22 September 2020 Tue 5 pm

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The structural stability is a technical name for the intuitive notion that behavior of the some systems is essentially unchanged for relatively large variation of the system parameters. However, in the vicinity of the critical values of the parameters, system can changes its properties or its behavior significantly and abruptly. The Catastrophe theory was established by precise mathematical formulation of these ideas, and by rigorous pursuit of their implications. In the language of the catastrophe theory described abrupt change is called the catastrophe.
Full range of the catastrophe theory application is too broad and diverse to be surveyed in one lecture. Therefore, focus here would be on the usefulness of this approach for modeling of quantum mechanical phenomena. For demonstration wave packet motion in the potential of the one dimensional optical trap purposes was analyzed. The classical and the quantum aspects of the particle dynamics were investigated in detail, and it was shown how to construct models of classical and quantum caustic lines. Obtained catastrophic models describe equally well structurally stable wave patterns, and classical singularities of the differential scattering cross-section.
Quantum and classical dynamics in the phase space was also investigated. Existence of the Wigner catastrophes was shown, and the relationship between their bifurcation sets, and the classical rainbow diagrams was investigated.