One Dimensional Dynamics

See here for an explanation of the software and sample lab assignment using it.

One-dimensional maps are the simplest dynamical systems that may be chaotic. Textbooks can give a static picture of such dynamics, but since dynamics involves time, a student can get a much better understanding of evolution and chaos by using hands on software. There is no substitute for seeing dynamics evolve before your eyes. While there are many general purpose tools that can be programmed to show these dynamics, having a single purpose application allows the instructor and the student to focus on the phenomena, avoiding the programming details.

James Meiss developed and has used the Macintosh application “1DMaps” in both in an undergraduate engineering focused course on dynamics (usually using Steve Strogatz’s wonderful text), and in a graduate course on discrete dynamical systems. Randell Callahan was a student in the undergraduate class last fall, and has been porting the 1DMaps program to iOS so it will run on the iPad, and iPhone platforms.

Goals of the lectures, labs and homework assignments that use 1DMaps are (1) to develop an intuition for how one iterates maps (2) visualize the concepts of stability, bifurcation, and asymptotic behavior, and (3) investigate Feigenbaum’s universality.

nonlinear dynamics 1: geometry of chaos

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

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

    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 aimed at PhD students, postdoctoral fellows and (very) advanced undergraduates in physics, mathematics, chemistry and engineering.

Interactive Materials for Teaching Elementary Dynamical Systems

Prize winner, Teaching DS Competition, 2013

The web site Math Insight contains expository material, interactive applets, videos, and exercises intended to be used either in a classroom setting or as an online resource for the greater community. The focus is on qualitative description rather than getting all technical details precise. Many of the pages were designed to be read even before students attend lecture on the topic, so they are intended to be somewhat readable introductions to the basic ideas.

This entry contains materials for an elementary dynamical systems material for a course Calculus and Dynamical Systems in Biology. The dynamical systems material is located at the website below!

The materials are designed to facilitate an inverted (or “flipped”) method of instruction. In this approach, the lecture expository material is delivered via videos or other material that students view at home. Then, the time in class is used to work through problems or projects in groups, where the instructor guides the students through this key stage of the learning process.

The web site also contains randomly generated study problems which are the same questions as those given on the paper and pencil exams. In a few cases (as development of this aspect is just beginning), students can enter their answers into the web page to check if they are correct. For most problems, they can also receive hints or view the solution.

All material is accessed from the threads, i.e., the links given above, by searching the site, or by using the index. As many pages contain interactive applets, they are intended to be accessed live on the site. The exception is the worksheets, which are designed to be printed so that students can fill in the answers by hand.

Graduate (MAGIC) course Equivariant Bifurcation Theory

These are course materials for a ten lecture course for first year PhD students in mathematics. It is not aimed necessarily at those who will specialise or use Equivariant Bifurcation Theory, but it is designed to be a “broadening” training for example for those doing more conventional mathematical modelling as part of their PhD.

Materials include:
Examples and solutions, course slides, sample exam are under the "Files" tab.
Syllabus and general course information are also tabs.

Graduate (MAGIC) course Dynamical Systems: Flows

These are lecture notes (slides) for a 10-hour course delivered as part of the MAGIC suite of graduate-level courses in nonlinear dynamics and dynamical systems. The course was all about flows; there is a follow-on course on maps and another on equivariant bifurcation theory.

Topics covered (in 9 lectures) were:

Flows and the Poincare map; Linearisation; Stability of equilibrium points, periodic orbits and other invariant sets; Local and global bifurcation theory; Centre Manifold Theorem; Birkhoff normal form; Local bifurcations of periodic orbits; An in-depth example (the saddle-node--Hopf bifurcation); Homoclinic bifurcations.

There was then an option in the last lecture for students to hear about PDEs, patterns, and the role of symmetry (leading in to a further 10 lectures on equivariant bifurcation theory by Pete Ashwin, or on hydrodynamic stability theory), or Numerical methods for dynamical systems (Symbolic algebra, Integrating ODEs and Continuation).