nonlinear dynamics 1 & 2: 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 (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 (see syllabus)
  • Transfer operators - statistical distributions in dynamics
  • Spectroscopy of chaotic systems
  • Dynamical zeta functions
  • Dynamical theory of turbulence
The course, which covers the same material and the same exercises as the Georgia Tech course PHYS 7224, is in part an advanced seminar in nonlinear dynamics, aimed at PhD students, postdoctoral fellows and advanced undergraduates in physics, mathematics, chemistry and engineering.

why this course

Most institutions have too few graduate students in any narrow speciality to offer a high level specialized course tailored to them. This Specialized Open Online Course is an experiment in sharing such advanced course with the off-campus research cohort.

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. These are on-demand online courses: you can start at any date, and complete the course at your own pace.

textbook

The course is based on P. Cvitanović  et al. ChaosBook.org. 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". Your active participation in improving the book is very much appreciated.

instructor

We are grateful to the family of late G. Robinson, Jr. and National Science Fundation, grant DMS-1211827 for support