Chapters 1-3

Wednesday

• Topic: Chapters 1-3: Introduction. We introduce the concept of a dynamical system and some terminology (iteration, orbits, fixed points).
• Class materials: Lecture notes, worksheet, iterator.m script, time_series.m script
• After class: Work on and submit Homework 0. Please make sure to install MATLAB and come ask for help if you have trouble.

Chapters 3-4

Monday

• Topic: Chapter 3: Iteration and Orbits. We discuss fixed points, periodic points, and eventually periodic points. We also introduce and study the doubling map.
• Class materials: Lecture notes, worksheet, doubling_map.m script
  
• After class: Read sections 4.1 and 4.2.

Wednesday

• Topic: Chapter 4: Graphical analysis. We introduce cobweb diagrams in order to visualize orbits of a dynamical system. We also discuss complete orbit analysis and phase portraits.
• Class materials: Lecture notes, worksheet, Desmos cobweb plotter
• After class: Think about the role slope plays in today’s worksheet examples and its relationship to attracting and repelling fixed points.

Chapters 4, 5

Monday

• 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: Read Section 5.1.

Wednesday

• 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, worksheet
• After class: Finish Problem 1 from today’s worksheet and prepare to discuss it at the start of next class.

Chapter 5

Monday

• 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 next time.

Wednesday

• 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, worksheet
• After class: Finish Problem 2 on today’s worksheet and prepare to discuss it at the start of our next class.

Chapters 5, 6

Monday

• 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, worksheet
• After class: Read Section 6.1.

Wednesday

• 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, worksheet
• After class: Read the end of Section 6.1 about the second kind of bifurcation.

Chapter 6

Monday

• 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, worksheet
• After class: Work on Homework 5.

Wednesday

• 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, worksheet
• After class: Finish today’s worksheet.

Chapter 6

Monday

• Topic: Exam 1.

Wednesday

• 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
• After class: Enjoy spring break!

Spring Break

Monday

• Topic: Spring break, no class.

Wednesday

• Topic: Spring break, no class.

Chapter 7

Monday

• 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: Read Section 7.3.

Wednesday

• 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.

Chapter 7

Monday

• 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: .

Wednesday

• 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: .

Chapters 7, 9

Monday

• 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

Wednesday

• 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.

Chapters 9, 10

Monday

• 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:.

Wednesday

• Topic: TBA
• Class materials: Lecture notes
• After class:.

Exam, Group work

Monday

• Topic: Exam 2.
• After class: Think about your presentation topic in preparation for the work time in the next class meeting.

Wednesday

• 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. Begin preparing ideas for presentation. Think about what will be presented (you can’t say everything you think of!) and how the presented material will be divided between group mates.

Group work time, presentations

Monday

• 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.

Wednesday

• Topic: Presentations. We’ll spend the day on the first group presentations.
• After class: Wrap up your presentation if presenting next week.