**Instructor**: Tim Chumley

**Office**: Clapp 423

**Phone**: 413-538-2525

**e-mail**: tchumley

**Office Hours**: Mondays & Wednesdays 1:00-2:00; Tuesdays & Thursdays 2:00-3:00 and 4:00-5:00; additional availability by appointment

**Textbook**: *A First Course in Chaotic Dynamical Systems* by Robert L. Devaney, ISBN: 9780429280665;

available as a free e-text

Announcements will be posted on the Course Announcements Moodle forum throughout the semester, but essentially all other materials will be posted on this page.

Check the syllabus for all the important class policies (grades, attendance, etc.).

There will be weekly homework assignments throughout the semester to be turned in. Please read these guidelines for writing in our class.

**General information**. A selection of problems will be assigned to be written up individually and turned in each week.- These problems will be due
**Wednesdays at 5 pm**. - You may work with others but the writing should be done on your own.

- These problems will be due
**Gradescope**. Homework will be turned in through Gradescope.- You should be enrolled automatically. Please let me know if you have any issues logging in.
- Gradescope has made a short tutorial on submitting homework.

**Collaboration**. I want you to work together on the homework! The process of explaining your ideas to one another really helps in learning math. However, you must write up the assignments on your own and avoid copying others’ work directly. Also, please only write what you understand so that I know where to help, and please avoid using online forums like Math StackExchange, solutions manuals, or similar resources. A huge part of learning in this class comes from figuring out how to get unstuck by spending time thinking on your own or talking to me and classmates; these other resources can end up being counter-productive in the long term.**Rewrites**. Homework is for practice, and you are not expected to write perfect answers from the start!- You will be allowed to submit revisions of most problems for full credit each week.
- Your revisions will be due on
**Fridays at 5 pm**. - Please resubmit (only the problems you’re revising) on Gradescope.

Assignment | Due |
---|---|

Homework 0 | Jan 28 |

Homework 1 | Feb 2 |

Homework 2 | Feb 9 |

Homework 3 | Feb 16 |

Homework 4 | Feb 23 |

Homework 5 | Mar 2 |

Homework 6 | Mar 30 |

Homework 7 | Apr 6 |

Homework 8 | Apr 13 |

Homework 9 | Apr 19 |

There will be two midterm exams. The dates for the exams are subject to change slightly.

Exam | Due Date | Format | Material |
---|---|---|---|

Exam 1 | Mar 11 | Take home | Chapters 3-5 |

Exam 2 | May 9 | Take home | Chapters 6-7, 9 |

In lieu of a final exam, we’ll devote the last week of the semester to a mini-symposium of short group presentations. Since the field of dynamical systems is rich with interesting examples and topics, more than we could cover in a single semester, each group of students will choose a topic that we might otherwise not have time for in class.

- Project notes related to your topic are due in advance of class on April 19.
- Presentation on your topic will be April 28 or May 3.

- Julia sets
- Julia sets 2
- Newton’s method
- Chaos and Feigenbaum’s constant
- Topological and fractal dimension
- Iterated function systems

Our plan is to cover parts of the first 11 chapters of the textbook, as well as parts of later chapters, time permitting. Below is a rough outline of what is to be covered week by week through the semester. Please check back regularly for precise details on what is covered, as well as postings of class materials like lecture notes.

**Topic**: Chapter 2: Examples. We introduce the concept of a dynamical system and some terminology (iteration, orbits, fixed points) through the example of the logistic map.**Class materials**: Lecture notes, Introduction**After class**: Work on Homework 0, making sure to try installing MATLAB.

**Topic**: Sections 3.1-3.3: Iteration and Orbits. We introduce the idea of periodic points and begin to think about the idea of attraction and repulsion, again using the logistic map as our main example.**Class materials**: Lecture notes, Exploring orbits, time_series.m script**After class**: Work on Homework 1.

**Topic**: Sections 4.1-4.2: Graphical analysis. We introduce the concept of a cobweb diagram in order to visualize orbits of a dynamical system. Our aim for each example system will be to summarize all the possible behaviors of orbits.**Class materials**: Lecture notes, Graphical analysis, cobweb.m script**After class**: Think about the role slope plays in today’s worksheet examples and its relationship to attracting and repelling fixed points.

