MATH 3451
Chaos and Fractals
Autumn 2015

This is the homepage for MATH 3451 (Chaos and Fractals). This page will be updated throughout the term with important information for our course, including homework assignments, review materials, and more.


  • I've posted a list of topics for the final exam below.
  • Solutions for Homework 5 have been posted.
  • Homework 6 has been posted and is due Friday, November 20th.
Course Information

Course meets every MW from 2:00 p.m. - 3:50 p.m. in the basement of Aspen Hall.

Instructor: Ronnie Pavlov
Office: Aspen Hall 715C
Phone: (303)-871-4001
Office hours: Tuesday 10-11, Thursday 2-3, or by appointment (try to give 24 hours notice)


Text: An Introduction to Chaotic Dynamical Systems (2nd edition) by Robert Devaney.

This book is available at the DU Bookstore.

Course summary

In this course we will study dynamical systems, or deterministic maps on "structured" spaces which preserve "structure." For our purposes, the structured space will almost always be the real numbers, a subset of the real numbers, or something very close (like the unit circle), and the maps will be continuous. We will mainly be interested in the question of what happens when the map is iterated over and over from a certain starting point. The iterates may converge, they may repeat periodically, they may approach infinity, or they might have some more complicated or "chaotic" behavior. We'll learn many techniques for how to classify the behavior of points given a certain starting map, which use fundamental theorems from real analysis such as the Intermediate Value Theorem, Mean Value Theorem, and Implicit Function Theorem. Eventually, we'll work with entire families of maps parametrized in some way (for instance, the logistic family is given by f(x) = Cx(1-x) for some parameter C), and learn about how varying the parameter changes the behavior of the system.

The prerequisite for this course is Real Analysis (MATH 3161); you'll need a very good grasp on the concepts of that course to succeed in this one. If you are not sure whether you are a good candidate for the class, please feel free to come talk to me.

Grading scheme

Your term grade will consist of homework assignments, one midterm exam, and one final exam, broken down in the following way:

40% Homework
25% Midterm Exam
35% Final Exam


Your weekly homework assignments will be posted here. Students receiving graduate credit for the course will have some extra "graduate problems," denoted by an asterisk. (See Course Policies below.)


You will have a midterm exam Monday, October 19th, and final exam on Friday, November 20th. Both exams will be in our classroom during classtime. (2:00 p.m. - 3:50 p.m.) Makeup exams will only be offered in the event of extreme circumstances. If you think you have a problem which will force you to miss an exam, come talk to me as soon as possible!!!

Course Policies

Graduate/Undergraduate Crosslisting:
Both undergraduate and graduate students can earn credit toward their degrees in this course. The university requires that such courses must include distinct graduate-level course requirements that demonstrate the rigor of a true graduate academic experience. As such, throughout the quarter I will assign “graduate student problems.” These will test students’ ability to prove results about dynamical systems at a deeper level than the remaining problems and will be required for graduate students only.

Students with Disabilities:
If you qualify for academic accommodations because of a disability or medical issue, please submit a faculty letter to me from Disability Services Program (DSP) in a timely manner so that your needs may be addressed. DSP determines accommodations based on documented disabilities/medical issues. DSP is located on the 4th floor of Ruffatto Hall, 1999 E. Evans Ave, 303.871.2278. Information is also available online at; see the Handbook for Students with Disabilities.

Honor Code:
Follow the Honor Code in all activities related to this course. Incidents of academic misconduct will be reported to and investigated by the office of Student Conduct.