Instructor: Alena Erchenko

Course on canvas.dartmouth.edu.⇗

Syllabus

Week Date Topic Lecture Notes
Week 1
(March 31
-April 4)
 Tuesday Overview and examples Lecture 1 Download Lecture 1
Thursday More examples of dynamical systems. Contraction maps. Terminology: orbit, fixed point, periodic point of a map. Lecture 2  Download Lecture 2 
Week 2
(April 7-11)
 Tuesday

 

Contraction principle.

 

Lecture 3 Download Lecture 3
Thursday

 

Application of a contraction principle. Attracting and repelling fixed points. 

 Lecture 4 Download Lecture 4
Week 3
(April 14-18)
 Tuesday Bifurcation. Logistic maps. Circle maps. Lecture 5 Download Lecture 5
Thursday Properties of irrational rotation on a circle.

 

Lecture 6 Download Lecture 6
Week 4
(April 21-25)
 Tuesday

 

Benford's law for first digits of powers of 2. Review of linear algebra. Linear systems on a plane.

 Lecture 7 Download Lecture 7
Thursday

 

Linear systems on a plane: phase portraits, solutions, and connection to dynamical systems.

 Lecture 8 Download Lecture 8
Week 5
(April 28-May 2)
 Tuesday

 

Nonlinear systems on a plane. 

 

Lecture 9 Download Lecture 9
Thursday

 

Linear expanding maps on a circle.

 Lecture 10 Download Lecture 10
Week 6
(May 5-9)
 Tuesday

 

Midterm

 

Midterm Solutions Download Midterm Solutions
Thursday

 

Space of sequences and shifts. Subshifts of finite type.

 

Lecture 11 Download Lecture 11
Week 7
(May 12-16)
 Tuesday

 

More on subshifts of finite type. Logistic map for the parameter larger than 4.

 Lecture 12 Download Lecture 12
Thursday

 

Continue on logistic map for the parameter larger than 4. Topological conjugacy. Billiards in a circle.

 Lecture 13 Download Lecture 13
May 16  (3:30-4:20pm)
 Friday- X-Hour (0.5 make-up for Tuesday May 26th) Cancelled
Week 8
(May 19-23)
 Tuesday

 

 Continue on billiards in a circle. Billiards in a polygon.

 

Lecture 14 Download Lecture 14
Thursday Continue on billiards in a polygon. "Unfolding" method.

 

Lecture 15 Download Lecture 15
Week 9
(May 26-30)
 Tuesday

 

No class
Thursday

 

Convergence of Newton's method applied to f(z)=z^2-1. Caley's theorem. Mandelbrot set.

 

Lecture 16 Download Lecture 16
May 30
(3:30-4:20pm)
 Friday- X-Hour (0.5 make-up for Tuesday May 26th) Continue on the Mandelbrot set.

 

Lecture 17 Download Lecture 17

Week 10
(June 2-4)

 Tuesday Julia set. Sharkovsky theorem.

 

Lecture 18
Final Exam (Take home) due June 9 (Monday) 5pm, submit on Gradescope.