**Topic**: Sections 4.2-4.3: More on graphical analysis. We discuss the worksheet from last time and introduce the concept of a phase diagram.**Class materials**: Lecture notes**After class**: Work on Homework 2.

**Topic**: Sections 5.2-5.3: Attraction and repulsion. We begin to formalize the observations we have made on attraction and repulsion using calculus. Our aim is to summarize the possible behaviors of orbits based on the value of \(|F'(p)|\) and we begin with case of a positive derivative that is less than 1.**Class materials**: Lecture notes, Attraction, repulsion, and derivatives**After class**: Finish Problem 1 from today’s worksheet and prepare to discuss it at the start of next class.

**Topic**: Section 5.2-5.4: More on attraction and repulsion. We continue our discussion from last time and outline a proof for the attracting fixed point theorem. We introduce the Mean Value Theorem from calculus as our primary tool.**Class materials**: Lecture notes**After class**: Write up your ideas on how to understand the behavior of orbits that start near a repelling fixed point. You should mostly repeat the argument we worked on together for the attracting case, but we’ll discuss a small subtlety that’s different on Tuesdays.

**Topic**: Section 5.4: Neutral fixed points. We consider the behavior of orbits that begin near a fixed point \(p\) where \(|F'(p)|=1\). Our main general observation will be that the second and third derivatives play an important role in classifying the convergence and divergence behavior of orbits that start near \(p\).**Class materials**: Lecture notes, Neutral fixed points**After class**: Finish Problem 2 on today’s worksheet and prepare to discuss it at the start of our next class.

**Topic**: Section 5.5: Periodic points. We discuss attracting, repelling, and neutral periodic points. The key here will be using the chain rule to simplify our calculations.**Class materials**: Lecture notes, Periodic points**After class**: Work on Homework 4.

**Topic**: Section 6.1: Introduction to bifurcations. We discuss the saddle node bifurcation for the quadratic family \(Q_c(x)=x^2+c\), a bifurcation where the system transitions from 0 to 1 to 2 fixed points.**Class materials**: Lecture notes, Saddle node bifurcation**After class**: Read the end of Section 6.1 about the second kind of bifurcation.

**Topic**: Section 6.1: More on bifurcations. We discuss the period doubling bifurcation for the quadratic family. Here, we see the appearance of period-2 points as we change the parameter \(c\).**Class materials**: Lecture notes, Period doubling bifurcation**After class**: Work on Homework 5.

**Topic**: Sections 6.2, 6.3: Bifurcations more generally. Our goal is to continue our exploration of bifurcations through more examples. We’d like to understand what properties, algebraically and geometrically, give rise to saddle node and period doubling bifurcations.**Class materials**: Lecture notes, Bifurcations of the logistic map**After class**: Finish today’s worksheet.

**Topic**: Sections 6.2, 6.3: Formal definitions. We wrap up our discussion of last time’s worksheet and lay out the formal definitions of saddle node and period doubling bifurcations.**Class materials**: Lecture notes**After class**: Work on the exam!

**Topic**: Section 6.4: Bifurcation diagrams. We discuss a method of visualizing the bifurcations that occur in the quadratic map. A bifurcation diagram has the parameter values along the horizontal axis and fixed point and periodic point values along the vertical axis. We construct a bifurcation diagram for the quadratic map through numerical simulation.**Class materials**: Lecture notes, Bifurcation diagram spreadsheet**After class**: Work on the exam.

**Topic**: Exam work day, no class.**After class**: Finish your exam and enjoy spring break!

**Topic**: Spring break, no class.

**Topic**: Spring break, no class.

**Topic**: Section 7.2: More on the quadratic family. We discuss the behavior of orbits of the quadratic map that escape to infinity.**Class materials**: Lecture notes**After class**: Respond to the mid-semester feedback form

**Topic**: Section 7.3: The Cantor Middle-Thirds set. We discuss an

iteratively defined set with very interesting, bizarre properties. While it serves as an object worthy of study on its own, it will serve as a prototype for the sets of initial seeds with bounded orbits that studied in Section 7.2.**Class materials**: Lecture notes**After class**: Work on Homework 6.

