Math 557 -- Dynamical Systems and Chaos

Fall 2015

When: MW 3-4:15pm
Where: Math East 246 Math 514
Instructor: Kevin Lin

Office: Math 606
Phone: 626-6628
E-mail: klin@math.arizona.edu

Course web page (this page):
http://math.arizona.edu/~klin/557

D2L page:
https://d2l.arizona.edu/d2l/home/452102

Office hours:
see here

Announcements

(Updated on January 11, 2016.)
- <2015-12-16 Wed> presentations 3:30 - 5:30pm
- <2015-12-14 Mon> presentations 3:30 - 5:30pm (confirmed)
- <2015-12-09 Wed> PS#3 due
- Lec 29 <2015-12-09 Wed>: Section 6.1 in [GH]
- Lec 28 <2015-12-07 Mon>: Sections 3.5 and 6.1 in [GH]; see also Section 3.4 of [Ermentrout-Terman]
- Lec 27 <2015-12-02 Wed>: Section 3.5 in [GH]
- <2015-11-30 Mon> no class
- Lec 26 <2015-11-25 Wed>: Sections 3.4 and 3.5 in [GH]
- Lec 25 <2015-11-23 Mon>: Section 3.4 in [GH]
- Lec 24 <2015-11-18 Wed>: Sections 3.3 and 3.4 in [GH]
- Lec 23 <2015-11-16 Mon>: Section 3.3 in [GH]
- <2015-11-11 Wed> Veterans Day
- Lec 22 <2015-11-09 Mon>: Section 3.2 in [GH]; [Carr] Ch 1.
- Lec 21 <2015-11-04 Wed>: Sections 3.1 and 3.2 in [GH]
- Lec 20 <2015-11-02 Mon>: Section 3.1 in [GH]
- Lec 19 <2015-10-28 Wed>: Section 3.1 in [GH]
- Lec 18 <2015-10-26 Mon>: See Guckenheimer (1975) on D2L +
  handouts on phase and phase response; PS#2 due
- Lec 17 <2015-10-19 Mon>: See Guckenheimer (1975) on D2L +
  handouts on phase and phase response
- Lec 16 <2015-10-16 Fri>: See Guckenheimer (1975) on D2L
- Lec 15 <2015-10-14 Wed>: Section 1.6 [GH] (but see also Poincare-Bendixson in 1.8)
- Lec 14 <2015-10-12 Mon>: Section 1.6 [GH] (but see also Poincare-Bendixson in 1.8)
- Lec 13 <2015-10-07 Wed>: Sections 1.3 & 1.4 in [GH]
- Lec 12 <2015-10-05 Mon>: See notes in D2L.
- Lec 11 <2015-09-30 Wed>: Section 1.4 in [GH]
- Lec 10 <2015-09-28 Mon>: Section 1.3 in [GH]
- Lec 9 <2015-09-23 Wed>: Burstall notes and Milnor excerpts; PS#1 due
- Lec 8 <2015-09-21 Mon>: Section 1.5 from [Guillemin-Pollack]
- Lec 7 <2015-09-16 Wed>: Section 1.4 from [Guillemin-Pollack]
- Lec 6 <2015-09-14 Mon>: Section 1.3 from [Guillemin-Pollack]
- Lec 5 <2015-09-09 Wed>: Section 1.2 from [Guillemin-Pollack]
- Lec 4 <2015-09-02 Wed>: Section 1.1, 1.2 from [Guillemin-Pollack]
- Lec 3 <2015-08-31 Mon>: Section 1.5 in [GH]
- Lec 2 <2015-08-26 Wed>: read Section 1.5 in [GH]
- Lec 1 <2015-08-24 Mon>: read Sections 1.0, 1.1 in [GH]
- Course policy (last revised <2015-08-24 Mon>)

About this course

Syllabus (this may be adjusted as we go along):

Part I: Fixed points, periodic orbits, and invariant manifolds
- Crash course in manifolds
- Differentiable dynamics
- Fixed points and periodic orbits
- Hyperbolic fixed points and periodic orbits
- Stable and unstable manifolds
- Center manifolds
- Attractors and genericity

Part II: Bifurcations of fixed points and periodic orbits
- Details TBA, but will include an analysis of the simpleset
  bifurcations.

Part III: Additional topics (time permitting of course)
- Geometric singular perturbations?
- Symmetries?
- Bifurcations in the presence of symmetries?
- Something else?

If the second semester course (557B) runs, I plan to
introduce some of the modern mathematical tools useful for
studying chaotic dynamics, e.g., ergodic theory.  I may also
cover some additional topics, depending in part on how much
time we have and on interest (yours and mine).

Grading. See course policy

Computing. Computers are vital tools for understanding nonlinear dynamical systems. As part of this course, you are expected to write programs to compute and analyze concrete examples. If you do not already know a programming language, it is fairly easy to learn a suitable high-level scientific programming language, e.g., Matlab, Python, R, or Julia. I will not spend lecture time on programming, but am happy to provide resources to help you learn these things.

Main references

  1. J Guckenheimer and P Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer

  2. D Ruelle, Elements of Differentiable Dynamics and Bifurcation Theory, Academic Press

  3. S Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer

#1 is our main text; I will not follow it closely, but everything I cover will be in there. #2 will be closer to Parts I and II of the course, but is unfortunately out of print as far as I know. #3 covers a superset of #1 and can be a useful reference. I'll also hand out notes (or copies of papers) when useful.

Additional general references

  1. R Devaney, An Introduction to Chaotic Dynamical Systems, Westview Press

  2. R Devaney, M Hirsch, S Smale, Differential Equations, Dynamical Systems, and An Introduction to Chaos, Springer

  3. S Strogatz, Nonlinear Dynamics and Chaos, Addison-Wesley

References with a focus on Hamiltonian systems

  1. G J Sussman and J Wisdom, with M Mayer, Structure and Interpretation of Classical Mechanics, MIT Press

  2. M Tabor, Chaos and Integrability in Nonlinear Dynamics: An Introduction, Addison-Wesley

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This page was last updated on September 16, 2018.