Thom’s catastrophe theory

The Thom s catastrophe theory (Thom, 1973) basically describes how, for a given system, a continuous action on the control space (C ), parameterized by c s, provides a suddenly change on its behavior space (/ ), described by variables x s, through the stable singularities of the smooth map... 

Presents the Fokker-Planck quantum description of open systems, by considering the drift and diffusion contributions in quantum evolution, as based on a non-equilibrium Lagrangian, lading with generating of the so called Markovian families of electronic localization functions, accommodating the Thom s catastrophe theory in an iimovative maimer, thus directly describing the atomic valence distribution as well as the molecular formation. 

We have selected two examples (i) QTAIM study on the AgVOj, for the simulation of Ag nucleation and formation on AgV03 provoked in this crystal by the electron-beam irradiation, (ii) An ELF and Thom s catastrophe theory (BET) study for different reaction pathways associated with the decomposition of glycolic acid decomposition. 

Aerts, D., Czachor, M., Gabora, L., Kuna, M., Posiewnik, A., Pykaez, J., Syty, M. (2003). Quantum morphogenesis a variation on Thom s catastrophe theory. Phys. Rev. E Cl,... 

The dominant practice in Quantum chemistry is optimization. If the geometry optimization, for instance through analytic gradients, leads to symmetry-broken conformations, we publish and do not examine the departure from symmetry, the way it goes. This is a pity since symmetry breaking is a catastrophe (in the sense of Thom s theory) and the critical region deserves attention. There are trivial problems (the planar three-fold symmetry conformation of NH3 is a saddle point between the two pyramidal equilibrium conformations). Other processes appear as bifurcations for instance in the electron transfer... 

This fine balance between the protection and destruction of the processes maintaining life is a reminiscent of the catastrophe theory proposed by Rene Thom in the early 1970 s [323]. We are currently applying mathematical modelling to this biological feedback system in order to establish its relationship with the catastrophe hypothesis. 

We have demonstrated that Thom s theory of elementary catastrophes finds a direct application in the analysis of structural instabilities which correspond to the making and/or opening of a ring structure. The usefulness of Thom s classification theorem is a consequence of the fact that all the changes in Vp that are involved in such a process occur on a. two-dimensional submanifold of the behaviour space of the electronic coordinates. Clearly, more complex cases of structural changes are to be expected, cases whose complete description will necessitate the use of the full three-dimensional behaviour space. Such a case is illustrated by the formation of a cage structure. 

A fundamental concept in catastrophe theory is the structurally stable form (this can be a form of thing or process). Such a concept is close to the Platonic concept of idea or to the Aristotelian concept of form. In the Plato system the basic concept is an invariable and absolute idea, while things are imitations of the idea. In the Aristotle system it is assumed that matter and form exist combined into entirety. Form is more important (being an equivalent of the Plato s idea), since only form, an essential and permanent component of things, is cognizable. Hence, structural stability of forms in the Thom theory corresponds, on the one hand, to the invariable and indestructible ideas of Plato and, on the other, to the forms of Aristotle. 

The program of catastrophe theory has been formulated in the Thom s book Structural Stability and Morphogenesis. The philosophical, mathematical and experimental incentives to the program of catastrophe theory may be found there. On the position of catastrophe theory among methods of investigation of stability of solutions of non-linear equations (models) one may learn from a book by Thompson. Stewart s and Zeeman s papers on the perspectives of the development of generalized catastrophe theory are also worth study. 

Thom s theory of catastrophes has acquired much popularity for its simple yet valuable description ofthe system-environment interaction that includes phenomena such as steady state equilibrium and life cycles (Thom, 1973). In particular, biological systems come first under catastrophe modeling because they display a causal action-reaction response to various natural or imposed constraining limits. As an example, the reactions of oiganisms to vital toxicological threats were developed into the survival attractor... 

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