Math 557 -- Dynamical Systems and Chaos

Announcements

(Last revised on May 04, 2016.)

- <2016-05-09 Mon> 3--5pm presentations (usual room)
- <2016-05-04 Wed>
  1) For more details on entropy, see [BS] Ch 9 and also the
     1993 lecture notes by L-S Young.  The latter also go
     into the theory of Sinai-Ruelle-Bowen measures in some
     detail.

  2) Markov partitions are discussed in Section 5.12 in [BS].

  3) See also paper posted in D2L.

- <2016-05-02 Mon> References on D2L.
  1) Later start, at 4pm, in usual room.
  2) References on D2L.
- <2016-04-29 Fri>
  1) 3:30pm make-up class (usual room, note time change).
  2) References on D2L.
- <2016-04-27 Wed> Computing exponents: see posted papers on D2L
- <2016-04-25 Mon> Lyapunov exponents: see posted paper on D2L
- <2016-04-20 Wed> Lyapunov exponents: see posted paper on D2L (dated 4/25)
- <2016-04-18 Mon> Horseshoes continued
- <2016-04-13 Wed> Horseshoes
  1. [BS] Section 5.8.
  2. [GH] Section 6.1.
  3. For the curious, [GH] Section 4.5 discusses Melnikov's
     method.
- <2016-04-06 Wed> [BS] Sections 5.6, 5.8.
- <2016-04-04 Mon> [BS] Sections 5.5, 5.6.
- <2016-03-30 Wed> [BS] Sections 5.3, 5.4, 5.5.
- <2016-03-28 Mon>
  1. [BS] Sections 5.1, 5.2, 5.3.
  2. Shadowing references on D2L.
- <2016-03-23 Wed> [BS] Sections 5.1, 5.2.
- <2016-03-21 Mon> Papers on D2L.
- <2016-03-11 Fri> problem set due
- <2016-03-09 Wed> See reference on D2L.
- <2016-03-07 Mon> [BS] Sections 4.5 (on mean ergodic theorem).
- <2016-03-02 Wed> [BS] Sections 4.5 (on mean ergodic theorem).
- <2016-02-29 Mon> [BS] Sections 4.6, 4.7.
- <2016-02-26 Fri> make-up class; notes on D2L.
- <2016-02-24 Wed> [BS] Sections 4.3, 4.4.
- <2016-02-22 Mon> [BS] Sections 4.3, 4.4, 4.5.
- <2016-02-17 Wed> No class (KL out of town).
- <2016-02-15 Mon> [BS] Section 4.3, 4.4.
- <2016-02-10 Wed> [BS] Section 4.2.
- <2016-02-08 Mon> [BS] Section 4.1.
- <2016-02-03 Wed>
  1. [BS] Section 4.1.
  2. Optional: [Flaschka] Sections 1.5.2, 2.2.2.
  3. Optional: Watson and Wayman on non-measurable sets
- <2016-02-01 Mon> [BS] Sections 1.12, 1.13.
- <2016-01-27 Wed> [BS] Sections 1.8, 1.9.
- <2016-01-25 Mon> [BS] Sections 1.3, 1.5, 1.7.  Optional: 1.4 (on shifts).
- <2016-01-20 Wed> [BS] Sections 1.1--1.3.
- <2016-01-13 Wed> first class!
  1) You might want to read Lorenz's 1963 paper.
  2) [GH] Section 2.3 presents a slightly more up-to-date
     view of the Lorenz model.  (But keep in mind a fair bit
     of progress has occurred since!)
  3) The simulations I showed in class are on D2L.
- <2016-01-11 Mon> Course policy

About this course

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

Last semester focused on (mostly local) bifurcations.  This
semester, we will shift focus to mathematical tools for
describing and analyzing global, qualitative behavior,
especially chaotic dynamics.  Tentative list of topics
includes
- examples of chaotic dynamical systems
- an introduction to hyperbolic dynamics
- basic ergodic theory
- additional, special topics may include introductions to
  attractor reconstruction, Lyapunov exponents, transfer
  operators, fractal dimension, geometric singular
  perturbations, and others (to be determined by your
  interests and mine).

Grading. See the course policy.

Main references

1. M Brin and G Stuck, Introduction to Dynamical Systems, Cambridge

2. E Ott, Chaos in Dynamical Systems, Cambridge

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

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

#1 and #3 are our main references. #2 covers some of the same material, but from a more physical point of view. #4 is a concise exposition of the material in #3. I'll also hand out notes (or copies of papers) when useful.

Additional general references

1. KT Alligood, TD Sauer, JA Yorke, Chaos: an Introduction to Dynamical Systems, Springer

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

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

4. 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.