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AMATH 575 A: Dynamical Systems

Meeting Time: 
MWF 3:30pm - 4:50pm
Location: 
LOW 216
SLN: 
10221
Joint Sections: 
AMATH 575 B
Instructor:
Bernard Deconinck
Bernard Deconinck

Syllabus Description:

Dynamical Systems

SLN 10221

LOW 216, MW 3:30-4:50pm (NOTE: typically no class on Friday)

Prereqs: Amath 402/502 or Instructor Permission

Instructor: Bernard Deconinck

LEW 207

deconinc@uw.edu

Tel: 206-543-6069

Office Hours: M10-11am, T9-11am, M9-10pm (online only)

Course Description

Overview of ways in which complex dynamics arise in nonlinear dynamical systems. Topics include bifurcation theory, universality, Poincare maps, routes to chaos, horseshoe maps, Hamiltonian chaos, fractal dimensions, Liapunov exponents. Examples from biology, mechanics, and other fields. 

Textbook

The required textbook for this course is Stephen Wiggins' Introduction to Applied Nonlinear Dynamical Systems and Chaos (second edition), Springer 2003. This excellent text is available at all your popular and unpopular book vendors, online and other. 

My typed-up lecture notes are available.

Message Board

We're using Piazza for the class message board. 

Syllabus (subject to some changes)

  1. Introduction. Continuous and discrete systems. Flows and maps. 
  2. Equilibrium solutions. Stability. Linearization. 
  3. Invariant manifolds. Autonomous vector fields. Transverse intersections and homoclinic tangles. The center manifold reduction. 
  4. Periodic solutions. Bendixson's theorem. 
  5. General properties of flows and maps. Liouville's theorem. The Poincare recurrence theorem. The Poincare-Bendixson theorem. 
  6. Poincare maps. Poincare maps near periodic orbits. Time-periodic problems. The periodically forced oscillator. 
  7. Hamiltonian systems. Basic properties. Symplectic or canonical transformations. Completely integrable Hamiltonian systems. 
  8. Symbolic dynamics. Bernoulli shift maps. Smale's horseshoe. Horseshoes in dynamical systems. 
  9. Indicators of Chaos. Lyapunov exponents. Fractal dimension. 
  10. Normal forms. The Takens-Bogdanov normal form. The normal form theorem. 
  11. Bifurcations in vector fields. Saddle-node bifurcation. Transcritical bifurcation. Pitchfork bifurcation. Hopf bifurcation. 
  12. Bifurcations in maps. Period doubling. The logistic map. Renormalization.

Grading

In addition to homework, each of you will present their findings on a class-related project. We will set some days outside of regular class time aside for the presentation of these projects. You are expected to be present for the presentations of your colleagues. Your course grade will be calculated by weighing your homework and project work in the proportions 60% and 40%, respectively.

Homework sets are assigned biweekly. Homework is due at the beginning of class on its due date. Late homework is not accepted. Every homework set you hand in should have a header containing your name, student number, due date, course, and the homework number as a title. Your homework should be neat and readable. Your homework score may reflect the presentation of your homework set.

Please see specific homework guidelines here:

Catalog Description: 
Overview of ways in which complex dynamics arise in nonlinear dynamical systems. Topics include bifurcation theory, universality, Poincare maps, routes to chaos, horseshoe maps, Hamiltonian chaos, fractal dimensions, Liapunov exponents, and the analysis of time series. Examples from biology, mechanics, and other fields. Prerequisite: either AMATH 502 or permission of instructor. Offered: Sp, odd years.
Credits: 
5.0
Status: 
Active
Last updated: 
March 14, 2019 - 9:00pm
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