Outline of Prof. Nalini Joshi's lectures:

Hunting Nonlinear Mathematical Butterflies.

A series of 5 lectures by Prof. Nalini Joshi.

• Introduction - chaos and order
  - what are mathematical butterflies?
  - integrable ODEs
  - asymptotic analysis
• Linear Asymptotics
  - divergent asymptotic series
  - Borel summation
  - natural summation
• Nonlinear Asymptotics
  -major difficulties
  - how to approximate solutions
• Painleve Butterflies
  - divergent asymptotic series
  - finding the true solution
• Catching Butterflies
  - new approximations

The "butterfly effect" is a famous metaphor of chaos theory. According to this metaphor, a butterfly flapping its wings near the coast of Peru could influence whether (and where) a tropical cyclone lands on the coast of Queensland. The butterfly represents an extremely unstable solution of the differential equations modelling the weather.But such solutions exist in many differential equations, including ones that are ordered or integrable and have no chaos whatsoever. The aim of these lectures is to explain the mathematical methods for finding and describing such solutions of nonlinear differential equations in asymptotic limits.