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

**18 lectures**

education

problems
and lessons

**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.

**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.

**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.

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

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