**Topic**: Community day, no class.

**Topic**: Section 7.3: More on the Cantor Middle-Thirds set. We continue our discussion of the Cantor Middle-Thirds set, with a focus on how to find the ternary representation of a real number in \([0,1]\) and how these ideas allow us to prove that the Cantor set is uncountably infinite.**Class materials**: Lecture notes**After class**: Work on Homework 7.

**Topic**: Section 7.3. We discuss a refined method or algorithm for finding the ternary expansion of a number in \([0,1]\) and properties of ternary expansions of endpoints of the Cantor middle-thirds set.**Class materials**: Lecture notes, Ternary expansions**After class**: Finish Homework 7 and make sure to email me your group project preferences if you submit your homework late.

**Topic**: Dynamics on the Cantor set. We discuss the map \(T(x)\) from last week’s homework and how it acts on initial seeds from the Cantor middle-thirds set.**Class materials**: Lecture notes**After class**: Read section 10.1

**Topic**: Section 10.1: Chaos. We discuss Devaney’s definition of chaos and the intuitive meaning of each piece of the definition.**Class materials**: Lecture notes, Chaos**After class**: Read about itineraries in section 9.1.

**Topic**: Section 10.1: Chaos. We discuss the dynamics of the shift map on binary sequences and the itinerary of an orbit of the tent map.**Class materials**: Lecture notes**After class**:

**Topic**: Group work time. We spend the day working in groups learning about the group project topic, discussing questions, and beginning to think about presentations.**Class materials**: Presentation rubric**After class**: Spend some time working on the ideas discussed in your group today. Check in with your group about outside-of-class meeting time.

**Topic**: Group work time. We continue our group work time.**After class**: Check in with your group about outside-of-class meeting time. Begin preparing ideas for presentation. Think about what will be presented (you can’t say everything in the book!) and how the presented material will be divided between group mates.

**Topic**: Group work time. We spend our last day on group work time before the presentations begin.**After class**: Meet with your group. Do practice runs of your presentation, particularly if you’re presenting Thursday.

**Topic**: Presentations. We’ll spend the day on the first group presentations.**Class materials**: Julia sets, Newton’s method**After class**: Wrap up your presentation if presenting on Tuesday. Review material and come ask questions before Exam 2.

**Topic**: Presentations. We’ll spend the day on the remaining group presentations.**Class materials**: - Chaos and Feigenbaum’s constant, Topological and fractal dimension, Iterated function systems**After class**: Work on Exam 2. Enjoy your summer! Keep in touch!

Here are a few ways to get help:

**Office Hours**: Mondays & Wednesdays 1:00-2:00; Tuesdays & Thursdays 2:00-3:00 and 4:00-5:00; additional availability by appointment

**Study groups**: Other students in the class are a wonderful resource. I want our class to feel like a community of people working together. Please get in touch if you’d like me to help you find study partners, and please reach out if you’d like others to join your group. You may work together on homework, explain and check answers, but make sure you write what*you*know on homework in order to get good feedback.**Piazza**: I’ve set up an online forum for asking and answering questions about class material, including homework problems. My plan is to check the forum every day and answer questions as they come up, but I hope everyone in the class will pitch in and answer others’ questions when possible.

- We will be using MATLAB, a programming language and piece of software, to experiment with some of the theoretical ideas we learn. LITS has a short installation guide and MathWorks, the creators of MATLAB, have an onramp guide for learning the basics.
- Our textbook’s author has written a number of Java Applets to aid in our understanding. Unfortunately they seem to be out of date and it might not be possible to run them without some work but I’ve put a link here in case they can be used.
- Over the course of the semester, I’ll be posting MATLAB scripts that we will use as we learn new concepts. I’ll be posting them here:
- Here are some dynamical systems related videos. Please share anything interesting you come across.