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The theory of catastrophes

The author: Doctor of Sciences, Professor Kuznetsov A.P.

18 lectures

problems and lessons

Section 1. The basic concepts

1.1 Approximations in mathematics and physics and strategy of constructioning of multiparameter models. Idea of approximation in mathematics. Approximations in physics and physical models. A Taylor series and its use for constructioning multiparameter models, examples. Completeness and universality of the models obtained by means of Taylor approximations.

1.2 The notion of typicalness according to Poincare and the related concepts. Intuitive notion of typicalness. Generic case and degenerated cases. A method of "small perturbations" of parameters. Classification of the degenerated situations by a codimension. Examples of situations of a various codimension. Poincare's strategy of examination of dynamic systems in accordance with an increasing codimension. Concept of robust (structurally stable) systems.

1.3 The first informations about catastrophes. Examples of systems with catastrophes: a mass point in the one-dimensional and two-dimensional potential field, a weighted beam, stability of courts etc. Zeeman machine and gravitational machine of catastrophs. Intersections and differences between the theory of catastrophes and theory of bifurcations.

1.4 Critical points of functions of one variable. Critical points and their role at examination of catastrophes and bifurcations. Some elementary critical points of functions of one variable. The research program of the theory of catastrophes detection of essential parameters, a role of changes of variables, classification of critical points by a codimension - by an example of the analysis of the simplest polynomials.

1.5 Critical points of functions of two variables. Pictorial representation of functions of two variables with the help of contour maps. Examples: horizontals and maps, equipotential lines, phase portraits. Critical points of functions of two variables and their examples. Hessian and examples of its use for distinguishing the degenerated critical points. Quadratic forms and their classification.

1.6 Cubics and their classification. Typical critical points for maps. A case of n-variables and definition of the Morse critical point (Morse saddles).

1.7 Mathematical bases of the theory of catastrophes. A lemma of Morse. A lemma of decomposition. Thom's classification theorem. Germs and perturbations. Completeness and universality of models of the theory of catastrophes.

Section 2. Properties of catastrophes

2.1 Fold catastrophe. The scheme of analysis of catastrophes by an example of a fold (catastrophe's manifold, a bifurcation set, transformations of potential function). Catastrophe of a fold in a system of two attracted currents. Other examples of the systems demonstrating fold catastrophe.

2.2 Cusp catastrophe. Analysis of cusp catastrophe. Dual cusp. Examples of cusp catastrophe: " overturning pendulum " with a nonlinear spring, a critical point of Van der Waals gas, a tunnel diode in the circuit with tunable resistance and electromotive force. Other examples of catastrophe of cusp. Cusp catastrophe and the Whitney's theory of smooth maps. Cusps as singularities of the elementary curves. Evolute of the parabola and its description in terms of cusp catastrophe's manifold. Ruled surfaces.

2.3 Dovetail catastrophe. Analysis of a dovetail catastrophe. An example of the system showing catastrophe a dovetail.

2.4 Cuspoid catastrophes in two-dimensional systems. Fold catastrophe in a two-dimensional system by an example of a problem about a blob rolling out from the hole. Evolution of a relief of potential function, level lines and separatrixes at catastrophe of a fold in two-dimensional systems. Physically distinctive types of catastrophes of cusp in two-dimensional systems, examples. Evolution of level lines and separatrixes for the cusp catastrophe. Searching for the cuspoid catastrophes in two-dimensional systems and a peculiarities of their reduction to a canonical form (a possibility of cusp catastrophe in case of a cubic potential). Concept of nonlocal bifurcations by an example of a model potential. Transformations of a relief of potential function and level lines for the elementary nonlocal codimension-one bifurcation. Why nonlocal bifurcations do not concern to catastrophes? Concept of a nonlocal bifurcation set. A nonlocal bifurcation set for a cusp catastrophe in two-dimensional systems. A pendulum between two charged filaments, as an example of the system featuring a nonlocal bifurcation set.

2.5 Catastrophes of elliptic and hyperbolic umbilics. Analysis of catastrophe of an elliptic umbilic. A nonlocal bifurcation set of catastrophe of an elliptic umbilic. An example of the system demonstrating catastrophe of an elliptic umbilic. Taylor-Couette flows. Analysis of catastrophe of a hyperbolic umbilic.

Section 3. Physical applications of the theory of catastrophes

3.1 Swelling of an elastic rod. How to describe a problem about a swelling of a rod with the help of finite-dimensional potential function? (the method of Rayleigh-Ritz). The degenerated cusp catastrophe in a problem about a swelling of a rod. Power laws in a neighbourhood of a point of catastrophe. Influence of asymmetry. Bifurcation diagrams in case of the symmetric and asymmetrical swelling. A swelling of a rod in a multimode approximation.

3.2 Phase transitions and catastrophes. Landau's theory of phase transitions of the second order: a thermodynamic potential and an order parameter, expansion of a thermodynamic potential in a series with respect to an order parameter. The degenerated cusp catastrophe and phase transitions of the second order. Laws of a modification of thermodynamic functions in a neighbourhood of a point of catastrophe. Phase transitions in the presence of an external field and the complete cusp catastrophe ("blurring" of transitions of the second order, first-order transitions, metastable states). A critical point of phase transitions of the second order. Evolution of a thermodynamic potential as a functions of an order parameter in a neighbourhood of this point. Tricritical potential and tricritical point. The arrangement of parameter space in a neighbourhood of tricritical points. Concept of critical phenomena.

3.3 Catastrophes in geometrical optics. Caustics. Experiments with a cylindrical cup. Caustics and cusp at reflection of light. Application of a Fermat's principle to an explanation of a nature of cusp catastrophe at reflection of light from a cylindrical surface. Reduction to a canonical form in the symmetric case. Other examples of caustics and cusps in optics (transiting of a laser beam through the nonuniform drop, a spherical aberration, mirages, reflection of light from a crystal etc.) Large ocean waves and their caustics and cusps.

Recommended literature

  1. T.Poston, I.Stewart. The theory of catastrophes and its applications. Pitman: London, 1978
  2. R.Gilmor. Catastrophe theory for scientists and engineers. Wiley: London, 1981.
  3. V.I.Arnol'd. The theory of catastrophes. Moscow: Nauka, 1990.
  4. J.M.T.Thompson. Instabilities and catastrophes in a science and engineering. John Wiley & Sons, 1982.
  5. L.D.Landau, E.M.Lifshits. Statistical physics. Moscow: Nauka, 1976.

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