Course Syllabus
MATH 27 Dynamical Systems with Applications
Course Info:
- Course Description
Dynamical systems theory is the branch of mathematics that studies the properties of time-evolving phenomena. It finds a great variety of applications in areas spanning physics, engineering, ecology, finance, among many disciplines.
Math 27 is an introductory course, studying aspects of dynamical systems that evolve in discrete time, as well as continuous-time systems described by ordinary differential equations. A primary objective will be to explore and understand the qualitative properties of dynamics, such as the existence of attractors, periodic orbits and chaos. We will do this by means of mathematical analysis, as well as simple numerical experiments.
- Prerequisites
Math 22 and Math 23, or instructor approval
- Lectures
Tuesdays and Thursdays 10-10am-12pm
(X-hour) Fridays 3:30pm-4:20pm (used occasionally)
Location: 028 Haldeman Center - Instructor Information
Alena Erchenko
E-mail: alena.erchenko@dartmouth.edu
Office: 331 Kemeny Hall - Office Hours
Tuesdays and Wednesdays 9am - 10am and by appointment
Location: 331 Kemeny Hall - Recommended Texts (NOT required!):
1) An Introduction to Chaotic Dynamical Systems by Robert Devaney;
2) Chaos: An introduction to dynamical systems by Alligood, Sauer, and Yorke;
3) An Introduction to Dynamical Systems by R. Clark Robinson;
4) Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering by Steven H. Strogatz
5) A First Course in Dynamics with panorama of recent developments by Boris Hasselblatt and Anatole Katok - Learning objectives
1) Master mathematical techniques, including methods based on ODE and linear algebra, to solve problems in dynamical systems.
2) Set up mathematical statements from the word problems and given data.
3) Develop a sense how mathematics can be used in various fields.
4) Develop skills to write detailed solutions using appropriate mathematical language.
- Attendance:
Attendance is strongly encouraged but not required. If you miss a class, you are still responsible for the material due, for learning all concepts covered, and turning in assignments given. Class participation (answering and asking questions during class) is encouraged. - Grading:
Grades will be based on weekly homework (30%), midterm (30%) and final (40%). -
Homework:
Homework is due on Gradescope on the day indicated below before 11:59pm ET. Late submissions will not be accepted. Please upload your solutions by following the instructions in this videoLinks to an external site.. Feel free to submit handwritten or work typeset in LaTeX (OverleafLinks to an external site. is a great resource and tool for the latter). If you are skipping a problem, it is easier for the grader if you write "Skip" and upload that as your solution to the problem. You are more than welcome to work with others on the homework problems. However, the solutions should be written up on your own and reflect your understanding.
Due Date Homework Solutions April 3, 2025 Homework 0 N/A April 10, 2025 Homework 1 HW1 - Solution April 17, 2025 Homework 2 HW2 - Solution April 24, 2025 Homework 3 HW3 - Solution May 1, 2025 Homework 4 HW4 - Solution May 15, 2025 Homework 5 HW5 - Solution May 22, 2025 Homework 6 HW6 - Solution May 29, 2025 Homework 7 HW7 - Solution - Exams:
The midterm will be held on Thursday, 05/06/2025 in class. The final exam will be cumulative, and will be conducted at a time and location determined by the Registrar during the finals period (Friday June 6 to Tuesday June 9). Neither the midterm nor the final examination will be given early, unless there are extenuating circumstances.
- Makeup Examinations:
No alternative date will be given for the midterm or the final exam. If a student has a valid documented reason, such as illness or religious holiday, during examination times and informs the instructor, Alena Erchenko, beforehand, then they are permitted to schedule a makeup examination with no penalty. A missed midterm must be made up within 7 days of the midterm. Students must be prepared to verify the reason for requesting the makeup by providing the proper document(s) upon request. Conflicts with other exams, personal business such as travel, employment, weddings, graduations, or attendance at public events such as concerts and sporting events are not valid excuses. Transportation trouble – missing a bus or having a car breakdown on the way to exam is not a valid excuse either. Nor is forgetting the date, time or room of an examination a valid excuse. If a student misses an exam, does not have a valid documented excuse, and does not inform the instructor, Alena Erchenko, then they get 0 points for the exam. -
Schedule:
The specific topics covered in each class are tentative and will be updated as the term progresses. Moreover, the notes posted for each lecture may contain slightly more content than the lecture and may bleed a bit into the next lecture. Let me know if you find any errors.
Week Date Topic Lecture Notes Week 1
(March 31
-April 4)Tuesday Overview and examples Lecture 1 Thursday More examples of dynamical systems. Contraction maps. Terminology: orbit, fixed point, periodic point of a map. Lecture 2 Week 2
(April 7-11)Tuesday Contraction principle.
Thursday Application of a contraction principle. Attracting and repelling fixed points.
Lecture 4 Week 3
(April 14-18)Tuesday Bifurcation. Logistic maps. Circle maps. Lecture 5 Thursday Properties of irrational rotation on a circle. 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 Thursday Linear systems on a plane: phase portraits, solutions, and connection to dynamical systems.
Lecture 8 Week 5
(April 28-May 2)Tuesday Nonlinear systems on a plane.
Thursday Linear expanding maps on a circle.
Lecture 10 Week 6
(May 5-9)Tuesday Midterm
Thursday Space of sequences and shifts. Subshifts of finite type.
Week 7
(May 12-16)Tuesday More on subshifts of finite type. Logistic map for the parameter larger than 4.
Lecture 12 Thursday Continue on logistic map for the parameter larger than 4. Topological conjugacy. Billiards in a circle.
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.
Thursday Continue on billiards in a polygon. "Unfolding" method. 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.
May 30
(3:30-4:20pm)Friday- X-Hour (0.5 make-up for Tuesday May 26th) Continue on the Mandelbrot set. Week 10
(June 2-4)Tuesday Julia set. Sharkovsky theorem. Final Exam (Take home) due June 9 (Monday) 5pm, submit on Gradescope. -
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