## Dynamical Systems

SLN 10242 (Section A)

SLN 10243 (Section B; Online only)

SLN 20648 (Virtual Section)

SLN 10243 (Section B; Online only)

SLN 20648 (Virtual Section)

Low 216, MW 3:30-4:45pm

Prereqs: Amath 568, 502 or Instructor Permission

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**Instructor: Bernard Deconinck**

bernard@amath.washington.edu

Tel: 206-543-6069

Office Hours: M 10-11, T10-12

#### 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, Lyapunov exponents, and the analysis of time series. Examples from biology, mechanics, and other fields.#### Textbook

The textbook for this course is Steven Wiggins' "Introduction to Applied Nonlinear Dynamical Systems and Chaos" (2nd edition, 2003) (Springer Texts in Applied Mathematics 2)". This is an 800 page volume with much more material than we will be able to cover. It is unfortunately unreasonably priced ($91 at one of the better-known on-line vendors; a 75% price increase over 6 yrs!), but it should be a great resource for you far beyond this course. If you buy this book used (a good idea!), make sure you get the second edition, which is very different from and vastly superior to the first edition.Update: (thanks to Sonseeahray Rucker) Springer has the book on sale until July, see http://www.springer.com/mathematics/dynamical+systems/book/978-0-387-00177-7. It's 50% off, which makes it reasonable!

As of now, the book is also available electronically through the UW library, but with serious restrictions to the extent that you will probably still want to get your own copy.

Also, if you need to catch up on some introductory material, or would like to have a lower-level text at your disposal for the occasionally different view, I recommend Steven Strogatz's "Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering" (Perseus 2001). My lecture notes are available here.

#### Course Canvas Page

I will use Canvas to post homework sets, link to the class message board, etc. You will need a UW account and be enrolled in the course to access this page. https://canvas.uw.edu/courses/812945 (Ignore the UW header. The page is for Sections A, B and C).

#### Syllabus

**(0)**Introduction. Terminology. Flows and maps.

**(1)**Equilibrium solutions. Periodic orbits. Poincare maps.

**(2)**Linear Stability. Nonlinear stability definitions.

**(3)**Asymptotic behavior of solutions of dynamical systems. The Poincare-Bendixson theorem.

**(4)**Hamiltonian systems. The KAM theorem.

**(5)**Center manifolds.

**(6)**Normal forms. Bifurcation theory.

**(7)**The Smale Horseshoe. Symbolic dynamics.

**(8)**Stable and unstable manifolds. Homoclinic and Heteroclinic connections.

**(9)**Establishing chaos: Liapunov exponents, fractal dimensions.

#### 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. EDGE students who cannot be present for the presentations will do a term paper instead of a project presentation.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.