structural stability of surfaces representing and supporting actions called manifolds. For example, in a graph of a function, say F(x), such that a change in cause x determines what happens to the result y= F(x), the stability question involves what happens when one perturbs F(x) with a littleδ, i.e. δ + F(x). Do the topological properties of the surface representing the potential range of actions of the system (such as nearness of an originally close point set, continuity and connectedness of the surface, its dimensionality, its compactness as a generalization of finiteness)

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