Singularities, Bifurcations and Catastrophes

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Prerequisites:

  • Calculus of Several Variables (eg MATH20132)
  • Algebraic Structures 1 (eg MATH20201)

Recommended:

  • Algebraic Structures 2 (MATH20212)

Specific topics you will need: Critical points of functions (MATH20132), existence and uniqueness of solutions of ordinary differential equations (from MATH10222 or similar), and familiarity with the notion of fields and rings (MATH20201/MATH20212).

Description

This course is about bifurcations in families of functions or maps, and how aspects of their geometry changes from one member of the family to the next. This has many applications, as there are many systems that have some external parameters which can perhaps be controlled and as the parameter is varied so the properties of the system may change.

For example, consider the family \(f_u(x) = x^3 -3ux\) (here x is the variable and u the parameter). When u=0 this familiar function \(f_0(x)=x^3\) has a degenerate critical point at the origin. As u is increased, the critical point splits up into two non-degenerate ones, at \(x=\pm \sqrt{u}\). If on the other hand u is decreased, the two critical points disappear (or they become complex).

The course will look at such examples, and introduce techniques which tell us:

  1. what order polynomial can be used to study a given function (locally) - for example \(x^3+x^4\) can be replaced by \(x^3\) (this is called finite determinacy)
  2. for a given function, which deformations do we need to study (the so-called versal deformations)
  3. Applications of these methods

See the Reading list for information on texts and other books.