nonlinear dynamics 1: geometry of chaos

An advanced introduction to nonlinear dynamics, with emphasis on methods used to analyze chaotic dynamical systems encountered in science and engineering.


what is this course about?

The theory developed here (that you will not find in any other course :) has much in common with (and complements) statistical mechanics and field theory courses; partition functions and transfer operators are applied to computation of observables and spectra of chaotic systems.

Nonlinear dynamics 1: Geometry of chaos (this course - see syllabus)
  • Topology of flows - how to enumerate orbits, Smale horseshoes
  • Dynamics, quantitative - periodic orbits, local stability
  • Role of symmetries in dynamics
Nonlinear dynamics 2: Chaos rules (second course)
  • Transfer operators - statistical distributions in dynamics
  • Spectroscopy of chaotic systems
  • dynamical zeta functions
  • Dynamical theory of turbulence
The course is in part an advanced seminar in nonlinear dynamics, aimed at PhD students, postdoctoral fellows and advanced undergraduates in physics, mathematics, chemistry and engineering.

prerequisites

A basic background in linear algebra, calculus, ordinary differential equations, probability theory, classical and statistical mechanics: ability to work with equations involving vectors and matrices, differentiate simple functions, and understand what a probability distribution is. Weekly homework assignments require both analytic and numerical work, so we will teach you Python as we go along. Knowledge of Matlab or Octave or another programming language is a very helpful. For introductory literature, check the book.

part 1

part 2

format

Each of the two courses lasts 8 weeks and consists of book study, with links to explanatory videos, and weekly homework assignments (no midterm or final), which include some computer programming. While you are free to use any computational tool that you are comfortable with, we are going to provide you Python script templates for computational assignments.

textbook

The course is in its entirety based on P. Cvitanović  et al. ChaosBook.org (experimental format: e-textbook). In the first 8-week course we will cover Part I  of the book, or most of the material from chapter to "Flows" to "Fixed points, and how to get them". All chapter and section numbers refer to the version 13 of the web e-textbook, unless stated otherwise. This course is a prepublication test run of the web e-textbook. Your active participation in improving the book is very much encouraged.

instructor